TPTP Problem File: ITP270^4.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP270^4 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_DeleteBounds 00283_015243
% 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 : 0074_VEBT_DeleteBounds_00283_015243 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 9713 (2914 unt; 758 typ; 0 def)
% Number of atoms : 30877 (10476 equ; 1 cnn)
% Maximal formula atoms : 71 ( 3 avg)
% Number of connectives : 186666 (2661 ~; 343 |;2287 &;167339 @)
% ( 0 <=>;14036 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 7 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 4201 (4201 >; 0 *; 0 +; 0 <<)
% Number of symbols : 750 ( 746 usr; 24 con; 0-8 aty)
% Number of variables : 28094 (2650 ^;23840 !; 977 ?;28094 :)
% ( 627 !>; 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 12:10:18.585
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_t_VEBT__Definitions_OVEBT,type,
vEBT_VEBT: $tType ).
thf(ty_t_Code__Numeral_Ointeger,type,
code_integer: $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Extended__Nat_Oenat,type,
extended_enat: $tType ).
thf(ty_t_Complex_Ocomplex,type,
complex: $tType ).
thf(ty_t_String_Oliteral,type,
literal: $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_Filter_Ofilter,type,
filter: $tType > $tType ).
thf(ty_t_String_Ochar,type,
char: $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Rat_Orat,type,
rat: $tType ).
thf(ty_t_Num_Onum,type,
num: $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
% Explicit typings (739)
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Odvd,type,
dvd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom,type,
idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_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_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_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_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Osemiring__numeral,type,
semiring_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__modulo,type,
semidom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__modulo,type,
semiring_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__parity,type,
semiring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring__abs,type,
ordered_ring_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oalgebraic__semidom,type,
algebraic_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__cancel,type,
semiring_1_cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Odistrib__lattice,type,
distrib_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Obanach,type,
real_Vector_banach:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring,type,
linordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Partial__Order_Occpo,type,
comple9053668089753744459l_ccpo:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__distrib__lattice,type,
finite8700451911770168679attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Onormalization__semidom,type,
normal8620421768224518004emidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot2__space,type,
topological_t2_space:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_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_Topological__Spaces_Ouniformity,type,
topolo4638772830378233104ormity:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel2418104881723323429up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord5086331880401160121up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere6911136660526730532id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__monoid__mult,type,
topolo1898628316856586783d_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
real_V4867850818363320053vector:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim2362893244070406136eiling:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd__mult__normalize,type,
semiri6843258321239162965malize:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere166539214618696060dd_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere6658533253407199908up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
real_V7819770556892013058_space:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide__unit__factor,type,
semido2269285787275462019factor:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Operfect__space,type,
topolo8386298272705272623_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ouniform__space,type,
topolo7287701948861334536_space:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__semigroup__mult,type,
topolo4211221413907600880p_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ocomplete__space,type,
real_V8037385150606011577_space:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__algebra__1,type,
real_V2191834092415804123ebra_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord181362715937106298miring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
semiri2026040879449505780visors:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Oorder__topology,type,
topolo2564578578187576103pology:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Oring__bit__operations,type,
bit_ri3973907225187159222ations:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple6319245703460814977attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
linord4140545234300271783up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__comm__monoid__add,type,
topolo5987344860129210374id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ounbounded__dense__linorder,type,
unboun7993243217541854897norder:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ouniformity__dist,type,
real_V768167426530841204y_dist:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1__strict,type,
linord715952674999750819strict:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim462609752435547400_field:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple5582772986160207858norder:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__div__algebra,type,
real_V5047593784448816457lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
bounde4346867609351753570nf_top:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde4967611905675639751up_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__field,type,
real_V3459762299906320749_field:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Olinorder__topology,type,
topolo1944317154257567458pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Otopological__space,type,
topolo4958980785337419405_space:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
euclid3725896446679973847miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni5634975068530333245id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere8940638589300402666id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict7427464778891057005id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V822414075346904944vector:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
linord2810124833399127020strict:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
bit_se359711467146920520ations:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere2412721322843649153imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere580206878836729694up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
ordere1170586879665033532d_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict9044650504122735259up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oordered__real__vector,type,
real_V5355595471888546746vector:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra,type,
real_V4412858255891104859lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri6575147826004484403cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
euclid8851590272496341667cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra__1,type,
real_V2822296259951069270ebra_1:
!>[A: $tType] : $o ).
thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1627219031080169319umeral:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
comple592849572758109894attice:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__div__algebra,type,
real_V8999393235501362500lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Onormalization__semidom__multiplicative,type,
normal6328177297339901930cative:
!>[A: $tType] : $o ).
thf(sy_cl_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_Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,type,
counta4013691401010221786attice:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit6923001295902523014norder:
!>[A: $tType] : $o ).
thf(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim6421214686448440834_floor:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofrac,type,
archimedean_frac:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Archimedean__Field_Oround,type,
archimedean_round:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca3754400796208372196lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
thf(sy_c_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_Oand__not__num__rel,type,
bit_and_not_num_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
bit_or_not_num_neg: num > num > num ).
thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
bit_or3848514188828904588eg_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : ( A > nat > $o ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
thf(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: nat > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num,type,
bit_un1837492267222099188nd_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num__rel,type,
bit_un5425074673868309765um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oor__num,type,
bit_un2785000775030745342or_num: num > num > num ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oor__num__rel,type,
bit_un6909899581280750971um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num,type,
bit_un6178654185764691216or_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num__rel,type,
bit_un3595099601533988841um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
bit_un7362597486090784418nd_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
bit_un4731106466462545111um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oor__num,type,
bit_un6697907153464112080or_num: num > num > num ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oor__num__rel,type,
bit_un4773296044027857193um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
bit_un2480387367778600638or_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
bit_un2901131394128224187um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
boolea2506097494486148201lgebra:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
thf(sy_c_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_ONeg,type,
code_Neg: num > code_integer ).
thf(sy_c_Code__Numeral_OPos,type,
code_Pos: num > code_integer ).
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_Odup,type,
code_dup: code_integer > code_integer ).
thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
code_int_of_integer: code_integer > int ).
thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: int > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
code_integer_of_nat: nat > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__num,type,
code_integer_of_num: num > code_integer ).
thf(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
thf(sy_c_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: code_integer > num ).
thf(sy_c_Code__Numeral_Opcr__integer,type,
code_pcr_integer: int > code_integer > $o ).
thf(sy_c_Code__Numeral_Osub,type,
code_sub: num > 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_ONat,type,
code_Target_Nat: code_integer > nat ).
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_Complete__Partial__Order_Occpo_Oadmissible,type,
comple1908693960933563346ssible:
!>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > $o ) > $o ) ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp,type,
comple115746919287870866o_fixp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiterates,type,
comple6359979572994053840erates:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) ) ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp,type,
comple7512665784863727008ratesp:
!>[A: $tType] : ( ( A > A ) > A > $o ) ).
thf(sy_c_Complete__Partial__Order_Ochain,type,
comple1602240252501008431_chain:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Complete__Partial__Order_Omonotone,type,
comple7038119648293358887notone:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Complex_OArg,type,
arg: complex > real ).
thf(sy_c_Complex_Ocis,type,
cis: real > complex ).
thf(sy_c_Complex_Ocnj,type,
cnj: complex > complex ).
thf(sy_c_Complex_Ocomplex_OComplex,type,
complex2: real > real > complex ).
thf(sy_c_Complex_Ocomplex_OIm,type,
im: complex > real ).
thf(sy_c_Complex_Ocomplex_ORe,type,
re: complex > real ).
thf(sy_c_Complex_Ocsqrt,type,
csqrt: complex > complex ).
thf(sy_c_Complex_Oimaginary__unit,type,
imaginary_unit: complex ).
thf(sy_c_Complex_Orcis,type,
rcis: real > real > complex ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
condit941137186595557371_above:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
condit1013018076250108175_below:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Countable__Set_Ocountable,type,
countable_countable:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Deriv_Odifferentiable,type,
differentiable:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Deriv_Ohas__derivative,type,
has_derivative:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Deriv_Ohas__field__derivative,type,
has_field_derivative:
!>[A: $tType] : ( ( A > A ) > A > ( filter @ A ) > $o ) ).
thf(sy_c_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_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
euclid6346220572633701492n_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
euclid7384307370059645450egment:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Extended__Nat_OeSuc,type,
extended_eSuc: extended_enat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat,type,
extended_enat2: nat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat_OAbs__enat,type,
extended_Abs_enat: ( option @ nat ) > extended_enat ).
thf(sy_c_Extended__Nat_Oenat_ORep__enat,type,
extended_Rep_enat: extended_enat > ( option @ nat ) ).
thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
extended_case_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
extended_rec_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Oenat_Orec__set__enat,type,
extend4933016492236175606t_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T > $o ) ).
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
extend4730790105801354508finity:
!>[A: $tType] : A ).
thf(sy_c_Extended__Nat_Othe__enat,type,
extended_the_enat: extended_enat > nat ).
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_Oabstract__filter,type,
abstract_filter:
!>[A: $tType] : ( ( product_unit > ( filter @ A ) ) > ( filter @ 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_Ofiltermap,type,
filtermap:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Ofrequently,type,
frequently:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Finite__Set_Ofolding__on_OF,type,
finite_folding_F:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
thf(sy_c_Fun_Ostrict__mono__on,type,
strict_mono_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_Fun__Def_Ois__measure,type,
fun_is_measure:
!>[A: $tType] : ( ( A > nat ) > $o ) ).
thf(sy_c_Fun__Def_Omax__strict,type,
fun_max_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omax__weak,type,
fun_max_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omin__strict,type,
fun_min_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omin__weak,type,
fun_min_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Opair__leq,type,
fun_pair_leq: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_Fun__Def_Opair__less,type,
fun_pair_less: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_Fun__Def_Oreduction__pair,type,
fun_reduction_pair:
!>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
thf(sy_c_Fun__Def_Orp__inv__image,type,
fun_rp_inv_image:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
thf(sy_c_GCD_OGcd__class_OGcd,type,
gcd_Gcd:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > ( product_prod @ int @ int ) ).
thf(sy_c_GCD_Obezw__rel,type,
bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
bounde2362111253966948842tice_F:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( set @ A ) > A ) ).
thf(sy_c_GCD_Ogcd__class_Ogcd,type,
gcd_gcd:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__class_Olcm,type,
gcd_lcm:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__nat__rel,type,
gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
semiri4206861660011772517g_char:
!>[A: $tType] : ( ( itself @ A ) > nat ) ).
thf(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
semiring_gcd_Gcd_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
semiring_gcd_Lcm_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ocomm__monoid,type,
comm_monoid:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Ogroup,type,
group:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Omonoid,type,
monoid:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_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__monoid__list__set,type,
groups4802862169904069756st_set:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
groups4207007520872428315er_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
groups8242544230860333062m_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_Groups__List_Omonoid__list_OF,type,
groups_monoid_F:
!>[A: $tType] : ( ( A > A > A ) > A > ( list @ A ) > A ) ).
thf(sy_c_HOL_ONO__MATCH,type,
nO_MATCH:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_HOL_OUniq,type,
uniq:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp,type,
complete_lattice_gfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
complete_lattice_lfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
infini527867602293511546merate:
!>[A: $tType] : ( ( set @ A ) > nat > A ) ).
thf(sy_c_Int_OAbs__Integ,type,
abs_Integ: ( product_prod @ nat @ nat ) > int ).
thf(sy_c_Int_ORep__Integ,type,
rep_Integ: int > ( product_prod @ nat @ nat ) ).
thf(sy_c_Int_Ocr__int,type,
cr_int: ( product_prod @ nat @ nat ) > int > $o ).
thf(sy_c_Int_Oint__ge__less__than,type,
int_ge_less_than: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Ointrel,type,
intrel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Opcr__int,type,
pcr_int: ( product_prod @ nat @ nat ) > int > $o ).
thf(sy_c_Int_Opower__int,type,
power_int:
!>[A: $tType] : ( A > int > A ) ).
thf(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osemilattice__neutr,type,
semilattice_neutr:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Lattices_Osemilattice__neutr__order,type,
semila1105856199041335345_order:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices__Big_Olinorder_OMax,type,
lattices_Max:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder_OMin,type,
lattices_Min:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
lattic643756798350308766er_Min:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__max,type,
lattices_ord_arg_max:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__max__on,type,
lattic1883929316492267755max_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min,type,
lattices_ord_arg_min:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Ois__arg__max,type,
lattic501386751176901750rg_max:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
thf(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
lattic501386751177426532rg_min:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__order__set,type,
lattic4895041142388067077er_set:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__set,type,
lattic149705377957585745ce_set:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF,type,
lattic1715443433743089157tice_F:
!>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lifting_OQuotient,type,
quotient:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
thf(sy_c_Limits_OBfun,type,
bfun:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Limits_OZfun,type,
zfun:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Limits_Oat__infinity,type,
at_infinity:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oarg__min__list,type,
arg_min_list:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
thf(sy_c_List_Oarg__min__list__rel,type,
arg_min_list_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( list @ A ) ) > ( product_prod @ ( A > B ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oenumerate,type,
enumerate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
thf(sy_c_List_Ofilter,type,
filter2:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ofind,type,
find:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
thf(sy_c_List_Ofold,type,
fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ofolding__insort__key,type,
folding_insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > ( set @ B ) > ( B > A ) > $o ) ).
thf(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( ( B > A > B ) > B > ( list @ A ) > B ) ).
thf(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ogen__length,type,
gen_length:
!>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
linord329482645794927042rt_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__key,type,
linorder_insort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osort__key,type,
linorder_sort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord4507533701916653071of_set:
!>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Olist__all2,type,
list_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist_Osize__list,type,
size_list:
!>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist__ex,type,
list_ex:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
thf(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Omap__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omeasures,type,
measures:
!>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Onths,type,
nths:
!>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
thf(sy_c_List_Oord__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OremoveAll,type,
removeAll:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( nat > A > ( list @ A ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate,type,
rotate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Otake,type,
take:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Otranspose,type,
transpose:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > ( list @ nat ) ).
thf(sy_c_List_Oupto,type,
upto: int > int > ( list @ int ) ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > ( list @ int ) > ( list @ int ) ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_List_Ozip,type,
zip:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Map_Odom,type,
dom:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Map_Ograph,type,
graph:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Map_Omap__add,type,
map_add:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Omap__comp,type,
map_comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > ( option @ C ) ) > ( A > ( option @ B ) ) > A > ( option @ C ) ) ).
thf(sy_c_Map_Omap__le,type,
map_le:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > $o ) ).
thf(sy_c_Map_Omap__of,type,
map_of:
!>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Omap__upds,type,
map_upds:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Oran,type,
ran:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Map_Orestrict__map,type,
restrict_map:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > A > ( option @ B ) ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Nat_Ofunpow,type,
funpow:
!>[A: $tType] : ( nat > ( A > A ) > A > A ) ).
thf(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
thf(sy_c_Nat_Onat_Opred,type,
pred: nat > nat ).
thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
rec_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
thf(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
rec_set_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T > $o ) ).
thf(sy_c_Nat_Osemiring__1__class_ONats,type,
semiring_1_Nats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri8178284476397505188at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Nat__Bijection_Olist__decode,type,
nat_list_decode: nat > ( list @ nat ) ).
thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
nat_list_decode_rel: nat > nat > $o ).
thf(sy_c_Nat__Bijection_Olist__encode,type,
nat_list_encode: ( list @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
nat_list_encode_rel: ( list @ nat ) > ( list @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__decode,type,
nat_prod_decode: nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__encode,type,
nat_prod_encode: ( product_prod @ nat @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > ( set @ nat ) ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: ( set @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_NthRoot_Oroot,type,
root: nat > real > real ).
thf(sy_c_NthRoot_Osqrt,type,
sqrt: real > real ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Onat__of__num,type,
nat_of_num: num > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Ois__num,type,
neg_numeral_is_num:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Num_Oneg__numeral__class_Osub,type,
neg_numeral_sub:
!>[A: $tType] : ( num > num > A ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onum_Ocase__num,type,
case_num:
!>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ).
thf(sy_c_Num_Onum_Orec__num,type,
rec_num:
!>[A: $tType] : ( A > ( num > A > A ) > ( num > A > A ) > num > A ) ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Num_Oring__1__class_Oiszero,type,
ring_1_iszero:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Num_Osqr,type,
sqr: num > num ).
thf(sy_c_Option_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( option @ A ) > ( A > ( option @ B ) ) > ( option @ B ) ) ).
thf(sy_c_Option_Ocombine__options,type,
combine_options:
!>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
thf(sy_c_Option_Ooption_Omap__option,type,
map_option:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( option @ A ) > ( option @ Aa ) ) ).
thf(sy_c_Option_Ooption_Opred__option,type,
pred_option:
!>[A: $tType] : ( ( A > $o ) > ( option @ A ) > $o ) ).
thf(sy_c_Option_Ooption_Orec__option,type,
rec_option:
!>[C: $tType,A: $tType] : ( C > ( A > C ) > ( option @ A ) > C ) ).
thf(sy_c_Option_Ooption_Orel__option,type,
rel_option:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( option @ A ) > ( option @ B ) > $o ) ).
thf(sy_c_Option_Ooption_Oset__option,type,
set_option:
!>[A: $tType] : ( ( option @ A ) > ( set @ A ) ) ).
thf(sy_c_Option_Ooption_Osize__option,type,
size_option:
!>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
thf(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : ( ( option @ A ) > A ) ).
thf(sy_c_Option_Othese,type,
these:
!>[A: $tType] : ( ( set @ ( option @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Order__Continuity_Ocountable__complete__lattice__class_Occlfp,type,
order_532582986084564980_cclfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Order__Continuity_Oinf__continuous,type,
order_inf_continuous:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Order__Continuity_Osup__continuous,type,
order_sup_continuous:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Order__Relation_OunderS,type,
order_underS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord_OLeast,type,
least:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord_Omax,type,
max:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord_Omin,type,
min:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Omono,type,
order_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oordering__top,type,
ordering_top:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Power_Opower_Opower,type,
power2:
!>[A: $tType] : ( A > ( A > A > A ) > A > nat > A ) ).
thf(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Rat_OAbs__Rat,type,
abs_Rat: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_OFract,type,
fract: int > int > rat ).
thf(sy_c_Rat_OFrct,type,
frct: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_ORep__Rat,type,
rep_Rat: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Ofield__char__0__class_ORats,type,
field_char_0_Rats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
field_char_0_of_rat:
!>[A: $tType] : ( rat > A ) ).
thf(sy_c_Rat_Onormalize,type,
normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Opcr__rat,type,
pcr_rat: ( product_prod @ int @ int ) > rat > $o ).
thf(sy_c_Rat_Opositive,type,
positive: rat > $o ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oratrel,type,
ratrel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Real_OReal,type,
real2: ( nat > rat ) > real ).
thf(sy_c_Real_Ocauchy,type,
cauchy: ( nat > rat ) > $o ).
thf(sy_c_Real_Ocr__real,type,
cr_real: ( nat > rat ) > real > $o ).
thf(sy_c_Real_Opcr__real,type,
pcr_real: ( nat > rat ) > real > $o ).
thf(sy_c_Real_Opositive,type,
positive2: real > $o ).
thf(sy_c_Real_Orealrel,type,
realrel: ( nat > rat ) > ( nat > rat ) > $o ).
thf(sy_c_Real_Orep__real,type,
rep_real: real > nat > rat ).
thf(sy_c_Real_Ovanishes,type,
vanishes: ( nat > rat ) > $o ).
thf(sy_c_Real__Vector__Spaces_OReals,type,
real_Vector_Reals:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__bilinear,type,
real_V2442710119149674383linear:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear,type,
real_V3181309239436604168linear:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear__axioms,type,
real_V4916620083959148203axioms:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Oconstruct,type,
real_V4425403222259421789struct:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B ) ).
thf(sy_c_Real__Vector__Spaces_Odependent,type,
real_V358717886546972837endent:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Odim,type,
real_Vector_dim:
!>[A: $tType] : ( ( set @ A ) > nat ) ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
real_V557655796197034286t_dist:
!>[A: $tType] : ( A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_Oextend__basis,type,
real_V4986007116245087402_basis:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Real__Vector__Spaces_Olinear,type,
real_Vector_linear:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
real_V7770717601297561774m_norm:
!>[A: $tType] : ( A > real ) ).
thf(sy_c_Real__Vector__Spaces_Oof__real,type,
real_Vector_of_real:
!>[A: $tType] : ( real > A ) ).
thf(sy_c_Real__Vector__Spaces_Orepresentation,type,
real_V7696804695334737415tation:
!>[A: $tType] : ( ( set @ A ) > A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
real_V8093663219630862766scaleR:
!>[A: $tType] : ( real > A > A ) ).
thf(sy_c_Real__Vector__Spaces_Ospan,type,
real_Vector_span:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_Orelcomp,type,
relcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ).
thf(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
algebr8660921524188924756oprime:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
unit_f5069060285200089521factor:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Series_Osuminf,type,
suminf:
!>[A: $tType] : ( ( nat > A ) > A ) ).
thf(sy_c_Series_Osummable,type,
summable:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Series_Osums,type,
sums:
!>[A: $tType] : ( ( nat > A ) > A > $o ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[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_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_Ovimage,type,
vimage:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
set_fo6178422350223883121st_nat:
!>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
set_ord_atLeast:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
set_or1337092689740270186AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
set_or7035219750837199246ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatMost,type,
set_ord_atMost:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
set_ord_greaterThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
set_or3652927894154168847AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
set_or5935395276787703475ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_String_OCode_Oabort,type,
abort:
!>[A: $tType] : ( literal > ( product_unit > A ) > A ) ).
thf(sy_c_String_OLiteral,type,
literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).
thf(sy_c_String_Oascii__of,type,
ascii_of: char > char ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
comm_s6883823935334413003f_char:
!>[A: $tType] : ( char > A ) ).
thf(sy_c_String_Ointeger__of__char,type,
integer_of_char: char > code_integer ).
thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
unique5772411509450598832har_of:
!>[A: $tType] : ( A > char ) ).
thf(sy_c_Sum__Type_OPlus,type,
sum_Plus:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
thf(sy_c_Topological__Spaces_Ocontinuous,type,
topolo3448309680560233919inuous:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( A > B ) > $o ) ).
thf(sy_c_Topological__Spaces_Ocontinuous__on,type,
topolo81223032696312382ous_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > $o ) ).
thf(sy_c_Topological__Spaces_Omonoseq,type,
topological_monoseq:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
topolo1002775350975398744n_open:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ot2__space__class_OLim,type,
topolo3827282254853284352ce_Lim:
!>[F: $tType,A: $tType] : ( ( filter @ F ) > ( F > A ) > A ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
topolo174197925503356063within:
!>[A: $tType] : ( A > ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Ocompact,type,
topolo2193935891317330818ompact:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oconnected,type,
topolo1966860045006549960nected:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent,type,
topolo6863149650580417670ergent:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type,
topolo7230453075368039082e_nhds:
!>[A: $tType] : ( A > ( filter @ A ) ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy,type,
topolo3814608138187158403Cauchy:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ocauchy__filter,type,
topolo6773858410816713723filter:
!>[A: $tType] : ( ( filter @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
topolo6688025880775521714ounded:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity,type,
topolo7806501430040627800ormity:
!>[A: $tType] : ( filter @ ( product_prod @ A @ A ) ) ).
thf(sy_c_Topological__Spaces_Ouniformly__continuous__on,type,
topolo6026614971017936543ous_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > $o ) ).
thf(sy_c_Transcendental_Oarccos,type,
arccos: real > real ).
thf(sy_c_Transcendental_Oarcosh,type,
arcosh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oarcsin,type,
arcsin: real > real ).
thf(sy_c_Transcendental_Oarctan,type,
arctan: real > real ).
thf(sy_c_Transcendental_Oarsinh,type,
arsinh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oartanh,type,
artanh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Ocos,type,
cos:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Ocos__coeff,type,
cos_coeff: nat > real ).
thf(sy_c_Transcendental_Ocosh,type,
cosh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Ocot,type,
cot:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Odiffs,type,
diffs:
!>[A: $tType] : ( ( nat > A ) > nat > A ) ).
thf(sy_c_Transcendental_Oexp,type,
exp:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oln__class_Oln,type,
ln_ln:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Olog,type,
log: real > real > real ).
thf(sy_c_Transcendental_Opi,type,
pi: real ).
thf(sy_c_Transcendental_Opowr,type,
powr:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Transcendental_Opowr__real,type,
powr_real: real > real > real ).
thf(sy_c_Transcendental_Osin,type,
sin:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Osin__coeff,type,
sin_coeff: nat > real ).
thf(sy_c_Transcendental_Osinh,type,
sinh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Otan,type,
tan:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Otanh,type,
tanh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transfer_Obi__total,type,
bi_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transitive__Closure_Ontrancl,type,
transitive_ntrancl:
!>[A: $tType] : ( nat > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Ortrancl,type,
transitive_rtrancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_VEBT__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_Oelim__dead,type,
vEBT_VEBT_elim_dead: vEBT_VEBT > extended_enat > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
vEBT_V312737461966249ad_rel: ( product_prod @ vEBT_VEBT @ extended_enat ) > ( product_prod @ vEBT_VEBT @ extended_enat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
vEBT_V5917875025757280293ildren: nat > ( list @ vEBT_VEBT ) > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
vEBT_V4351362008482014158ma_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
vEBT_V5765760719290551771er_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
vEBT_VEBT_valid_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_Oset__vebt,type,
vEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
vEBT_v4011308405150292612up_rel: nat > nat > $o ).
thf(sy_c_VEBT__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__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__Insert_Ovebt__insert,type,
vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
vEBT_vebt_insert_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ).
thf(sy_c_VEBT__Member_Ovebt__member,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
vEBT_vebt_member_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
vEBT_VEBT_add: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
vEBT_VEBT_greater: ( option @ nat ) > ( option @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
vEBT_VEBT_less: ( option @ nat ) > ( option @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
vEBT_VEBT_lesseq: ( option @ nat ) > ( option @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
vEBT_VEBT_max_in_set: ( set @ nat ) > nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
vEBT_VEBT_min_in_set: ( set @ nat ) > nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
vEBT_VEBT_mul: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__comp__shift,type,
vEBT_V6923181176774028177_shift:
!>[A: $tType] : ( ( A > A > $o ) > ( option @ A ) > ( option @ A ) > $o ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__comp__shift__rel,type,
vEBT_V4810408830578336424ft_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) > ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) > $o ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift,type,
vEBT_V2048590022279873568_shift:
!>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel,type,
vEBT_V459564278314245337ft_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) > ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) > $o ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
vEBT_VEBT_power: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
vEBT_vebt_maxt: vEBT_VEBT > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
vEBT_vebt_mint: vEBT_VEBT > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
vEBT_is_pred_in_set: ( set @ nat ) > nat > nat > $o ).
thf(sy_c_VEBT__Pred_Ovebt__pred,type,
vEBT_vebt_pred: vEBT_VEBT > nat > ( option @ nat ) ).
thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
vEBT_vebt_pred_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Succ_Ois__succ__in__set,type,
vEBT_is_succ_in_set: ( set @ nat ) > nat > nat > $o ).
thf(sy_c_VEBT__Succ_Ovebt__succ,type,
vEBT_vebt_succ: vEBT_VEBT > nat > ( option @ nat ) ).
thf(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
vEBT_vebt_succ_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Wellfounded_Oless__than,type,
less_than: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__ext,type,
max_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Omin__ext,type,
min_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omlex__prod,type,
mlex_prod:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_fChoice,type,
fChoice:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_deg____,type,
deg: nat ).
thf(sy_v_m____,type,
m: nat ).
thf(sy_v_ma____,type,
ma: nat ).
thf(sy_v_mi____,type,
mi: nat ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_summary____,type,
summary: vEBT_VEBT ).
thf(sy_v_treeList____,type,
treeList: list @ vEBT_VEBT ).
thf(sy_v_x,type,
x: nat ).
% Relevant facts (8180)
thf(fact_0_False,axiom,
x != mi ).
% False
thf(fact_1__C5_Ohyps_C_I8_J,axiom,
ord_less @ nat @ ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ deg ) ).
% "5.hyps"(8)
thf(fact_2__092_060open_062mi_A_060_Ax_092_060close_062,axiom,
ord_less @ nat @ mi @ x ).
% \<open>mi < x\<close>
thf(fact_3_less__exp,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% less_exp
thf(fact_4_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one2 ) ).
% semiring_norm(85)
thf(fact_5_semiring__norm_I83_J,axiom,
! [N: num] :
( one2
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_6_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_7__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ deg ).
% \<open>2 \<le> deg\<close>
thf(fact_8_verit__eq__simplify_I8_J,axiom,
! [X2: num,Y2: num] :
( ( ( bit0 @ X2 )
= ( bit0 @ Y2 ) )
= ( X2 = Y2 ) ) ).
% verit_eq_simplify(8)
thf(fact_9_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_10_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_11_verit__eq__simplify_I10_J,axiom,
! [X2: num] :
( one2
!= ( bit0 @ X2 ) ) ).
% verit_eq_simplify(10)
thf(fact_12_local_Opower__def,axiom,
( vEBT_VEBT_power
= ( vEBT_V2048590022279873568_shift @ nat @ ( power_power @ nat ) ) ) ).
% local.power_def
thf(fact_13__C5_Ohyps_C_I7_J,axiom,
ord_less_eq @ nat @ mi @ ma ).
% "5.hyps"(7)
thf(fact_14__092_060open_062_092_060not_062_A_Ix_A_061_Ami_A_092_060and_062_Ax_A_061_Ama_J_092_060close_062,axiom,
~ ( ( x = mi )
& ( x = ma ) ) ).
% \<open>\<not> (x = mi \<and> x = ma)\<close>
thf(fact_15_max__in__set__def,axiom,
( vEBT_VEBT_max_in_set
= ( ^ [Xs: set @ nat,X: nat] :
( ( member @ nat @ X @ Xs )
& ! [Y: nat] :
( ( member @ nat @ Y @ Xs )
=> ( ord_less_eq @ nat @ Y @ X ) ) ) ) ) ).
% max_in_set_def
thf(fact_16_min__in__set__def,axiom,
( vEBT_VEBT_min_in_set
= ( ^ [Xs: set @ nat,X: nat] :
( ( member @ nat @ X @ Xs )
& ! [Y: nat] :
( ( member @ nat @ Y @ Xs )
=> ( ord_less_eq @ nat @ X @ Y ) ) ) ) ) ).
% min_in_set_def
thf(fact_17__092_060open_062x_A_092_060le_062_Ama_A_092_060and_062_Ami_A_092_060le_062_Ax_092_060close_062,axiom,
( ( ord_less_eq @ nat @ x @ ma )
& ( ord_less_eq @ nat @ mi @ x ) ) ).
% \<open>x \<le> ma \<and> mi \<le> x\<close>
thf(fact_18_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_19_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_20_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less @ num @ M @ one2 ) ).
% semiring_norm(75)
thf(fact_21_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_22_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(2)
thf(fact_23_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_24_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B3: B,A3: B] :
( ( ~ ( ord_less_eq @ B @ B3 @ A3 ) )
= ( ord_less @ B @ A3 @ B3 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_25_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_26_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_27_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_28_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(1)
thf(fact_29_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_30_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_31_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_32_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_33_power__shift,axiom,
! [X3: nat,Y3: nat,Z: nat] :
( ( ( power_power @ nat @ X3 @ Y3 )
= Z )
= ( ( vEBT_VEBT_power @ ( some @ nat @ X3 ) @ ( some @ nat @ Y3 ) )
= ( some @ nat @ Z ) ) ) ).
% power_shift
thf(fact_34_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: nat,I: num,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less_eq @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_35_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X3: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X3 ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X3 ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_36_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: nat,I: num,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_37_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X3: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X3 ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X3 ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_38_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less @ nat @ ( F2 @ Y4 ) @ B2 ) )
=> ? [X4: A] :
( ( P @ X4 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( F2 @ Y5 ) @ ( F2 @ X4 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_39_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 )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ K2 @ I2 )
=> ( P @ I2 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_40_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ord_less @ nat @ ( F2 @ I3 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_41_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_42_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_43_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_46_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X4: A] :
( ( F2 @ X4 )
= ( G @ X4 ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_47_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% semiring_norm(68)
thf(fact_48_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_49_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_50_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_51_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_52_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [B2: nat,W: nat,X3: nat] :
( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W )
= ( semiring_1_of_nat @ A @ X3 ) )
= ( ( power_power @ nat @ B2 @ W )
= X3 ) ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_53_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X3: nat,B2: nat,W: nat] :
( ( ( semiring_1_of_nat @ A @ X3 )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
= ( X3
= ( power_power @ nat @ B2 @ W ) ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_54_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% semiring_norm(69)
thf(fact_55_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_56_lesseq__shift,axiom,
( ( ord_less_eq @ nat )
= ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some @ nat @ X ) @ ( some @ nat @ Y ) ) ) ) ).
% lesseq_shift
thf(fact_57_numeral__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X3: num,N: nat,Y3: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N )
= ( semiring_1_of_nat @ A @ Y3 ) )
= ( ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N )
= Y3 ) ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_58_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [Y3: nat,X3: num,N: nat] :
( ( ( semiring_1_of_nat @ A @ Y3 )
= ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N ) )
= ( Y3
= ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N ) ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_59_of__nat__less__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W: nat,X3: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) @ ( semiring_1_of_nat @ A @ X3 ) )
= ( ord_less @ nat @ ( power_power @ nat @ B2 @ W ) @ X3 ) ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_60_of__nat__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: nat,B2: nat,W: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
= ( ord_less @ nat @ X3 @ ( power_power @ nat @ B2 @ W ) ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_61_of__nat__le__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W: nat,X3: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) @ ( semiring_1_of_nat @ A @ X3 ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W ) @ X3 ) ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_62_of__nat__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: nat,B2: nat,W: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
= ( ord_less_eq @ nat @ X3 @ ( power_power @ nat @ B2 @ W ) ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_63_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_64_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_65_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_66_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B4: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_67_le__num__One__iff,axiom,
! [X3: num] :
( ( ord_less_eq @ num @ X3 @ one2 )
= ( X3 = one2 ) ) ).
% le_num_One_iff
thf(fact_68_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_69_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_70_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_71_pow_Osimps_I1_J,axiom,
! [X3: num] :
( ( pow @ X3 @ one2 )
= X3 ) ).
% pow.simps(1)
thf(fact_72_order__antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ) ).
% order_antisym_conv
thf(fact_73_linorder__le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% linorder_le_cases
thf(fact_74_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 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_75_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 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_76_linorder__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% linorder_linear
thf(fact_77_order__eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( X3 = Y3 )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% order_eq_refl
thf(fact_78_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 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ C @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_79_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 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_80_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_81_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F3 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_82_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ).
% le_funI
thf(fact_83_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funE
thf(fact_84_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funD
thf(fact_85_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_86_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_87_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_88_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_89_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A,B5: A] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_90_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z )
=> ( ord_less_eq @ A @ X3 @ Z ) ) ) ) ).
% order_trans
thf(fact_91_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_92_order__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ) ).
% order_antisym
thf(fact_93_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_94_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_95_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
& ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_96_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_97_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_98_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X3: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X3 ) ) ).
% lt_ex
thf(fact_99_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X3: A] :
? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ).
% gt_ex
thf(fact_100_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ? [Z3: A] :
( ( ord_less @ A @ X3 @ Z3 )
& ( ord_less @ A @ Z3 @ Y3 ) ) ) ) ).
% dense
thf(fact_101_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ) ).
% less_imp_neq
thf(fact_102_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_103_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_104_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_105_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X4 )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_106_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y3: A,X3: A] :
( ~ ( ord_less @ A @ Y3 @ X3 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ) ).
% antisym_conv3
thf(fact_107_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ( X3 != Y3 )
=> ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% linorder_cases
thf(fact_108_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_109_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_110_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X5: A] : ( P2 @ X5 ) )
= ( ^ [P3: A > $o] :
? [N3: A] :
( ( P3 @ N3 )
& ! [M3: A] :
( ( ord_less @ A @ M3 @ N3 )
=> ~ ( P3 @ M3 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_111_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A6: A,B5: A] :
( ( ord_less @ A @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A] : ( P @ A6 @ A6 )
=> ( ! [A6: A,B5: A] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_112_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_113_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
= ( ( ord_less @ A @ Y3 @ X3 )
| ( X3 = Y3 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_114_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_115_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_116_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_117_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y5: A] :
( ( ord_less @ B @ ( F2 @ Y5 ) @ ( F2 @ X4 ) )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ) ).
% measure_induct
thf(fact_118_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y5: A] :
( ( ord_less @ B @ ( F2 @ Y5 ) @ ( F2 @ X4 ) )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ) ).
% measure_induct_rule
thf(fact_119_linorder__neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( X3 != Y3 )
=> ( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% linorder_neqE
thf(fact_120_order__less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% order_less_asym
thf(fact_121_linorder__neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( X3 != Y3 )
= ( ( ord_less @ A @ X3 @ Y3 )
| ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% linorder_neq_iff
thf(fact_122_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_123_order__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ Z )
=> ( ord_less @ A @ X3 @ Z ) ) ) ) ).
% order_less_trans
thf(fact_124_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 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_125_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 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ B @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_126_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 ) ) ).
% order_less_irrefl
thf(fact_127_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 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_128_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 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ C @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_129_order__less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% order_less_not_sym
thf(fact_130_order__less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A,P: $o] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ X3 )
=> P ) ) ) ).
% order_less_imp_triv
thf(fact_131_linorder__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
| ( X3 = Y3 )
| ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% linorder_less_linear
thf(fact_132_order__less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ) ).
% order_less_imp_not_eq
thf(fact_133_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( Y3 != X3 ) ) ) ).
% order_less_imp_not_eq2
thf(fact_134_order__less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% order_less_imp_not_less
thf(fact_135_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_136_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_137_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_138_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_139_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_140_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_141_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_142_linorder__neqE__nat,axiom,
! [X3: nat,Y3: nat] :
( ( X3 != Y3 )
=> ( ~ ( ord_less @ nat @ X3 @ Y3 )
=> ( ord_less @ nat @ Y3 @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_143_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X3: A] :
( ! [X4: A] :
( ~ ( P @ X4 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X4 ) )
& ~ ( P @ Y5 ) ) )
=> ( P @ X3 ) ) ).
% infinite_descent_measure
thf(fact_144_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ? [X4: A] :
( ( P @ X4 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M @ X4 ) @ ( M @ Y5 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_145_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B2 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_146_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_147_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_148_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_149_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_150_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_151_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ~ ( ord_less @ A @ X3 @ Y3 ) ) ) ).
% leD
thf(fact_152_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% leI
thf(fact_153_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_154_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ) ).
% antisym_conv1
thf(fact_155_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ) ).
% antisym_conv2
thf(fact_156_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,Y3: A] :
( ! [X4: A] :
( ( ord_less @ A @ Z @ X4 )
=> ( ord_less_eq @ A @ Y3 @ X4 ) )
=> ( ord_less_eq @ A @ Y3 @ Z ) ) ) ).
% dense_ge
thf(fact_157_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y3: A,Z: A] :
( ! [X4: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ord_less_eq @ A @ Y3 @ Z ) ) ) ).
% dense_le
thf(fact_158_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
& ~ ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ).
% less_le_not_le
thf(fact_159_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y3: A,X3: A] :
( ~ ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ord_less @ A @ X3 @ Y3 ) ) ) ).
% not_le_imp_less
thf(fact_160_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_161_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_162_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_163_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_164_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ~ ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_165_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,X3: A,Y3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ! [W2: A] :
( ( ord_less @ A @ Z @ W2 )
=> ( ( ord_less @ A @ W2 @ X3 )
=> ( ord_less_eq @ A @ Y3 @ W2 ) ) )
=> ( ord_less_eq @ A @ Y3 @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_166_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ! [W2: A] :
( ( ord_less @ A @ X3 @ W2 )
=> ( ( ord_less @ A @ W2 @ Y3 )
=> ( ord_less_eq @ A @ W2 @ Z ) ) )
=> ( ord_less_eq @ A @ Y3 @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_167_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less @ A @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_168_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_169_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_170_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_171_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ~ ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_172_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_173_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_174_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ) ).
% order_le_less
thf(fact_175_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
& ( X != Y ) ) ) ) ) ).
% order_less_le
thf(fact_176_linorder__not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ~ ( ord_less_eq @ A @ X3 @ Y3 ) )
= ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% linorder_not_le
thf(fact_177_linorder__not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% linorder_not_less
thf(fact_178_order__less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% order_less_imp_le
thf(fact_179_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_180_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_181_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ Z )
=> ( ord_less @ A @ X3 @ Z ) ) ) ) ).
% order_le_less_trans
thf(fact_182_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z )
=> ( ord_less @ A @ X3 @ Z ) ) ) ) ).
% order_less_le_trans
thf(fact_183_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 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_184_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 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ C @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_185_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 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_186_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 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ C @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_187_linorder__le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
| ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% linorder_le_less_linear
thf(fact_188_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less @ A @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_189_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_190_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_191_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_192_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_193_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_194_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_195_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_196_numeral__less__real__of__nat__iff,axiom,
! [W: num,N: nat] :
( ( ord_less @ real @ ( numeral_numeral @ real @ W ) @ ( semiring_1_of_nat @ real @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ W ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_197_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W: num] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( numeral_numeral @ real @ W ) )
= ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ W ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_198_greater__shift,axiom,
( ( ord_less @ nat )
= ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some @ nat @ X ) @ ( some @ nat @ Y ) ) ) ) ).
% greater_shift
thf(fact_199_less__shift,axiom,
( ( ord_less @ nat )
= ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some @ nat @ X ) @ ( some @ nat @ Y ) ) ) ) ).
% less_shift
thf(fact_200_option_Oinject,axiom,
! [A: $tType,X2: A,Y2: A] :
( ( ( some @ A @ X2 )
= ( some @ A @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_201_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
! [A: $tType,F2: A > A > A,A2: A,B2: A] :
( ( vEBT_V2048590022279873568_shift @ A @ F2 @ ( some @ A @ A2 ) @ ( some @ A @ B2 ) )
= ( some @ A @ ( F2 @ A2 @ B2 ) ) ) ).
% VEBT_internal.option_shift.simps(3)
thf(fact_202_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_203_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [N2: nat] : ( ord_less @ A @ X3 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% reals_Archimedean2
thf(fact_204_real__arch__simple,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [N2: nat] : ( ord_less_eq @ A @ X3 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% real_arch_simple
thf(fact_205_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ~ ( ord_less_eq @ A @ T2 @ X6 ) ) ) ).
% minf(8)
thf(fact_206_nat__int__comparison_I1_J,axiom,
( ( ^ [Y6: nat,Z2: nat] : Y6 = Z2 )
= ( ^ [A5: nat,B4: nat] :
( ( semiring_1_of_nat @ int @ A5 )
= ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_207_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_208_complete__real,axiom,
! [S2: set @ real] :
( ? [X6: real] : ( member @ real @ X6 @ S2 )
=> ( ? [Z4: real] :
! [X4: real] :
( ( member @ real @ X4 @ S2 )
=> ( ord_less_eq @ real @ X4 @ Z4 ) )
=> ? [Y4: real] :
( ! [X6: real] :
( ( member @ real @ X6 @ S2 )
=> ( ord_less_eq @ real @ X6 @ Y4 ) )
& ! [Z4: real] :
( ! [X4: real] :
( ( member @ real @ X4 @ S2 )
=> ( ord_less_eq @ real @ X4 @ Z4 ) )
=> ( ord_less_eq @ real @ Y4 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_209_verit__la__generic,axiom,
! [A2: int,X3: int] :
( ( ord_less_eq @ int @ A2 @ X3 )
| ( A2 = X3 )
| ( ord_less_eq @ int @ X3 @ A2 ) ) ).
% verit_la_generic
thf(fact_210_less__eq__real__def,axiom,
( ( ord_less_eq @ real )
= ( ^ [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
| ( X = Y ) ) ) ) ).
% less_eq_real_def
thf(fact_211_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_212_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_213_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_214_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_215_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( X6 != T2 ) ) ) ).
% pinf(3)
thf(fact_216_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( X6 != T2 ) ) ) ).
% pinf(4)
thf(fact_217_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ~ ( ord_less @ A @ X6 @ T2 ) ) ) ).
% pinf(5)
thf(fact_218_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( ord_less @ A @ T2 @ X6 ) ) ) ).
% pinf(7)
thf(fact_219_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F4: D] :
? [Z3: C] :
! [X6: C] :
( ( ord_less @ C @ Z3 @ X6 )
=> ( F4 = F4 ) ) ) ).
% pinf(11)
thf(fact_220_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_221_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_222_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( X6 != T2 ) ) ) ).
% minf(3)
thf(fact_223_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( X6 != T2 ) ) ) ).
% minf(4)
thf(fact_224_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( ord_less @ A @ X6 @ T2 ) ) ) ).
% minf(5)
thf(fact_225_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ~ ( ord_less @ A @ T2 @ X6 ) ) ) ).
% minf(7)
thf(fact_226_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F4: D] :
? [Z3: C] :
! [X6: C] :
( ( ord_less @ C @ X6 @ Z3 )
=> ( F4 = F4 ) ) ) ).
% minf(11)
thf(fact_227_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_228_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ~ ( ord_less_eq @ A @ X6 @ T2 ) ) ) ).
% pinf(6)
thf(fact_229_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( ord_less_eq @ A @ T2 @ X6 ) ) ) ).
% pinf(8)
thf(fact_230_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( ord_less_eq @ A @ X6 @ T2 ) ) ) ).
% minf(6)
thf(fact_231_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A2: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
& ( ord_less_eq @ A @ C3 @ B2 )
& ! [X6: A] :
( ( ( ord_less_eq @ A @ A2 @ X6 )
& ( ord_less @ A @ X6 @ C3 ) )
=> ( P @ X6 ) )
& ! [D2: A] :
( ! [X4: A] :
( ( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less @ A @ X4 @ D2 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq @ A @ D2 @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_232_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_233_power2__eq__iff__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( X3 = Y3 ) ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_234_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_235_power2__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less @ A @ X3 @ Y3 ) ) ) ) ).
% power2_less_imp_less
thf(fact_236_nat__le__numeral__power__cancel__iff,axiom,
! [A2: int,X3: num,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_237_numeral__power__le__nat__cancel__iff,axiom,
! [X3: num,N: nat,A2: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N ) @ ( nat2 @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) @ A2 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_238_numeral__power__less__nat__cancel__iff,axiom,
! [X3: num,N: nat,A2: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N ) @ ( nat2 @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) @ A2 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_239_nat__less__numeral__power__cancel__iff,axiom,
! [A2: int,X3: num,N: nat] :
( ( ord_less @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_240_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X3: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_241_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: num,N: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) @ A2 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_242_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_243_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_244_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_245_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_246_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_247_of__int__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W: int,Z: int] :
( ( ( ring_1_of_int @ A @ W )
= ( ring_1_of_int @ A @ Z ) )
= ( W = Z ) ) ) ).
% of_int_eq_iff
thf(fact_248_i0__less,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( N
!= ( zero_zero @ extended_enat ) ) ) ).
% i0_less
thf(fact_249_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_250_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_251_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_252_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_253_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int] :
( ( ( zero_zero @ A )
= ( ring_1_of_int @ A @ Z ) )
= ( Z
= ( zero_zero @ int ) ) ) ) ).
% of_int_0_eq_iff
thf(fact_254_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int] :
( ( ( ring_1_of_int @ A @ Z )
= ( zero_zero @ A ) )
= ( Z
= ( zero_zero @ int ) ) ) ) ).
% of_int_eq_0_iff
thf(fact_255_nat__zero__less__power__iff,axiom,
! [X3: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X3 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_256_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_257_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_258_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
= N ) ).
% nat_int
thf(fact_259_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_260_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_261_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_262_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_263_of__int__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int,N: num] :
( ( ( ring_1_of_int @ A @ Z )
= ( numeral_numeral @ A @ N ) )
= ( Z
= ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_264_of__int__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: int,Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ W @ Z ) ) ) ).
% of_int_le_iff
thf(fact_265_of__int__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: int,Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ W @ Z ) ) ) ).
% of_int_less_iff
thf(fact_266_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral @ int @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% nat_numeral
thf(fact_267_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ).
% zero_less_nat_eq
thf(fact_268_zless__nat__conj,axiom,
! [W: int,Z: int] :
( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
& ( ord_less @ int @ W @ Z ) ) ) ).
% zless_nat_conj
thf(fact_269_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_270_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
= ( zero_zero @ int ) ) ) ) ).
% int_nat_eq
thf(fact_271_of__int__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int,N: nat] :
( ( ring_1_of_int @ A @ ( power_power @ int @ Z @ N ) )
= ( power_power @ A @ ( ring_1_of_int @ A @ Z ) @ N ) ) ) ).
% of_int_power
thf(fact_272_of__int__eq__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [B2: int,W: nat,X3: int] :
( ( ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W )
= ( ring_1_of_int @ A @ X3 ) )
= ( ( power_power @ int @ B2 @ W )
= X3 ) ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_273_of__int__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X3: int,B2: int,W: nat] :
( ( ( ring_1_of_int @ A @ X3 )
= ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
= ( X3
= ( power_power @ int @ B2 @ W ) ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_274_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_275_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_276_of__int__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% of_int_le_0_iff
thf(fact_277_of__int__0__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% of_int_0_le_iff
thf(fact_278_of__int__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
= ( ord_less @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% of_int_less_0_iff
thf(fact_279_of__int__0__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% of_int_0_less_iff
thf(fact_280_of__nat__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ A @ ( nat2 @ Z ) )
= ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_nat_nat
thf(fact_281_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_282_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X3 ) @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_283_of__int__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z: int] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ ( numeral_numeral @ int @ N ) @ Z ) ) ) ).
% of_int_numeral_le_iff
thf(fact_284_of__int__le__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int,N: num] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ int @ Z @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_le_numeral_iff
thf(fact_285_of__int__less__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int,N: num] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ int @ Z @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_less_numeral_iff
thf(fact_286_of__int__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z: int] :
( ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ ( numeral_numeral @ int @ N ) @ Z ) ) ) ).
% of_int_numeral_less_iff
thf(fact_287_numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X3: num,N: nat,Y3: int] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N )
= ( ring_1_of_int @ A @ Y3 ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N )
= Y3 ) ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_288_of__int__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y3: int,X3: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y3 )
= ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N ) )
= ( Y3
= ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_289_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: int,B2: int,W: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X3 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
= ( ord_less_eq @ int @ X3 @ ( power_power @ int @ B2 @ W ) ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_290_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W: nat,X3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) @ ( ring_1_of_int @ A @ X3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ B2 @ W ) @ X3 ) ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_291_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W: nat,X3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) @ ( ring_1_of_int @ A @ X3 ) )
= ( ord_less @ int @ ( power_power @ int @ B2 @ W ) @ X3 ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_292_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: int,B2: int,W: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X3 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
= ( ord_less @ int @ X3 @ ( power_power @ int @ B2 @ W ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_293_nat__eq__numeral__power__cancel__iff,axiom,
! [Y3: int,X3: num,N: nat] :
( ( ( nat2 @ Y3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N ) )
= ( Y3
= ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_294_numeral__power__eq__nat__cancel__iff,axiom,
! [X3: num,N: nat,Y3: int] :
( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N )
= ( nat2 @ Y3 ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N )
= Y3 ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_295_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: num,N: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) @ A2 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_296_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X3: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_297_nat__zero__as__int,axiom,
( ( zero_zero @ nat )
= ( nat2 @ ( zero_zero @ int ) ) ) ).
% nat_zero_as_int
thf(fact_298_eq__nat__nat__iff,axiom,
! [Z: int,Z5: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z5 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z5 ) )
= ( Z = Z5 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_299_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
! [X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% all_nat
thf(fact_300_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% ex_nat
thf(fact_301_int__ops_I1_J,axiom,
( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ) ).
% int_ops(1)
thf(fact_302_of__int__nonneg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_nonneg
thf(fact_303_of__int__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_pos
thf(fact_304_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M
= ( nat2 @ W ) )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( W
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff2
thf(fact_305_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( ( nat2 @ W )
= M )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( W
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff
thf(fact_306_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_307_nat__mono__iff,axiom,
! [Z: int,W: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ W @ Z ) ) ) ).
% nat_mono_iff
thf(fact_308_nat__power__eq,axiom,
! [Z: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( nat2 @ ( power_power @ int @ Z @ N ) )
= ( power_power @ nat @ ( nat2 @ Z ) @ N ) ) ) ).
% nat_power_eq
thf(fact_309_int__eq__iff,axiom,
! [M: nat,Z: int] :
( ( ( semiring_1_of_nat @ int @ M )
= Z )
= ( ( M
= ( nat2 @ Z ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% int_eq_iff
thf(fact_310_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_311_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [Z3: int] : ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_le_of_int
thf(fact_312_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [Z3: int] : ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_less_of_int
thf(fact_313_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [Z3: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ X3 ) ) ).
% ex_of_int_less
thf(fact_314_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_315_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_316_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_317_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_318_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_319_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_320_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_321_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_322_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_323_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_324_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_325_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_326_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_327_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X3: A] :
( ! [X4: A] :
( ( ( V @ X4 )
= ( zero_zero @ nat ) )
=> ( P @ X4 ) )
=> ( ! [X4: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X4 ) )
=> ( ~ ( P @ X4 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X4 ) )
& ~ ( P @ Y5 ) ) ) )
=> ( P @ X3 ) ) ) ).
% infinite_descent0_measure
thf(fact_328_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_329_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_330_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_331_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_332_conj__le__cong,axiom,
! [X3: int,X7: int,P: $o,P4: $o] :
( ( X3 = X7 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
& P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_333_imp__le__cong,axiom,
! [X3: int,X7: int,P: $o,P4: $o] :
( ( X3 = X7 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_334_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_335_less__int__code_I1_J,axiom,
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% less_int_code(1)
thf(fact_336_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_less @ extended_enat @ N @ ( zero_zero @ extended_enat ) ) ).
% not_iless0
thf(fact_337_nat__less__eq__zless,axiom,
! [W: int,Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ W @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_338_nat__le__eq__zle,axiom,
! [W: int,Z: int] :
( ( ( ord_less @ int @ ( zero_zero @ int ) @ W )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) )
=> ( ( ord_less_eq @ nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_eq @ int @ W @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_339_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_340_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_341_i0__lb,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ).
% i0_lb
thf(fact_342_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_343_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_344_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_345_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_346_nat__numeral__as__int,axiom,
( ( numeral_numeral @ nat )
= ( ^ [I4: num] : ( nat2 @ ( numeral_numeral @ int @ I4 ) ) ) ) ).
% nat_numeral_as_int
thf(fact_347_nat__less__iff,axiom,
! [W: int,M: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( ( ord_less @ nat @ ( nat2 @ W ) @ M )
= ( ord_less @ int @ W @ ( semiring_1_of_nat @ int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_348_nat__mono,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq @ int @ X3 @ Y3 )
=> ( ord_less_eq @ nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ).
% nat_mono
thf(fact_349_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_350_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_351_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_352_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_353_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_354_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_355_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_356_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_357_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_358_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_359_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K2 )
=> ~ ( P @ I2 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_360_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_361_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_362_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_363_zless__nat__eq__int__zless,axiom,
! [M: nat,Z: int] :
( ( ord_less @ nat @ M @ ( nat2 @ Z ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ M ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_364_nat__le__iff,axiom,
! [X3: int,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ X3 ) @ N )
= ( ord_less_eq @ int @ X3 @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% nat_le_iff
thf(fact_365_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit5016429287641298734tinuum @ A )
=> ! [A2: A] :
? [B5: A] :
( ( ord_less @ A @ A2 @ B5 )
| ( ord_less @ A @ B5 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_366_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_367_of__nat__less__of__int__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,X3: int] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_of_int @ A @ X3 ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ X3 ) ) ) ).
% of_nat_less_of_int_iff
thf(fact_368_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_369_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_370_power2__eq__imp__eq,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,Y3: A] :
( ( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( X3 = Y3 ) ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_371_power2__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ).
% power2_le_imp_le
thf(fact_372_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_373_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_374_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_375_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_376_pos2,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ).
% pos2
thf(fact_377_realpow__pos__nth__unique,axiom,
! [N: nat,A2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [X4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X4 )
& ( ( power_power @ real @ X4 @ N )
= A2 )
& ! [Y5: real] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y5 )
& ( ( power_power @ real @ Y5 @ N )
= A2 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_378_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_379_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_380_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_381_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_382_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_383_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X3: A] :
( ( ( zero_zero @ A )
= X3 )
= ( X3
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_384_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) ).
% zero_le
thf(fact_385_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_386_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_387_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_388_XOR__upper,axiom,
! [X3: int,N: nat,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( ( ord_less @ int @ X3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X3 @ Y3 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_389_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W: num] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W ) )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( A2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_390_OR__upper,axiom,
! [X3: int,N: nat,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( ( ord_less @ int @ X3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X3 @ Y3 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_391_real__sqrt__pow2__iff,axiom,
! [X3: real] :
( ( ( power_power @ real @ ( sqrt @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X3 )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) ) ).
% real_sqrt_pow2_iff
thf(fact_392_real__sqrt__pow2,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( power_power @ real @ ( sqrt @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X3 ) ) ).
% real_sqrt_pow2
thf(fact_393_real__less__lsqrt,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less @ real @ X3 @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sqrt @ X3 ) @ Y3 ) ) ) ) ).
% real_less_lsqrt
thf(fact_394_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_395_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_396_power__one__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( one_one @ nat ) )
= A2 ) ) ).
% power_one_right
thf(fact_397_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_398_or_Oidem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ A2 )
= A2 ) ) ).
% or.idem
thf(fact_399_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_400_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_401_real__sqrt__one,axiom,
( ( sqrt @ ( one_one @ real ) )
= ( one_one @ real ) ) ).
% real_sqrt_one
thf(fact_402_real__sqrt__eq__iff,axiom,
! [X3: real,Y3: real] :
( ( ( sqrt @ X3 )
= ( sqrt @ Y3 ) )
= ( X3 = Y3 ) ) ).
% real_sqrt_eq_iff
thf(fact_403_real__sqrt__eq__1__iff,axiom,
! [X3: real] :
( ( ( sqrt @ X3 )
= ( one_one @ real ) )
= ( X3
= ( one_one @ real ) ) ) ).
% real_sqrt_eq_1_iff
thf(fact_404_bit_Oxor__left__self,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A,Y3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X3 @ ( bit_se5824344971392196577ns_xor @ A @ X3 @ Y3 ) )
= Y3 ) ) ).
% bit.xor_left_self
thf(fact_405_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_406_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_407_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_408_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_409_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_410_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_411_of__int__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int] :
( ( ( ring_1_of_int @ A @ Z )
= ( one_one @ A ) )
= ( Z
= ( one_one @ int ) ) ) ) ).
% of_int_eq_1_iff
thf(fact_412_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_413_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_414_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_415_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_416_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_417_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_418_bit_Oxor__self,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X3 @ X3 )
= ( zero_zero @ A ) ) ) ).
% bit.xor_self
thf(fact_419_real__sqrt__zero,axiom,
( ( sqrt @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% real_sqrt_zero
thf(fact_420_real__sqrt__eq__zero__cancel__iff,axiom,
! [X3: real] :
( ( ( sqrt @ X3 )
= ( zero_zero @ real ) )
= ( X3
= ( zero_zero @ real ) ) ) ).
% real_sqrt_eq_zero_cancel_iff
thf(fact_421_real__sqrt__lt__1__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( sqrt @ X3 ) @ ( one_one @ real ) )
= ( ord_less @ real @ X3 @ ( one_one @ real ) ) ) ).
% real_sqrt_lt_1_iff
thf(fact_422_real__sqrt__gt__1__iff,axiom,
! [Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( sqrt @ Y3 ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y3 ) ) ).
% real_sqrt_gt_1_iff
thf(fact_423_real__sqrt__less__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) )
= ( ord_less @ real @ X3 @ Y3 ) ) ).
% real_sqrt_less_iff
thf(fact_424_real__sqrt__ge__1__iff,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ Y3 ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y3 ) ) ).
% real_sqrt_ge_1_iff
thf(fact_425_real__sqrt__le__1__iff,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( sqrt @ X3 ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X3 @ ( one_one @ real ) ) ) ).
% real_sqrt_le_1_iff
thf(fact_426_real__sqrt__le__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) )
= ( ord_less_eq @ real @ X3 @ Y3 ) ) ).
% real_sqrt_le_iff
thf(fact_427_log__one,axiom,
! [A2: real] :
( ( log @ A2 @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% log_one
thf(fact_428_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_429_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_430_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_431_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_432_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_433_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_434_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_435_real__sqrt__gt__0__iff,axiom,
! [Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ Y3 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y3 ) ) ).
% real_sqrt_gt_0_iff
thf(fact_436_real__sqrt__lt__0__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( sqrt @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ).
% real_sqrt_lt_0_iff
thf(fact_437_real__sqrt__ge__0__iff,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ Y3 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 ) ) ).
% real_sqrt_ge_0_iff
thf(fact_438_real__sqrt__le__0__iff,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( sqrt @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) ) ) ).
% real_sqrt_le_0_iff
thf(fact_439_zero__less__log__cancel__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X3 ) )
= ( ord_less @ real @ ( one_one @ real ) @ X3 ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_440_log__less__zero__cancel__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( log @ A2 @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X3 @ ( one_one @ real ) ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_441_one__less__log__cancel__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( log @ A2 @ X3 ) )
= ( ord_less @ real @ A2 @ X3 ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_442_log__less__one__cancel__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( log @ A2 @ X3 ) @ ( one_one @ real ) )
= ( ord_less @ real @ X3 @ A2 ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_443_log__less__cancel__iff,axiom,
! [A2: real,X3: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ Y3 ) )
= ( ord_less @ real @ X3 @ Y3 ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_444_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_445_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_446_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_447_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_448_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_449_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X3: nat,Y3: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ X3 ) @ ( power_power @ A @ B2 @ Y3 ) )
= ( ord_less @ nat @ X3 @ Y3 ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_450_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_451_of__int__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) )
= ( ord_less_eq @ int @ Z @ ( one_one @ int ) ) ) ) ).
% of_int_le_1_iff
thf(fact_452_of__int__1__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ ( one_one @ int ) @ Z ) ) ) ).
% of_int_1_le_iff
thf(fact_453_of__int__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) )
= ( ord_less @ int @ Z @ ( one_one @ int ) ) ) ) ).
% of_int_less_1_iff
thf(fact_454_of__int__1__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z ) ) ) ).
% of_int_1_less_iff
thf(fact_455_real__sqrt__four,axiom,
( ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).
% real_sqrt_four
thf(fact_456_zero__le__log__cancel__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X3 ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X3 ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_457_log__le__zero__cancel__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X3 @ ( one_one @ real ) ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_458_one__le__log__cancel__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( log @ A2 @ X3 ) )
= ( ord_less_eq @ real @ A2 @ X3 ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_459_log__le__one__cancel__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X3 ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X3 @ A2 ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_460_log__le__cancel__iff,axiom,
! [A2: real,X3: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ Y3 ) )
= ( ord_less_eq @ real @ X3 @ Y3 ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_461_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_462_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_463_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X3: nat,Y3: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X3 ) @ ( power_power @ A @ B2 @ Y3 ) )
= ( ord_less_eq @ nat @ X3 @ Y3 ) ) ) ) ).
% power_increasing_iff
thf(fact_464_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_465_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_466_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_467_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_468_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_469_zero__le__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_470_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_471_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W: num] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_472_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_473_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) )
= ( ( ( numeral_numeral @ nat @ W )
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( A2
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_474_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_475_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_476_or_Ocommute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se1065995026697491101ons_or @ A )
= ( ^ [A5: A,B4: A] : ( bit_se1065995026697491101ons_or @ A @ B4 @ A5 ) ) ) ) ).
% or.commute
thf(fact_477_xor_Ocommute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5824344971392196577ns_xor @ A )
= ( ^ [A5: A,B4: A] : ( bit_se5824344971392196577ns_xor @ A @ B4 @ A5 ) ) ) ) ).
% xor.commute
thf(fact_478_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_479_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_480_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X3: A] :
( ( ( one_one @ A )
= X3 )
= ( X3
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_481_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_482_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_483_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ M @ N )
=> ( ( dvd_dvd @ nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_484_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_485_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_486_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_487_real__sqrt__ge__one,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ X3 ) ) ) ).
% real_sqrt_ge_one
thf(fact_488_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_489_int__ops_I2_J,axiom,
( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
= ( one_one @ int ) ) ).
% int_ops(2)
thf(fact_490_nat__one__as__int,axiom,
( ( one_one @ nat )
= ( nat2 @ ( one_one @ int ) ) ) ).
% nat_one_as_int
thf(fact_491_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_492_bit_Odisj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se1065995026697491101ons_or @ A @ X3 @ ( zero_zero @ A ) )
= X3 ) ) ).
% bit.disj_zero_right
thf(fact_493_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_494_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_495_real__sqrt__less__mono,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ X3 @ Y3 )
=> ( ord_less @ real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).
% real_sqrt_less_mono
thf(fact_496_real__sqrt__le__mono,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ X3 @ Y3 )
=> ( ord_less_eq @ real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).
% real_sqrt_le_mono
thf(fact_497_odd__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% odd_one
thf(fact_498_real__sqrt__power,axiom,
! [X3: real,K: nat] :
( ( sqrt @ ( power_power @ real @ X3 @ K ) )
= ( power_power @ real @ ( sqrt @ X3 ) @ K ) ) ).
% real_sqrt_power
thf(fact_499_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_500_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_501_dvd__power__same,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X3: A,Y3: A,N: nat] :
( ( dvd_dvd @ A @ X3 @ Y3 )
=> ( dvd_dvd @ A @ ( power_power @ A @ X3 @ N ) @ ( power_power @ A @ Y3 @ N ) ) ) ) ).
% dvd_power_same
thf(fact_502_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_503_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_504_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_505_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_506_zero__one__enat__neq_I1_J,axiom,
( ( zero_zero @ extended_enat )
!= ( one_one @ extended_enat ) ) ).
% zero_one_enat_neq(1)
thf(fact_507_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X3: A,M: nat,N: nat] :
( ( X3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X3 @ M ) @ ( power_power @ A @ X3 @ N ) )
= ( ( dvd_dvd @ A @ X3 @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M @ N ) ) ) ) ) ).
% dvd_power_iff
thf(fact_508_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat,X3: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
| ( X3
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X3 @ ( power_power @ A @ X3 @ N ) ) ) ) ).
% dvd_power
thf(fact_509_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_510_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_511_real__sqrt__gt__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ X3 ) ) ) ).
% real_sqrt_gt_zero
thf(fact_512_real__sqrt__ge__zero,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ X3 ) ) ) ).
% real_sqrt_ge_zero
thf(fact_513_real__sqrt__eq__zero__cancel,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ( sqrt @ X3 )
= ( zero_zero @ real ) )
=> ( X3
= ( zero_zero @ real ) ) ) ) ).
% real_sqrt_eq_zero_cancel
thf(fact_514_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_515_OR__lower,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ X3 @ Y3 ) ) ) ) ).
% OR_lower
thf(fact_516_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_517_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_518_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_519_XOR__lower,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ X3 @ Y3 ) ) ) ) ).
% XOR_lower
thf(fact_520_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_521_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_522_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_523_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_524_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_525_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_526_dvd__power__le,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X3: A,Y3: A,N: nat,M: nat] :
( ( dvd_dvd @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ A @ ( power_power @ A @ X3 @ N ) @ ( power_power @ A @ Y3 @ M ) ) ) ) ) ).
% dvd_power_le
thf(fact_527_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_528_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_529_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_530_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_531_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_532_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_533_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_534_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_535_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_536_real__arch__pow,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ? [N2: nat] : ( ord_less @ real @ Y3 @ ( power_power @ real @ X3 @ N2 ) ) ) ).
% real_arch_pow
thf(fact_537_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_538_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_539_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_540_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_541_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_542_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_543_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_544_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_545_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_546_real__arch__pow__inv,axiom,
! [Y3: real,X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ? [N2: nat] : ( ord_less @ real @ ( power_power @ real @ X3 @ N2 ) @ Y3 ) ) ) ).
% real_arch_pow_inv
thf(fact_547_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ Z )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_548_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_549_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_550_sqrt2__less__2,axiom,
ord_less @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% sqrt2_less_2
thf(fact_551_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_552_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_553_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_554_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_555_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_556_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_557_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_558_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_559_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_560_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_561_real__less__rsqrt,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y3 )
=> ( ord_less @ real @ X3 @ ( sqrt @ Y3 ) ) ) ).
% real_less_rsqrt
thf(fact_562_sqrt__le__D,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( sqrt @ X3 ) @ Y3 )
=> ( ord_less_eq @ real @ X3 @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% sqrt_le_D
thf(fact_563_real__le__rsqrt,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y3 )
=> ( ord_less_eq @ real @ X3 @ ( sqrt @ Y3 ) ) ) ).
% real_le_rsqrt
thf(fact_564_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_565_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_566_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_567_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_568_real__le__lsqrt,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less_eq @ real @ X3 @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sqrt @ X3 ) @ Y3 ) ) ) ) ).
% real_le_lsqrt
thf(fact_569_real__sqrt__unique,axiom,
! [Y3: real,X3: real] :
( ( ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( sqrt @ X3 )
= Y3 ) ) ) ).
% real_sqrt_unique
thf(fact_570_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_571_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_572_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_573_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_574_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_575_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_576_arcosh__1,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% arcosh_1
thf(fact_577_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_578_dvd__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ A2 @ ( zero_zero @ A ) ) ) ).
% dvd_0_right
thf(fact_579_artanh__0,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% artanh_0
thf(fact_580_arsinh__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% arsinh_0
thf(fact_581_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_582_log__ceil__idem,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X3 )
=> ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ ( archimedean_ceiling @ real @ X3 ) ) ) ) ) ) ).
% log_ceil_idem
thf(fact_583_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_584_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_585_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_586_ceiling__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int] :
( ( archimedean_ceiling @ A @ ( ring_1_of_int @ A @ Z ) )
= Z ) ) ).
% ceiling_of_int
thf(fact_587_of__int__ceiling__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X3 ) )
= X3 )
= ( ? [N3: int] :
( X3
= ( ring_1_of_int @ A @ N3 ) ) ) ) ) ).
% of_int_ceiling_cancel
thf(fact_588_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_589_ceiling__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% ceiling_zero
thf(fact_590_ceiling__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archimedean_ceiling @ A @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ int @ V2 ) ) ) ).
% ceiling_numeral
thf(fact_591_ceiling__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_592_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_593_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_594_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_595_ceiling__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) ) ) ).
% ceiling_le_zero
thf(fact_596_zero__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) ) ) ).
% zero_less_ceiling
thf(fact_597_ceiling__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% ceiling_le_numeral
thf(fact_598_ceiling__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) ) ) ).
% ceiling_less_one
thf(fact_599_numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( numeral_numeral @ A @ V2 ) @ X3 ) ) ) ).
% numeral_less_ceiling
thf(fact_600_one__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) ) ) ).
% one_le_ceiling
thf(fact_601_ceiling__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X3 @ ( one_one @ A ) ) ) ) ).
% ceiling_le_one
thf(fact_602_one__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( one_one @ A ) @ X3 ) ) ) ).
% one_less_ceiling
thf(fact_603_ceiling__numeral__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: num,N: nat] :
( ( archimedean_ceiling @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) ) ) ).
% ceiling_numeral_power
thf(fact_604_nat__ceiling__le__eq,axiom,
! [X3: real,A2: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X3 ) ) @ A2 )
= ( ord_less_eq @ real @ X3 @ ( semiring_1_of_nat @ real @ A2 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_605_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_606_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_607_ceiling__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y3 ) @ ( archimedean_ceiling @ A @ X3 ) ) ) ) ).
% ceiling_mono
thf(fact_608_le__of__int__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X3 ) ) ) ) ).
% le_of_int_ceiling
thf(fact_609_ceiling__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( archimedean_ceiling @ A @ Y3 ) )
=> ( ord_less @ A @ X3 @ Y3 ) ) ) ).
% ceiling_less_cancel
thf(fact_610_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_611_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_612_real__nat__ceiling__ge,axiom,
! [X3: real] : ( ord_less_eq @ real @ X3 @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ X3 ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_613_of__nat__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R2: A] : ( ord_less_eq @ A @ R2 @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archimedean_ceiling @ A @ R2 ) ) ) ) ) ).
% of_nat_ceiling
thf(fact_614_ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,A2: int] :
( ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ A2 ) )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ A2 ) ) ) ).
% ceiling_le
thf(fact_615_ceiling__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z: int] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ Z )
= ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% ceiling_le_iff
thf(fact_616_less__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X3: A] :
( ( ord_less @ int @ Z @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ X3 ) ) ) ).
% less_ceiling_iff
thf(fact_617_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_618_zdvd__imp__le,axiom,
! [Z: int,N: int] :
( ( dvd_dvd @ int @ Z @ N )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ Z @ N ) ) ) ).
% zdvd_imp_le
thf(fact_619_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y3: A] :
( ( X3 != Y3 )
=> ( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_620_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_621_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_622_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_623_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_624_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_625_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_626_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_627_nat__dvd__iff,axiom,
! [Z: int,M: nat] :
( ( dvd_dvd @ nat @ ( nat2 @ Z ) @ M )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( dvd_dvd @ int @ Z @ ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_dvd_iff
thf(fact_628_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_629_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_630_gcd__nat_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_631_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_632_gcd__nat_Oextremum__unique,axiom,
! [A2: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_unique
thf(fact_633_gcd__nat_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
& ( ( zero_zero @ nat )
!= A2 ) ) ).
% gcd_nat.extremum_strict
thf(fact_634_gcd__nat_Oextremum,axiom,
! [A2: nat] : ( dvd_dvd @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% gcd_nat.extremum
thf(fact_635_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_636_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_637_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_638_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_639_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_640_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_641_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_642_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_643_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_644_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_645_ln__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_646_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_647_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_648_real__sqrt__power__even,axiom,
! [N: nat,X3: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( power_power @ real @ ( sqrt @ X3 ) @ N )
= ( power_power @ real @ X3 @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_power_even
thf(fact_649_arsinh__real__aux,axiom,
! [X3: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X3 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% arsinh_real_aux
thf(fact_650_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% numeral_le_ceiling
thf(fact_651_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_652_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_653_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_654_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_655_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_656_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= N ) ).
% idiff_0_right
thf(fact_657_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( zero_zero @ extended_enat ) ) ).
% idiff_0
thf(fact_658_power__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 ) ) ) ) ).
% power_minus_is_div
thf(fact_659_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_660_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_661_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_662_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add_0
thf(fact_663_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X3: A,Y3: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X3 @ Y3 ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_664_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X3: A,Y3: A] :
( ( ( plus_plus @ A @ X3 @ Y3 )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_665_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_666_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_667_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_668_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_669_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_670_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_671_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_672_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_673_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_674_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_675_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_676_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_677_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_678_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_679_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_680_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_681_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_682_add__numeral__left,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [V2: num,W: num,Z: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Z ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W ) ) @ Z ) ) ) ).
% add_numeral_left
thf(fact_683_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_684_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_685_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_686_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_687_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_688_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_689_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_690_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_691_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_692_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_693_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_694_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_695_ln__inj__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ( ln_ln @ real @ X3 )
= ( ln_ln @ real @ Y3 ) )
= ( X3 = Y3 ) ) ) ) ).
% ln_inj_iff
thf(fact_696_ln__less__cancel__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y3 ) )
= ( ord_less @ real @ X3 @ Y3 ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_697_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_698_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_699_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_700_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_701_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_702_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_703_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_704_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_705_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_706_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_707_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_708_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_709_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_710_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_711_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_712_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_713_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_714_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_715_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_716_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_717_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_718_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_719_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_720_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_721_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_722_ln__le__cancel__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y3 ) )
= ( ord_less_eq @ real @ X3 @ Y3 ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_723_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_724_ln__less__zero__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X3 @ ( one_one @ real ) ) ) ) ).
% ln_less_zero_iff
thf(fact_725_ln__gt__zero__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) )
= ( ord_less @ real @ ( one_one @ real ) @ X3 ) ) ) ).
% ln_gt_zero_iff
thf(fact_726_ln__eq__zero__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ( ln_ln @ real @ X3 )
= ( zero_zero @ real ) )
= ( X3
= ( one_one @ real ) ) ) ) ).
% ln_eq_zero_iff
thf(fact_727_of__int__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z: int] :
( ( ring_1_of_int @ A @ ( plus_plus @ int @ W @ Z ) )
= ( plus_plus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_add
thf(fact_728_of__int__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z: int] :
( ( ring_1_of_int @ A @ ( minus_minus @ int @ W @ Z ) )
= ( minus_minus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_diff
thf(fact_729_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_730_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_731_ln__le__zero__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X3 @ ( one_one @ real ) ) ) ) ).
% ln_le_zero_iff
thf(fact_732_ln__ge__zero__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X3 ) ) ) ).
% ln_ge_zero_iff
thf(fact_733_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_734_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less @ int @ W @ ( plus_plus @ int @ Z @ ( one_one @ int ) ) )
= ( ord_less_eq @ int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_735_ceiling__add__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z: int] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ ( ring_1_of_int @ A @ Z ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ Z ) ) ) ).
% ceiling_add_of_int
thf(fact_736_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq @ int @ W @ ( minus_minus @ int @ Z @ ( one_one @ int ) ) )
= ( ord_less @ int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_737_ceiling__diff__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z: int] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X3 @ ( ring_1_of_int @ A @ Z ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X3 ) @ Z ) ) ) ).
% ceiling_diff_of_int
thf(fact_738_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_739_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_740_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_741_ceiling__add__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_add_numeral
thf(fact_742_ceiling__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( one_one @ int ) ) ) ) ).
% ceiling_add_one
thf(fact_743_ceiling__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_diff_numeral
thf(fact_744_ceiling__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X3 @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( one_one @ int ) ) ) ) ).
% ceiling_diff_one
thf(fact_745_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_746_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_747_sum__power2__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y3: A] :
( ( ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_748_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_749_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_750_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_751_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_752_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_753_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_754_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X3 @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_755_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_756_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_757_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_758_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_759_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_760_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_761_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_762_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_763_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_764_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_765_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_766_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_767_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_768_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_769_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_770_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_771_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_772_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_773_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_774_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_775_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: A,K: A,A2: A,B2: A] :
( ( A4
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A4 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_776_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B6: A,K: A,B2: A,A2: A] :
( ( B6
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B6 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_777_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,K: A,A2: A,B2: A] :
( ( A4
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( minus_minus @ A @ A4 @ B2 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_778_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_779_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_780_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_781_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_782_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_783_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_784_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_785_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( A2 = B2 )
= ( C2 = D3 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_786_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_787_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ B4 @ A5 ) ) ) ) ).
% add.commute
thf(fact_788_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_789_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_790_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_791_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_792_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_793_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_794_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_795_add__diff__add,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,C2: A,B2: A,D3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) )
= ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( minus_minus @ A @ C2 @ D3 ) ) ) ) ).
% add_diff_add
thf(fact_796_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_797_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_798_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_799_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_800_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_801_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_802_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_803_ceiling__add__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ Y3 ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( archimedean_ceiling @ A @ Y3 ) ) ) ) ).
% ceiling_add_le
thf(fact_804_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
= ( ord_less_eq @ A @ C2 @ D3 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_805_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_806_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_807_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D3: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ D3 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_mono
thf(fact_808_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [A5: A,B4: A] :
( ( minus_minus @ A @ A5 @ B4 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_809_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D3: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ D3 @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_810_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
= ( ord_less @ A @ C2 @ D3 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_811_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_812_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_813_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_814_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_815_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
? [C4: A] :
( B4
= ( plus_plus @ A @ A5 @ C4 ) ) ) ) ) ).
% le_iff_add
thf(fact_816_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_817_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ! [C3: A] :
( B2
!= ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).
% less_eqE
thf(fact_818_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_819_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_mono
thf(fact_820_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_821_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_822_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_823_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% verit_sum_simplify
thf(fact_824_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_825_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_826_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_827_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_828_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_829_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_830_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_strict_mono
thf(fact_831_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_832_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_833_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_834_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_835_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_836_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_837_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_838_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_839_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_840_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% minus_int_code(1)
thf(fact_841_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_842_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_843_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_844_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_845_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_846_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_847_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_848_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_849_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_850_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% plus_int_code(1)
thf(fact_851_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus @ int @ ( zero_zero @ int ) @ L )
= L ) ).
% plus_int_code(2)
thf(fact_852_int__diff__cases,axiom,
! [Z: int] :
~ ! [M4: nat,N2: nat] :
( Z
!= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M4 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_853_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_854_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_855_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_856_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_857_ln__eq__minus__one,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ( ln_ln @ real @ X3 )
= ( minus_minus @ real @ X3 @ ( one_one @ real ) ) )
=> ( X3
= ( one_one @ real ) ) ) ) ).
% ln_eq_minus_one
thf(fact_858_ln__add__one__self__le__self,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) ) @ X3 ) ) ).
% ln_add_one_self_le_self
thf(fact_859_artanh__def,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A )
= ( ^ [X: A] : ( divide_divide @ A @ ( ln_ln @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% artanh_def
thf(fact_860_field__sum__of__halves,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ A @ X3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= X3 ) ) ).
% field_sum_of_halves
thf(fact_861_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_862_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_863_nat__diff__distrib,axiom,
! [Z5: int,Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z5 )
=> ( ( ord_less_eq @ int @ Z5 @ Z )
=> ( ( nat2 @ ( minus_minus @ int @ Z @ Z5 ) )
= ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_864_nat__diff__distrib_H,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( nat2 @ ( minus_minus @ int @ X3 @ Y3 ) )
= ( minus_minus @ nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_865_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_866_dbl__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A )
= ( ^ [X: A] : ( plus_plus @ A @ X @ X ) ) ) ) ).
% dbl_def
thf(fact_867_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_868_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_869_ln__le__minus__one,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ ( minus_minus @ real @ X3 @ ( one_one @ real ) ) ) ) ).
% ln_le_minus_one
thf(fact_870_field__less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ A @ X3 @ ( divide_divide @ A @ ( plus_plus @ A @ X3 @ Y3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% field_less_half_sum
thf(fact_871_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_872_ln__less__self,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less @ real @ ( ln_ln @ real @ X3 ) @ X3 ) ) ).
% ln_less_self
thf(fact_873_zdiff__int__split,axiom,
! [P: int > $o,X3: nat,Y3: nat] :
( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X3 @ Y3 ) ) )
= ( ( ( ord_less_eq @ nat @ Y3 @ X3 )
=> ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X3 ) @ ( semiring_1_of_nat @ int @ Y3 ) ) ) )
& ( ( ord_less @ nat @ X3 @ Y3 )
=> ( P @ ( zero_zero @ int ) ) ) ) ) ).
% zdiff_int_split
thf(fact_874_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_875_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A5 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_876_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] : ( ord_less @ A @ ( minus_minus @ A @ A5 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_877_divide__numeral__1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ one2 ) )
= A2 ) ) ).
% divide_numeral_1
thf(fact_878_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_879_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_880_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_881_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_882_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_883_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_884_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ( plus_plus @ A @ X3 @ Y3 )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_885_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X3 @ Y3 )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_886_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_887_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_888_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_889_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_890_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_891_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_892_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ! [C3: A] :
( ( B2
= ( plus_plus @ A @ A2 @ C3 ) )
=> ( C3
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_893_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_894_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X3 @ Y3 ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X3 @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y3 @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_895_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_896_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_897_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_898_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_899_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_900_one__plus__numeral__commute,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X3: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X3 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X3 ) @ ( one_one @ A ) ) ) ) ).
% one_plus_numeral_commute
thf(fact_901_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_902_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X6: B] :
( ( ord_less @ B @ Z3 @ X6 )
=> ( ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X6 @ S ) )
= ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X6 @ S ) ) ) ) ) ).
% pinf(9)
thf(fact_903_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X6: B] :
( ( ord_less @ B @ Z3 @ X6 )
=> ( ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X6 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X6 @ S ) ) ) ) ) ) ).
% pinf(10)
thf(fact_904_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X6: B] :
( ( ord_less @ B @ X6 @ Z3 )
=> ( ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X6 @ S ) )
= ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X6 @ S ) ) ) ) ) ).
% minf(9)
thf(fact_905_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X6: B] :
( ( ord_less @ B @ X6 @ Z3 )
=> ( ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X6 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X6 @ S ) ) ) ) ) ) ).
% minf(10)
thf(fact_906_ln__one__plus__pos__lower__bound,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ X3 @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) ) ) ) ) ).
% ln_one_plus_pos_lower_bound
thf(fact_907_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_908_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_909_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_910_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_911_arcosh__real__def,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X3 )
=> ( ( arcosh @ real @ X3 )
= ( ln_ln @ real @ ( plus_plus @ real @ X3 @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arcosh_real_def
thf(fact_912_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_913_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq @ int @ I @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_914_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less @ int @ I @ K )
=> ( ( P @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I3: int] :
( ( ord_less @ int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_915_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_916_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_917_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_918_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) @ Z )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_919_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq @ int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_920_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less @ int @ W @ ( plus_plus @ int @ Z @ ( one_one @ int ) ) )
= ( ( ord_less @ int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_921_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less @ int @ K @ I )
=> ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I3: int] :
( ( ord_less @ int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_922_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W3: int,Z6: int] :
? [N3: nat] :
( Z6
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_923_ln__bound,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ X3 ) ) ).
% ln_bound
thf(fact_924_ln__gt__zero__imp__gt__one,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less @ real @ ( one_one @ real ) @ X3 ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_925_ln__less__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( ord_less @ real @ ( ln_ln @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ).
% ln_less_zero
thf(fact_926_ln__gt__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) ) ) ).
% ln_gt_zero
thf(fact_927_ln__ge__zero,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) ) ) ).
% ln_ge_zero
thf(fact_928_arsinh__real__def,axiom,
( ( arsinh @ real )
= ( ^ [X: real] : ( ln_ln @ real @ ( plus_plus @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arsinh_real_def
thf(fact_929_ceiling__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: int,X3: A] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ N ) @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ N ) @ ( one_one @ A ) ) )
=> ( ( archimedean_ceiling @ A @ X3 )
= ( plus_plus @ int @ N @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_eq
thf(fact_930_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_931_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_932_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_933_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_934_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_935_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_936_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_937_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_938_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_939_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_940_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_941_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_942_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_943_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_944_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) @ Z ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ Z @ ( zero_zero @ int ) ) ) ).
% odd_less_0_iff
thf(fact_945_sqrt__add__le__add__sqrt,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ X3 @ Y3 ) ) @ ( plus_plus @ real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ) ) ).
% sqrt_add_le_add_sqrt
thf(fact_946_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less @ int @ W @ Z )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ W @ ( one_one @ int ) ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_947_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ W @ ( one_one @ int ) ) @ Z )
= ( ord_less @ int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_948_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_949_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_950_ln__ge__zero__imp__ge__one,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ X3 ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_951_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_952_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_953_bit__eq__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [A5: A,B4: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B4 ) )
& ( ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ B4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_954_power2__commute,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X3: A,Y3: A] :
( ( power_power @ A @ ( minus_minus @ A @ X3 @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ ( minus_minus @ A @ Y3 @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_commute
thf(fact_955_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_956_of__int__ceiling__diff__one__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R2 ) ) @ ( one_one @ A ) ) @ R2 ) ) ).
% of_int_ceiling_diff_one_le
thf(fact_957_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [X4: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X4 ) @ X3 )
& ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X4 @ ( one_one @ int ) ) ) )
& ! [Y5: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y5 ) @ X3 )
& ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y5 @ ( one_one @ int ) ) ) ) )
=> ( Y5 = X4 ) ) ) ) ).
% floor_exists1
thf(fact_958_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X3 )
& ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_959_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_960_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_961_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_962_of__int__ceiling__le__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R2: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R2 ) ) @ ( plus_plus @ A @ R2 @ ( one_one @ A ) ) ) ) ).
% of_int_ceiling_le_add_one
thf(fact_963_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_964_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_965_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) ) ) ).
% le_imp_0_less
thf(fact_966_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_967_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_968_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_969_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_970_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_971_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_972_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_973_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,A2: int] :
( ( ( archimedean_ceiling @ A @ X3 )
= A2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) @ X3 )
& ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ A2 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_974_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X3: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ Z ) )
=> ( ( archimedean_ceiling @ A @ X3 )
= Z ) ) ) ) ).
% ceiling_unique
thf(fact_975_ceiling__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X3 ) ) @ ( one_one @ A ) ) @ X3 )
& ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X3 ) ) ) ) ) ).
% ceiling_correct
thf(fact_976_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_977_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ Z )
= ( ord_less_eq @ A @ X3 @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_978_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X3: A] :
( ( ord_less_eq @ int @ Z @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% le_ceiling_iff
thf(fact_979_sum__power2__ge__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_980_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_981_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y3: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_power2_lt_zero
thf(fact_982_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( ( X3
!= ( zero_zero @ A ) )
| ( Y3
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_983_real__sqrt__sum__squares__eq__cancel,axiom,
! [X3: real,Y3: real] :
( ( ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= X3 )
=> ( Y3
= ( zero_zero @ real ) ) ) ).
% real_sqrt_sum_squares_eq_cancel
thf(fact_984_real__sqrt__sum__squares__eq__cancel2,axiom,
! [X3: real,Y3: real] :
( ( ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= Y3 )
=> ( X3
= ( zero_zero @ real ) ) ) ).
% real_sqrt_sum_squares_eq_cancel2
thf(fact_985_real__sqrt__sum__squares__ge1,axiom,
! [X3: real,Y3: real] : ( ord_less_eq @ real @ X3 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge1
thf(fact_986_real__sqrt__sum__squares__ge2,axiom,
! [Y3: real,X3: real] : ( ord_less_eq @ real @ Y3 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge2
thf(fact_987_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A2: real,C2: real,B2: real,D3: 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 @ D3 ) @ ( 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 @ D3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_triangle_ineq
thf(fact_988_sqrt__sum__squares__le__sum,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ X3 @ Y3 ) ) ) ) ).
% sqrt_sum_squares_le_sum
thf(fact_989_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_990_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_991_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_992_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_993_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_994_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_995_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_996_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_997_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_998_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_999_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_1000_pow__sum,axiom,
! [A2: nat,B2: nat] :
( ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ A2 @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) ) ).
% pow_sum
thf(fact_1001_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_1002_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_1003_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_1004_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_1005_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_1006_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_1007_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_1008_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_1009_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_1010_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_1011_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_1012_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_1013_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_1014_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_1015_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_1016_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_1017_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_1018_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_1019_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_1020_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_1021_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_1022_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_1023_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_1024_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_1025_semiring__norm_I2_J,axiom,
( ( plus_plus @ num @ one2 @ one2 )
= ( bit0 @ one2 ) ) ).
% semiring_norm(2)
thf(fact_1026_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_1027_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_1028_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_1029_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_1030_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_1031_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_1032_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_1033_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_1034_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_1035_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1036_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1037_real__of__int__div4,axiom,
! [N: int,X3: int] : ( ord_less_eq @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X3 ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X3 ) ) ) ).
% real_of_int_div4
thf(fact_1038_real__of__int__div,axiom,
! [D3: int,N: int] :
( ( dvd_dvd @ int @ D3 @ N )
=> ( ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ D3 ) )
= ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ D3 ) ) ) ) ).
% real_of_int_div
thf(fact_1039_add__One__commute,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ N )
= ( plus_plus @ num @ N @ one2 ) ) ).
% add_One_commute
thf(fact_1040_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A6: nat,B5: nat] :
( ( P @ A6 @ B5 )
= ( P @ B5 @ A6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ ( zero_zero @ nat ) )
=> ( ! [A6: nat,B5: nat] :
( ( P @ A6 @ B5 )
=> ( P @ A6 @ ( plus_plus @ nat @ A6 @ B5 ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_1041_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1042_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_1043_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_1044_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_1045_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_1046_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_1047_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_1048_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1049_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1050_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_1051_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_1052_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_1053_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_1054_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_1055_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_1056_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_1057_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_1058_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_1059_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_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_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_1068_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_1069_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_1070_real__sqrt__divide,axiom,
! [X3: real,Y3: real] :
( ( sqrt @ ( divide_divide @ real @ X3 @ Y3 ) )
= ( divide_divide @ real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).
% real_sqrt_divide
thf(fact_1071_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_1072_add__diff__assoc__enat,axiom,
! [Z: extended_enat,Y3: extended_enat,X3: extended_enat] :
( ( ord_less_eq @ extended_enat @ Z @ Y3 )
=> ( ( plus_plus @ extended_enat @ X3 @ ( minus_minus @ extended_enat @ Y3 @ Z ) )
= ( minus_minus @ extended_enat @ ( plus_plus @ extended_enat @ X3 @ Y3 ) @ Z ) ) ) ).
% add_diff_assoc_enat
thf(fact_1073_real__of__int__div2,axiom,
! [N: int,X3: 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 @ X3 ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X3 ) ) ) ) ).
% real_of_int_div2
thf(fact_1074_real__of__int__div3,axiom,
! [N: int,X3: int] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X3 ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X3 ) ) ) @ ( one_one @ real ) ) ).
% real_of_int_div3
thf(fact_1075_ln__div,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ln_ln @ real @ ( divide_divide @ real @ X3 @ Y3 ) )
= ( minus_minus @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y3 ) ) ) ) ) ).
% ln_div
thf(fact_1076_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_1077_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 ) ) )
| ? [D4: nat] :
( ( A2
= ( plus_plus @ nat @ B2 @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1078_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 ) ) )
& ! [D4: nat] :
( ( A2
= ( plus_plus @ nat @ B2 @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1079_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_1080_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,N: nat] :
( ( P @ K )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ? [Y5: A] :
( ( P @ Y5 )
& ~ ( ord_less_eq @ nat @ ( F2 @ Y5 ) @ ( F2 @ X4 ) ) ) )
=> ? [Y4: A] :
( ( P @ Y4 )
& ~ ( ord_less @ nat @ ( F2 @ Y4 ) @ ( plus_plus @ nat @ ( F2 @ K ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_1081_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_1082_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_1083_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_1084_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ Z ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_1085_ln__diff__le,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y3 ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ X3 @ Y3 ) @ Y3 ) ) ) ) ).
% ln_diff_le
thf(fact_1086_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X6: A] :
? [X_1: A] : ( ord_less @ A @ X6 @ X_1 ) ) ).
% linordered_field_no_ub
thf(fact_1087_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X6: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X6 ) ) ).
% linordered_field_no_lb
thf(fact_1088_real__div__sqrt,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( divide_divide @ real @ X3 @ ( sqrt @ X3 ) )
= ( sqrt @ X3 ) ) ) ).
% real_div_sqrt
thf(fact_1089_real__of__nat__div4,axiom,
! [N: nat,X3: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X3 ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X3 ) ) ) ).
% real_of_nat_div4
thf(fact_1090_real__of__nat__div,axiom,
! [D3: nat,N: nat] :
( ( dvd_dvd @ nat @ D3 @ N )
=> ( ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ D3 ) )
= ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ D3 ) ) ) ) ).
% real_of_nat_div
thf(fact_1091_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_1092_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_1093_real__of__nat__div2,axiom,
! [N: nat,X3: 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 @ X3 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X3 ) ) ) ) ).
% real_of_nat_div2
thf(fact_1094_nat__div__distrib_H,axiom,
! [Y3: int,X3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( nat2 @ ( divide_divide @ int @ X3 @ Y3 ) )
= ( divide_divide @ nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ).
% nat_div_distrib'
thf(fact_1095_nat__div__distrib,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( ( nat2 @ ( divide_divide @ int @ X3 @ Y3 ) )
= ( divide_divide @ nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ).
% nat_div_distrib
thf(fact_1096_real__of__nat__div3,axiom,
! [N: nat,X3: nat] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X3 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X3 ) ) ) @ ( one_one @ real ) ) ).
% real_of_nat_div3
thf(fact_1097_log__base__change,axiom,
! [A2: real,B2: real,X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ B2 @ X3 )
= ( divide_divide @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ B2 ) ) ) ) ) ).
% log_base_change
thf(fact_1098_log__divide,axiom,
! [A2: real,X3: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( log @ A2 @ ( divide_divide @ real @ X3 @ Y3 ) )
= ( minus_minus @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ Y3 ) ) ) ) ) ) ) ).
% log_divide
thf(fact_1099_nat__add__distrib,axiom,
! [Z: int,Z5: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z5 )
=> ( ( nat2 @ ( plus_plus @ int @ Z @ Z5 ) )
= ( plus_plus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1100_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_1101_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_1102_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_1103_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_1104_log__base__pow,axiom,
! [A2: real,N: nat,X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( log @ ( power_power @ real @ A2 @ N ) @ X3 )
= ( divide_divide @ real @ ( log @ A2 @ X3 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log_base_pow
thf(fact_1105_lemma__real__divide__sqrt__less,axiom,
! [U: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ U )
=> ( ord_less @ real @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ U ) ) ).
% lemma_real_divide_sqrt_less
thf(fact_1106_ln__sqrt,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ln_ln @ real @ ( sqrt @ X3 ) )
= ( divide_divide @ real @ ( ln_ln @ real @ X3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% ln_sqrt
thf(fact_1107_dvd__field__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A5: A,B4: A] :
( ( A5
= ( zero_zero @ A ) )
=> ( B4
= ( zero_zero @ A ) ) ) ) ) ) ).
% dvd_field_iff
thf(fact_1108_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_1109_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_1110_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_1111_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y3 ) ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1112_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1113_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1114_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y3 ) ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1115_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_1116_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_1117_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_1118_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_1119_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_1120_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_1121_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_1122_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_1123_divide__pos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y3 ) ) ) ) ) ).
% divide_pos_pos
thf(fact_1124_divide__pos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_pos_neg
thf(fact_1125_divide__neg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_neg_pos
thf(fact_1126_divide__neg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y3 ) ) ) ) ) ).
% divide_neg_neg
thf(fact_1127_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_1128_sqrt__sum__squares__half__less,axiom,
! [X3: real,U: real,Y3: real] :
( ( ord_less @ real @ X3 @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y3 @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ) ) ).
% sqrt_sum_squares_half_less
thf(fact_1129_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_1130_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A] :
( ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ( ord_less_eq @ A @ X3 @ ( plus_plus @ A @ Y3 @ E ) ) )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% field_le_epsilon
thf(fact_1131_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,X3: A,W: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Z ) @ ( divide_divide @ A @ Y3 @ W ) ) ) ) ) ) ) ).
% frac_le
thf(fact_1132_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A,W: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less @ A @ X3 @ Y3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X3 @ Z ) @ ( divide_divide @ A @ Y3 @ W ) ) ) ) ) ) ) ).
% frac_less
thf(fact_1133_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A,W: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less @ A @ W @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X3 @ Z ) @ ( divide_divide @ A @ Y3 @ W ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_1134_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_1135_divide__nonneg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_neg
thf(fact_1136_divide__nonneg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y3 ) ) ) ) ) ).
% divide_nonneg_pos
thf(fact_1137_divide__nonpos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y3 ) ) ) ) ) ).
% divide_nonpos_neg
thf(fact_1138_divide__nonpos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_pos
thf(fact_1139_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_1140_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_1141_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_1142_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_1143_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_1144_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_1145_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_1146_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_1147_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_1148_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_1149_add__def,axiom,
( vEBT_VEBT_add
= ( vEBT_V2048590022279873568_shift @ nat @ ( plus_plus @ nat ) ) ) ).
% add_def
thf(fact_1150_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_1151_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_1152__C5_Ohyps_C_I4_J,axiom,
( deg
= ( plus_plus @ nat @ na @ m ) ) ).
% "5.hyps"(4)
thf(fact_1153_div2__even__ext__nat,axiom,
! [X3: nat,Y3: nat] :
( ( ( divide_divide @ nat @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X3 )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Y3 ) )
=> ( X3 = Y3 ) ) ) ).
% div2_even_ext_nat
thf(fact_1154_zdiv__numeral__Bit0,axiom,
! [V2: num,W: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit0 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_1155_add__shift,axiom,
! [X3: nat,Y3: nat,Z: nat] :
( ( ( plus_plus @ nat @ X3 @ Y3 )
= Z )
= ( ( vEBT_VEBT_add @ ( some @ nat @ X3 ) @ ( some @ nat @ Y3 ) )
= ( some @ nat @ Z ) ) ) ).
% add_shift
thf(fact_1156__C5_Ohyps_C_I3_J,axiom,
( m
= ( suc @ na ) ) ).
% "5.hyps"(3)
thf(fact_1157_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_1158_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_1159_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_1160_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_1161_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_1162_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_1163_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_1164_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_1165_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_1166_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_1167_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_1168_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_1169_zdiv__mono2__neg,axiom,
! [A2: int,B3: int,B2: int] :
( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ B3 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B3 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1170_zdiv__mono1__neg,axiom,
! [A2: int,A3: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ A3 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1171_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_1172_zdiv__mono2,axiom,
! [A2: int,B3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ B3 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A2 @ B3 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1173_zdiv__mono1,axiom,
! [A2: int,A3: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A3 @ B2 ) ) ) ) ).
% zdiv_mono1
thf(fact_1174_int__div__less__self,axiom,
! [X3: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X3 )
=> ( ( ord_less @ int @ ( one_one @ int ) @ K )
=> ( ord_less @ int @ ( divide_divide @ int @ X3 @ K ) @ X3 ) ) ) ).
% int_div_less_self
thf(fact_1175_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_1176_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_1177_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_1178_Bolzano,axiom,
! [A2: real,B2: real,P: real > real > $o] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [A6: real,B5: real,C3: real] :
( ( P @ A6 @ B5 )
=> ( ( P @ B5 @ C3 )
=> ( ( ord_less_eq @ real @ A6 @ B5 )
=> ( ( ord_less_eq @ real @ B5 @ C3 )
=> ( P @ A6 @ C3 ) ) ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq @ real @ A2 @ X4 )
=> ( ( ord_less_eq @ real @ X4 @ B2 )
=> ? [D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
& ! [A6: real,B5: real] :
( ( ( ord_less_eq @ real @ A6 @ X4 )
& ( ord_less_eq @ real @ X4 @ B5 )
& ( ord_less @ real @ ( minus_minus @ real @ B5 @ A6 ) @ D2 ) )
=> ( P @ A6 @ B5 ) ) ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Bolzano
thf(fact_1179_round__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y3: int] :
( ( ord_less @ A @ ( minus_minus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y3 ) )
=> ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y3 ) @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
=> ( ( archimedean_round @ A @ X3 )
= Y3 ) ) ) ) ).
% round_unique
thf(fact_1180_high__bound__aux,axiom,
! [Ma: nat,N: nat,M: nat] :
( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% high_bound_aux
thf(fact_1181_tanh__ln__real,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( tanh @ real @ ( ln_ln @ real @ X3 ) )
= ( divide_divide @ real @ ( minus_minus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% tanh_ln_real
thf(fact_1182_ceiling__log__eq__powr__iff,axiom,
! [X3: real,B2: real,K: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archimedean_ceiling @ real @ ( log @ B2 @ X3 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) ) )
= ( ( ord_less @ real @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ K ) ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% ceiling_log_eq_powr_iff
thf(fact_1183__C5_Ohyps_C_I2_J,axiom,
( ( size_size @ ( list @ vEBT_VEBT ) @ treeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) ) ).
% "5.hyps"(2)
thf(fact_1184_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_1185_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X: nat,N3: nat] : ( divide_divide @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% high_def
thf(fact_1186_even__odd__cases,axiom,
! [X3: nat] :
( ! [N2: nat] :
( X3
!= ( plus_plus @ nat @ N2 @ N2 ) )
=> ~ ! [N2: nat] :
( X3
!= ( plus_plus @ nat @ N2 @ ( suc @ N2 ) ) ) ) ).
% even_odd_cases
thf(fact_1187_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1188_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_1189_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_1190_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1191_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1192_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_1193_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_1194_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_1195_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_1196_powr__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [Z: A] :
( ( powr @ A @ ( zero_zero @ A ) @ Z )
= ( zero_zero @ A ) ) ) ).
% powr_0
thf(fact_1197_powr__eq__0__iff,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [W: A,Z: A] :
( ( ( powr @ A @ W @ Z )
= ( zero_zero @ A ) )
= ( W
= ( zero_zero @ A ) ) ) ) ).
% powr_eq_0_iff
thf(fact_1198_floor__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int] :
( ( archim6421214686448440834_floor @ A @ ( ring_1_of_int @ A @ Z ) )
= Z ) ) ).
% floor_of_int
thf(fact_1199_of__int__floor__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) )
= X3 )
= ( ? [N3: int] :
( X3
= ( ring_1_of_int @ A @ N3 ) ) ) ) ) ).
% of_int_floor_cancel
thf(fact_1200_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_1201_tanh__real__zero__iff,axiom,
! [X3: real] :
( ( ( tanh @ real @ X3 )
= ( zero_zero @ real ) )
= ( X3
= ( zero_zero @ real ) ) ) ).
% tanh_real_zero_iff
thf(fact_1202_tanh__real__less__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( tanh @ real @ X3 ) @ ( tanh @ real @ Y3 ) )
= ( ord_less @ real @ X3 @ Y3 ) ) ).
% tanh_real_less_iff
thf(fact_1203_tanh__real__le__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( tanh @ real @ X3 ) @ ( tanh @ real @ Y3 ) )
= ( ord_less_eq @ real @ X3 @ Y3 ) ) ).
% tanh_real_le_iff
thf(fact_1204_round__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: int] :
( ( archimedean_round @ A @ ( ring_1_of_int @ A @ N ) )
= N ) ) ).
% round_of_int
thf(fact_1205_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_1206_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_1207_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1208_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_1209_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= M ) ).
% div_by_Suc_0
thf(fact_1210_power__Suc__0,axiom,
! [N: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_1211_nat__power__eq__Suc__0__iff,axiom,
! [X3: nat,M: nat] :
( ( ( power_power @ nat @ X3 @ M )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( X3
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1212_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% dvd_1_left
thf(fact_1213_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_1214_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_1215_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_1216_powr__zero__eq__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [X3: A] :
( ( ( X3
= ( zero_zero @ A ) )
=> ( ( powr @ A @ X3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) )
& ( ( X3
!= ( zero_zero @ A ) )
=> ( ( powr @ A @ X3 @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% powr_zero_eq_one
thf(fact_1217_floor__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% floor_zero
thf(fact_1218_floor__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archim6421214686448440834_floor @ A @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ int @ V2 ) ) ) ).
% floor_numeral
thf(fact_1219_floor__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_1220_powr__gt__zero,axiom,
! [X3: real,A2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( powr @ real @ X3 @ A2 ) )
= ( X3
!= ( zero_zero @ real ) ) ) ).
% powr_gt_zero
thf(fact_1221_powr__nonneg__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less_eq @ real @ ( powr @ real @ A2 @ X3 ) @ ( zero_zero @ real ) )
= ( A2
= ( zero_zero @ real ) ) ) ).
% powr_nonneg_iff
thf(fact_1222_powr__less__cancel__iff,axiom,
! [X3: real,A2: real,B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( ( ord_less @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ X3 @ B2 ) )
= ( ord_less @ real @ A2 @ B2 ) ) ) ).
% powr_less_cancel_iff
thf(fact_1223_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_1224_round__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% round_0
thf(fact_1225_tanh__real__pos__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X3 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ).
% tanh_real_pos_iff
thf(fact_1226_tanh__real__neg__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( tanh @ real @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ).
% tanh_real_neg_iff
thf(fact_1227_tanh__real__nonpos__iff,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( tanh @ real @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) ) ) ).
% tanh_real_nonpos_iff
thf(fact_1228_tanh__real__nonneg__iff,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X3 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) ) ).
% tanh_real_nonneg_iff
thf(fact_1229_round__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: num] :
( ( archimedean_round @ A @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ int @ N ) ) ) ).
% round_numeral
thf(fact_1230_round__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% round_1
thf(fact_1231_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_1232_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_1233_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_1234_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_1235_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_1236_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_1237_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% Suc_numeral
thf(fact_1238_powr__eq__one__iff,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ( powr @ real @ A2 @ X3 )
= ( one_one @ real ) )
= ( X3
= ( zero_zero @ real ) ) ) ) ).
% powr_eq_one_iff
thf(fact_1239_powr__one,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( powr @ real @ X3 @ ( one_one @ real ) )
= X3 ) ) ).
% powr_one
thf(fact_1240_powr__one__gt__zero__iff,axiom,
! [X3: real] :
( ( ( powr @ real @ X3 @ ( one_one @ real ) )
= X3 )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) ) ).
% powr_one_gt_zero_iff
thf(fact_1241_powr__le__cancel__iff,axiom,
! [X3: real,A2: real,B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ X3 @ B2 ) )
= ( ord_less_eq @ real @ A2 @ B2 ) ) ) ).
% powr_le_cancel_iff
thf(fact_1242_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_1243_floor__diff__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z: int] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X3 @ ( ring_1_of_int @ A @ Z ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ Z ) ) ) ).
% floor_diff_of_int
thf(fact_1244_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_1245_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_1246_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_1247_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_1248_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_1249_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_1250_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_1251_zero__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) ) ).
% zero_le_floor
thf(fact_1252_floor__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X3 @ ( zero_zero @ A ) ) ) ) ).
% floor_less_zero
thf(fact_1253_numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ V2 ) @ X3 ) ) ) ).
% numeral_le_floor
thf(fact_1254_zero__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ).
% zero_less_floor
thf(fact_1255_floor__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ).
% floor_le_zero
thf(fact_1256_floor__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% floor_less_numeral
thf(fact_1257_one__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ).
% one_le_floor
thf(fact_1258_floor__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) )
= ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ).
% floor_less_one
thf(fact_1259_floor__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% floor_diff_numeral
thf(fact_1260_floor__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X3 @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) ) ) ).
% floor_diff_one
thf(fact_1261_floor__numeral__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: num,N: nat] :
( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) ) ) ).
% floor_numeral_power
thf(fact_1262_powr__log__cancel,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( powr @ real @ A2 @ ( log @ A2 @ X3 ) )
= X3 ) ) ) ) ).
% powr_log_cancel
thf(fact_1263_log__powr__cancel,axiom,
! [A2: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ A2 @ ( powr @ real @ A2 @ Y3 ) )
= Y3 ) ) ) ).
% log_powr_cancel
thf(fact_1264_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_1265_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_1266_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_1267_one__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z ) ) ).
% one_less_nat_eq
thf(fact_1268_powr__numeral,axiom,
! [X3: real,N: num] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( powr @ real @ X3 @ ( numeral_numeral @ real @ N ) )
= ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% powr_numeral
thf(fact_1269_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_1270_numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% numeral_less_floor
thf(fact_1271_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_1272_one__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) ) ) ).
% one_less_floor
thf(fact_1273_floor__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) )
= ( ord_less @ A @ X3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% floor_le_one
thf(fact_1274_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_1275_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1276_Suc__inject,axiom,
! [X3: nat,Y3: nat] :
( ( ( suc @ X3 )
= ( suc @ Y3 ) )
=> ( X3 = Y3 ) ) ).
% Suc_inject
thf(fact_1277_floor__le__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archimedean_round @ A @ X3 ) ) ) ).
% floor_le_round
thf(fact_1278_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X3: A,Y3: A] :
( ( ( size_size @ A @ X3 )
!= ( size_size @ A @ Y3 ) )
=> ( X3 != Y3 ) ) ) ).
% size_neq_size_imp_neq
thf(fact_1279_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1280_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( ( zero_zero @ nat )
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_1281_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_1282_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1283_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_1284_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3
!= ( zero_zero @ nat ) )
=> ~ ! [Nat4: nat] :
( Y3
!= ( suc @ Nat4 ) ) ) ).
% old.nat.exhaust
thf(fact_1285_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_1286_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ ( zero_zero @ nat ) )
=> ( ! [Y4: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y4 ) )
=> ( ! [X4: nat,Y4: nat] :
( ( P @ X4 @ Y4 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1287_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_1288_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_1289_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1290_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1291_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1292_list__decode_Ocases,axiom,
! [X3: nat] :
( ( X3
!= ( zero_zero @ nat ) )
=> ~ ! [N2: nat] :
( X3
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_1293_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_1294_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1295_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1296_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_1297_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_1298_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1299_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_1300_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_1301_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1302_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_1303_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_1304_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1305_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1306_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_1307_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_1308_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1309_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_1310_nat__arith_Osuc1,axiom,
! [A4: nat,K: nat,A2: nat] :
( ( A4
= ( plus_plus @ nat @ K @ A2 ) )
=> ( ( suc @ A4 )
= ( plus_plus @ nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1311_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1312_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_1313_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1314_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_1315_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1316_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_1317_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_1318_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1319_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_1320_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_1321_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_1322_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R3: nat > nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ! [X4: nat] : ( R3 @ X4 @ X4 )
=> ( ! [X4: nat,Y4: nat,Z3: nat] :
( ( R3 @ X4 @ Y4 )
=> ( ( R3 @ Y4 @ Z3 )
=> ( R3 @ X4 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R3 @ N2 @ ( suc @ N2 ) )
=> ( R3 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1323_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_1324_floor__eq3,axiom,
! [N: nat,X3: real] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X3 ) )
= N ) ) ) ).
% floor_eq3
thf(fact_1325_floor__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) ) ) ).
% floor_mono
thf(fact_1326_of__int__floor__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) @ X3 ) ) ).
% of_int_floor_le
thf(fact_1327_floor__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y3 ) )
=> ( ord_less @ A @ X3 @ Y3 ) ) ) ).
% floor_less_cancel
thf(fact_1328_powr__less__mono2__neg,axiom,
! [A2: real,X3: real,Y3: real] :
( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ Y3 )
=> ( ord_less @ real @ ( powr @ real @ Y3 @ A2 ) @ ( powr @ real @ X3 @ A2 ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_1329_powr__non__neg,axiom,
! [A2: real,X3: real] :
~ ( ord_less @ real @ ( powr @ real @ A2 @ X3 ) @ ( zero_zero @ real ) ) ).
% powr_non_neg
thf(fact_1330_powr__ge__pzero,axiom,
! [X3: real,Y3: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( powr @ real @ X3 @ Y3 ) ) ).
% powr_ge_pzero
thf(fact_1331_powr__mono2,axiom,
! [A2: real,X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ Y3 )
=> ( ord_less_eq @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ Y3 @ A2 ) ) ) ) ) ).
% powr_mono2
thf(fact_1332_floor__eq4,axiom,
! [N: nat,X3: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X3 ) )
= N ) ) ) ).
% floor_eq4
thf(fact_1333_powr__less__mono,axiom,
! [A2: real,B2: real,X3: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( ord_less @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ X3 @ B2 ) ) ) ) ).
% powr_less_mono
thf(fact_1334_powr__less__cancel,axiom,
! [X3: real,A2: real,B2: real] :
( ( ord_less @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ X3 @ B2 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( ord_less @ real @ A2 @ B2 ) ) ) ).
% powr_less_cancel
thf(fact_1335_powr__mono,axiom,
! [A2: real,B2: real,X3: real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ X3 @ B2 ) ) ) ) ).
% powr_mono
thf(fact_1336_floor__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archimedean_ceiling @ A @ X3 ) ) ) ).
% floor_le_ceiling
thf(fact_1337_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_1338_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N5 )
=> ( ord_less @ A @ ( F2 @ N ) @ ( F2 @ N5 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1339_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N5 )
=> ( ord_less_eq @ A @ ( F2 @ N ) @ ( F2 @ N5 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1340_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq @ nat @ N @ N5 )
=> ( ord_less_eq @ A @ ( F2 @ N5 ) @ ( F2 @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1341_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_1342_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_1343_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1344_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_1345_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1346_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_1347_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_1348_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_1349_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ~ ! [Q4: nat] :
( N
!= ( suc @ ( plus_plus @ nat @ M @ Q4 ) ) ) ) ).
% less_natE
thf(fact_1350_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1351_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1352_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_1353_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_1354_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1355_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_1356_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_1357_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_1358_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_1359_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_1360_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_1361_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N3: nat] : ( ord_less_eq @ nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1362_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_1363_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_1364_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_1365_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1366_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_1367_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1368_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_1369_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_1370_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_1371_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_1372_tanh__real__lt__1,axiom,
! [X3: real] : ( ord_less @ real @ ( tanh @ real @ X3 ) @ ( one_one @ real ) ) ).
% tanh_real_lt_1
thf(fact_1373_le__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X3: A] :
( ( ord_less_eq @ int @ Z @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X3 ) ) ) ).
% le_floor_iff
thf(fact_1374_floor__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z: int] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ Z )
= ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% floor_less_iff
thf(fact_1375_powr__mono2_H,axiom,
! [A2: real,X3: real,Y3: real] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ Y3 )
=> ( ord_less_eq @ real @ ( powr @ real @ Y3 @ A2 ) @ ( powr @ real @ X3 @ A2 ) ) ) ) ) ).
% powr_mono2'
thf(fact_1376_powr__less__mono2,axiom,
! [A2: real,X3: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ Y3 )
=> ( ord_less @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ Y3 @ A2 ) ) ) ) ) ).
% powr_less_mono2
thf(fact_1377_le__floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y3 ) ) ) ) ).
% le_floor_add
thf(fact_1378_int__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X3: A] :
( ( plus_plus @ int @ Z @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ X3 ) ) ) ) ).
% int_add_floor
thf(fact_1379_floor__add__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z: int] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ Z )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ ( ring_1_of_int @ A @ Z ) ) ) ) ) ).
% floor_add_int
thf(fact_1380_powr__inj,axiom,
! [A2: real,X3: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ( powr @ real @ A2 @ X3 )
= ( powr @ real @ A2 @ Y3 ) )
= ( X3 = Y3 ) ) ) ) ).
% powr_inj
thf(fact_1381_gr__one__powr,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ord_less @ real @ ( one_one @ real ) @ ( powr @ real @ X3 @ Y3 ) ) ) ) ).
% gr_one_powr
thf(fact_1382_powr__le1,axiom,
! [A2: real,X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( powr @ real @ X3 @ A2 ) @ ( one_one @ real ) ) ) ) ) ).
% powr_le1
thf(fact_1383_powr__mono__both,axiom,
! [A2: real,B2: real,X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ Y3 )
=> ( ord_less_eq @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ Y3 @ B2 ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_1384_ge__one__powr__ge__zero,axiom,
! [X3: real,A2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( powr @ real @ X3 @ A2 ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_1385_powr__divide,axiom,
! [X3: real,Y3: real,A2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( powr @ real @ ( divide_divide @ real @ X3 @ Y3 ) @ A2 )
= ( divide_divide @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ Y3 @ A2 ) ) ) ) ) ).
% powr_divide
thf(fact_1386_floor__divide__of__int__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [K: int,L: int] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L ) ) )
= ( divide_divide @ int @ K @ L ) ) ) ).
% floor_divide_of_int_eq
thf(fact_1387_floor__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,N: nat] :
( ( X3
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ X3 @ N ) )
= ( power_power @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ N ) ) ) ) ).
% floor_power
thf(fact_1388_log__base__powr,axiom,
! [A2: real,B2: real,X3: real] :
( ( A2
!= ( zero_zero @ real ) )
=> ( ( log @ ( powr @ real @ A2 @ B2 ) @ X3 )
= ( divide_divide @ real @ ( log @ A2 @ X3 ) @ B2 ) ) ) ).
% log_base_powr
thf(fact_1389_round__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ int @ ( archimedean_round @ A @ X3 ) @ ( archimedean_round @ A @ Y3 ) ) ) ) ).
% round_mono
thf(fact_1390_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_1391_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_1392_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_1393_round__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% round_def
thf(fact_1394_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_1395_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less @ nat @ K2 @ N )
& ! [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ K2 )
=> ~ ( P @ I2 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1396_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_1397_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_1398_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_1399_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_1400_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_1401_ceiling__ge__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ int @ ( archimedean_round @ A @ X3 ) @ ( archimedean_ceiling @ A @ X3 ) ) ) ).
% ceiling_ge_round
thf(fact_1402_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_1403_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_1404_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W3: int,Z6: int] :
? [N3: nat] :
( Z6
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1405_of__nat__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R2 )
=> ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archim6421214686448440834_floor @ A @ R2 ) ) ) @ R2 ) ) ) ).
% of_nat_floor
thf(fact_1406_one__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) ) ) ) ).
% one_add_floor
thf(fact_1407_floor__log__eq__powr__iff,axiom,
! [X3: real,B2: real,K: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ B2 @ X3 ) )
= K )
= ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ K ) ) @ X3 )
& ( ord_less @ real @ X3 @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ) ) ) ) ) ).
% floor_log_eq_powr_iff
thf(fact_1408_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_1409_nat__floor__neg,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X3 ) )
= ( zero_zero @ nat ) ) ) ).
% nat_floor_neg
thf(fact_1410_powr__realpow,axiom,
! [X3: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( powr @ real @ X3 @ ( semiring_1_of_nat @ real @ N ) )
= ( power_power @ real @ X3 @ N ) ) ) ).
% powr_realpow
thf(fact_1411_powr__less__iff,axiom,
! [B2: real,X3: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( powr @ real @ B2 @ Y3 ) @ X3 )
= ( ord_less @ real @ Y3 @ ( log @ B2 @ X3 ) ) ) ) ) ).
% powr_less_iff
thf(fact_1412_less__powr__iff,axiom,
! [B2: real,X3: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( powr @ real @ B2 @ Y3 ) )
= ( ord_less @ real @ ( log @ B2 @ X3 ) @ Y3 ) ) ) ) ).
% less_powr_iff
thf(fact_1413_log__less__iff,axiom,
! [B2: real,X3: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( log @ B2 @ X3 ) @ Y3 )
= ( ord_less @ real @ X3 @ ( powr @ real @ B2 @ Y3 ) ) ) ) ) ).
% log_less_iff
thf(fact_1414_less__log__iff,axiom,
! [B2: real,X3: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ Y3 @ ( log @ B2 @ X3 ) )
= ( ord_less @ real @ ( powr @ real @ B2 @ Y3 ) @ X3 ) ) ) ) ).
% less_log_iff
thf(fact_1415_le__nat__floor,axiom,
! [X3: nat,A2: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X3 ) @ A2 )
=> ( ord_less_eq @ nat @ X3 @ ( nat2 @ ( archim6421214686448440834_floor @ real @ A2 ) ) ) ) ).
% le_nat_floor
thf(fact_1416_ceiling__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X: A] :
( if @ int
@ ( X
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) )
@ ( archim6421214686448440834_floor @ A @ X )
@ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_altdef
thf(fact_1417_ceiling__diff__floor__le__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) @ ( one_one @ int ) ) ) ).
% ceiling_diff_floor_le_1
thf(fact_1418_floor__eq,axiom,
! [N: int,X3: real] :
( ( ord_less @ real @ ( ring_1_of_int @ real @ N ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X3 )
= N ) ) ) ).
% floor_eq
thf(fact_1419_real__of__int__floor__add__one__gt,axiom,
! [R2: real] : ( ord_less @ real @ R2 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R2 ) ) @ ( one_one @ real ) ) ) ).
% real_of_int_floor_add_one_gt
thf(fact_1420_real__of__int__floor__add__one__ge,axiom,
! [R2: real] : ( ord_less_eq @ real @ R2 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R2 ) ) @ ( one_one @ real ) ) ) ).
% real_of_int_floor_add_one_ge
thf(fact_1421_real__of__int__floor__gt__diff__one,axiom,
! [R2: real] : ( ord_less @ real @ ( minus_minus @ real @ R2 @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R2 ) ) ) ).
% real_of_int_floor_gt_diff_one
thf(fact_1422_real__of__int__floor__ge__diff__one,axiom,
! [R2: real] : ( ord_less_eq @ real @ ( minus_minus @ real @ R2 @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R2 ) ) ) ).
% real_of_int_floor_ge_diff_one
thf(fact_1423_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_1424_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_1425_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_1426_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_1427_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_1428_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_1429_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_1430_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_1431_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_1432_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_1433_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X3: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X3 )
=> ( ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X3 )
= Z ) ) ) ) ).
% floor_unique
thf(fact_1434_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,A2: int] :
( ( ( archim6421214686448440834_floor @ A @ X3 )
= A2 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ X3 )
& ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_1435_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_1436_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X3: A] :
( ( ord_less @ int @ Z @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% less_floor_iff
thf(fact_1437_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ Z )
= ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_1438_floor__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) @ X3 )
& ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_correct
thf(fact_1439_le__log__iff,axiom,
! [B2: real,X3: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( log @ B2 @ X3 ) )
= ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y3 ) @ X3 ) ) ) ) ).
% le_log_iff
thf(fact_1440_log__le__iff,axiom,
! [B2: real,X3: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( log @ B2 @ X3 ) @ Y3 )
= ( ord_less_eq @ real @ X3 @ ( powr @ real @ B2 @ Y3 ) ) ) ) ) ).
% log_le_iff
thf(fact_1441_le__powr__iff,axiom,
! [B2: real,X3: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( powr @ real @ B2 @ Y3 ) )
= ( ord_less_eq @ real @ ( log @ B2 @ X3 ) @ Y3 ) ) ) ) ).
% le_powr_iff
thf(fact_1442_powr__le__iff,axiom,
! [B2: real,X3: real,Y3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y3 ) @ X3 )
= ( ord_less_eq @ real @ Y3 @ ( log @ B2 @ X3 ) ) ) ) ) ).
% powr_le_iff
thf(fact_1443_floor__eq2,axiom,
! [N: int,X3: real] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ N ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X3 )
= N ) ) ) ).
% floor_eq2
thf(fact_1444_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_1445_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_1446_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_1447_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_1448_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_1449_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_1450_Suc__nat__eq__nat__zadd1,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( suc @ ( nat2 @ Z ) )
= ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1451_ln__powr__bound,axiom,
! [X3: real,A2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ ( divide_divide @ real @ ( powr @ real @ X3 @ A2 ) @ A2 ) ) ) ) ).
% ln_powr_bound
thf(fact_1452_minus__log__eq__powr,axiom,
! [B2: real,X3: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( minus_minus @ real @ Y3 @ ( log @ B2 @ X3 ) )
= ( log @ B2 @ ( divide_divide @ real @ ( powr @ real @ B2 @ Y3 ) @ X3 ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_1453_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_1454_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_1455_powr__half__sqrt,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( powr @ real @ X3 @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( sqrt @ X3 ) ) ) ).
% powr_half_sqrt
thf(fact_1456_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_1457_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_1458_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_1459_of__int__round__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% of_int_round_le
thf(fact_1460_of__int__round__ge,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) ) ) ).
% of_int_round_ge
thf(fact_1461_of__int__round__gt,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less @ A @ ( minus_minus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) ) ) ).
% of_int_round_gt
thf(fact_1462_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X3: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X3 @ ( 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 @ X3 @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_1463_high__inv,axiom,
! [X3: nat,N: nat,Y3: nat] :
( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ Y3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ X3 ) @ N )
= Y3 ) ) ).
% high_inv
thf(fact_1464_Suc__if__eq,axiom,
! [A: $tType,F2: nat > A,H: nat > A,G: A,N: nat] :
( ! [N2: nat] :
( ( F2 @ ( suc @ N2 ) )
= ( H @ N2 ) )
=> ( ( ( F2 @ ( zero_zero @ nat ) )
= G )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( ( F2 @ N )
= G ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( F2 @ N )
= ( H @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% Suc_if_eq
thf(fact_1465_arcosh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A )
= ( ^ [X: A] : ( ln_ln @ A @ ( plus_plus @ A @ X @ ( powr @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% arcosh_def
thf(fact_1466_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X ) ) @ ( archimedean_ceiling @ A @ X ) @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ) ).
% round_altdef
thf(fact_1467_arsinh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A )
= ( ^ [X: A] : ( ln_ln @ A @ ( plus_plus @ A @ X @ ( powr @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% arsinh_def
thf(fact_1468_frac__frac,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( archimedean_frac @ A @ ( archimedean_frac @ A @ X3 ) )
= ( archimedean_frac @ A @ X3 ) ) ) ).
% frac_frac
thf(fact_1469_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_1470_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_1471_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_1472_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_1473_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_1474_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_1475_mult__numeral__left__semiring__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [V2: num,W: num,Z: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Z ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W ) ) @ Z ) ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1476_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.right_neutral
thf(fact_1477_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% mult_1
thf(fact_1478_of__int__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z: int] :
( ( ring_1_of_int @ A @ ( times_times @ int @ W @ Z ) )
= ( times_times @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_mult
thf(fact_1479_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_1480_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_1481_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_1482_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_1483_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_1484_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_1485_bit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H2: nat,L2: nat,D4: nat] : ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ D4 ) ) @ L2 ) ) ) ).
% bit_concat_def
thf(fact_1486_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_1487_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_1488_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_1489_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_1490_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X3: A,Y3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y3 @ Y3 ) )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1491_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_1492_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_1493_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_1494_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_1495_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_1496_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_1497_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_1498_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_1499_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_1500_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_1501_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_1502_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_1503_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_1504_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_1505_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_1506_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_1507_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_1508_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_1509_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_1510_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_1511_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_1512_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_1513_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_1514_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_1515_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_1516_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_1517_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_1518_frac__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int] :
( ( archimedean_frac @ A @ ( ring_1_of_int @ A @ Z ) )
= ( zero_zero @ A ) ) ) ).
% frac_of_int
thf(fact_1519_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W: num] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) @ B2 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_1520_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_1521_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W: num,A2: A] :
( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) )
= A2 )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1522_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,W: num] :
( ( A2
= ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) )
= B2 ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1523_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) @ A2 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_1524_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W: num] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) @ B2 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_1525_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_1526_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_1527_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_1528_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_1529_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_1530_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_1531_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_1532_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_1533_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_1534_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_1535_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_1536_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_1537_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_1538_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_1539_or__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% or_numerals(3)
thf(fact_1540_xor__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% xor_numerals(3)
thf(fact_1541_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_1542_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_1543_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_1544_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_1545_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_1546_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_1547_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A5: A,B4: A] : ( times_times @ A @ B4 @ A5 ) ) ) ) ).
% mult.commute
thf(fact_1548_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_1549_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_1550_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_1551_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_1552_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_1553_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_1554_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_1555_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_1556_inf__period_I2_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D5: A,Q: A > $o] :
( ! [X4: A,K2: A] :
( ( P @ X4 )
= ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K2 @ D5 ) ) ) )
=> ( ! [X4: A,K2: A] :
( ( Q @ X4 )
= ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K2 @ D5 ) ) ) )
=> ! [X6: A,K4: A] :
( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D5 ) ) )
| ( Q @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D5 ) ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1557_inf__period_I1_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D5: A,Q: A > $o] :
( ! [X4: A,K2: A] :
( ( P @ X4 )
= ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K2 @ D5 ) ) ) )
=> ( ! [X4: A,K2: A] :
( ( Q @ X4 )
= ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K2 @ D5 ) ) ) )
=> ! [X6: A,K4: A] :
( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D5 ) ) )
& ( Q @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D5 ) ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1558_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_1559_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_1560_power__commuting__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X3: A,Y3: A,N: nat] :
( ( ( times_times @ A @ X3 @ Y3 )
= ( times_times @ A @ Y3 @ X3 ) )
=> ( ( times_times @ A @ ( power_power @ A @ X3 @ N ) @ Y3 )
= ( times_times @ A @ Y3 @ ( power_power @ A @ X3 @ N ) ) ) ) ) ).
% power_commuting_commutes
thf(fact_1561_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_1562_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_1563_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_1564_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_1565_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X3: nat,Y3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X3 ) @ Y3 )
= ( times_times @ A @ Y3 @ ( semiring_1_of_nat @ A @ X3 ) ) ) ) ).
% mult_of_nat_commute
thf(fact_1566_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_1567_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_1568_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1569_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_1570_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_1571_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_1572_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_1573_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1574_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_1575_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_1576_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_1577_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_1578_mult__of__int__commute,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: int,Y3: A] :
( ( times_times @ A @ ( ring_1_of_int @ A @ X3 ) @ Y3 )
= ( times_times @ A @ Y3 @ ( ring_1_of_int @ A @ X3 ) ) ) ) ).
% mult_of_int_commute
thf(fact_1579_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_1580_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_1581_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_1582_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( 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 @ D3 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_1583_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( 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 @ D3 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1584_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_1585_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_1586_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_1587_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_1588_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_1589_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_1590_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_1591_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_1592_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_1593_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_1594_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_1595_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_1596_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_1597_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_1598_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_1599_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_1600_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_1601_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_1602_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_1603_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_1604_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_1605_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_1606_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_1607_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_1608_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_1609_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_1610_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_1611_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_1612_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_1613_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_1614_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_1615_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_1616_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_1617_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_1618_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_1619_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_1620_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_1621_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_1622_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_1623_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_1624_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_1625_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,Z: A,X3: A,W: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X3 @ Y3 )
= ( divide_divide @ A @ W @ Z ) )
= ( ( times_times @ A @ X3 @ Z )
= ( times_times @ A @ W @ Y3 ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1626_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_1627_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_1628_mult__diff__mult,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [X3: A,Y3: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( times_times @ A @ X3 @ Y3 ) @ ( times_times @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ X3 @ ( minus_minus @ A @ Y3 @ B2 ) ) @ ( times_times @ A @ ( minus_minus @ A @ X3 @ A2 ) @ B2 ) ) ) ) ).
% mult_diff_mult
thf(fact_1629_left__right__inverse__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X3: A,Y3: A,N: nat] :
( ( ( times_times @ A @ X3 @ Y3 )
= ( one_one @ A ) )
=> ( ( times_times @ A @ ( power_power @ A @ X3 @ N ) @ ( power_power @ A @ Y3 @ N ) )
= ( one_one @ A ) ) ) ) ).
% left_right_inverse_power
thf(fact_1630_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_1631_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_1632_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_1633_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_1634_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_1635_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_1636_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_1637_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_1638_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_1639_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_1640_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_1641_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_1642_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_1643_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_1644_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_1645_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_1646_frac__ge__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X3 ) ) ) ).
% frac_ge_0
thf(fact_1647_frac__lt__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X3 ) @ ( one_one @ A ) ) ) ).
% frac_lt_1
thf(fact_1648_option_Osize__neq,axiom,
! [A: $tType,X3: option @ A] :
( ( size_size @ ( option @ A ) @ X3 )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_1649_frac__1__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( archimedean_frac @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) )
= ( archimedean_frac @ A @ X3 ) ) ) ).
% frac_1_eq
thf(fact_1650_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( 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 @ D3 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1651_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( 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 @ D3 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1652_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_1653_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_1654_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_1655_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_1656_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_1657_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( 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 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1658_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_1659_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_1660_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( 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 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1661_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_1662_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_1663_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_1664_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y3 @ X3 ) @ X3 ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1665_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X3 @ Y3 ) @ X3 ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1666_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_1667_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_1668_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y3 @ Y3 ) ) @ ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1669_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y3 @ Y3 ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_1670_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y3 @ Y3 ) ) )
= ( ( X3
!= ( zero_zero @ A ) )
| ( Y3
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1671_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X3: A,Y3: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y3 @ Y3 ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_squares_lt_zero
thf(fact_1672_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_1673_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_1674_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_1675_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_1676_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_1677_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_1678_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_1679_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,X3: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less @ A @ X3 @ ( times_times @ A @ Z @ Y3 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y3 ) @ Z ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_1680_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,Z: A,X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less @ A @ ( times_times @ A @ Z @ Y3 ) @ X3 )
=> ( ord_less @ A @ Z @ ( divide_divide @ A @ X3 @ Y3 ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_1681_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_1682_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_1683_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ( divide_divide @ A @ B2 @ C2 )
= ( numeral_numeral @ A @ W ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_1684_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ( numeral_numeral @ A @ W )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_1685_ordered__ring__class_Ole__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E2: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E2 ) @ D3 ) ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1686_ordered__ring__class_Ole__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E2: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E2 ) @ C2 ) @ D3 ) ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1687_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X3: A,Y3: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ Z ) @ Y3 )
= ( divide_divide @ A @ ( plus_plus @ A @ X3 @ ( times_times @ A @ Y3 @ Z ) ) @ Z ) ) ) ) ).
% divide_add_eq_iff
thf(fact_1688_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X3: A,Y3: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X3 @ ( divide_divide @ A @ Y3 @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ) ).
% add_divide_eq_iff
thf(fact_1689_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,Z: A,X3: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z @ ( divide_divide @ A @ X3 @ Y3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X3 @ ( times_times @ A @ Z @ Y3 ) ) @ Y3 ) ) ) ) ).
% add_num_frac
thf(fact_1690_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,X3: A,Z: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ Y3 ) @ Z )
= ( divide_divide @ A @ ( plus_plus @ A @ X3 @ ( times_times @ A @ Z @ Y3 ) ) @ Y3 ) ) ) ) ).
% add_frac_num
thf(fact_1691_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,Z: A,X3: A,W: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ Y3 ) @ ( divide_divide @ A @ W @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ Z ) @ ( times_times @ A @ W @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_1692_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z ) )
= A2 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ Z ) @ B2 ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1693_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z ) @ B2 )
= B2 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1694_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E2: A,C2: A,B2: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E2 ) @ C2 ) @ D3 ) ) ) ).
% less_add_iff1
thf(fact_1695_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E2: A,C2: A,B2: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
= ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E2 ) @ D3 ) ) ) ) ).
% less_add_iff2
thf(fact_1696_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X3: A,Y3: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X3 @ Z ) @ Y3 )
= ( divide_divide @ A @ ( minus_minus @ A @ X3 @ ( times_times @ A @ Y3 @ Z ) ) @ Z ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_1697_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X3: A,Y3: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X3 @ ( divide_divide @ A @ Y3 @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_1698_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,Z: A,X3: A,W: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X3 @ Y3 ) @ ( divide_divide @ A @ W @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z ) @ ( times_times @ A @ W @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1699_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z ) )
= A2 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A2 @ Z ) @ B2 ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1700_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_1701_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_1702_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ? [N2: nat] : ( ord_less @ A @ Y3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ X3 ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_1703_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ~ ( ( A2
!= ( zero_zero @ A ) )
=> ! [C3: A] :
( B2
!= ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% unit_dvdE
thf(fact_1704_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L: A] :
( ( ? [X: A] : ( P @ ( times_times @ A @ L @ X ) ) )
= ( ? [X: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X @ ( zero_zero @ A ) ) )
& ( P @ X ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_1705_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A,B2: A,D3: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D3 )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= ( divide_divide @ A @ D3 @ C2 ) )
= ( ( times_times @ A @ B2 @ C2 )
= ( times_times @ A @ A2 @ D3 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_1706_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_1707_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_1708_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_1709_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D5: A,T2: A] :
( ( dvd_dvd @ A @ D3 @ D5 )
=> ! [X6: A,K4: A] :
( ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X6 @ T2 ) ) )
= ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D5 ) ) @ T2 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_1710_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D5: A,T2: A] :
( ( dvd_dvd @ A @ D3 @ D5 )
=> ! [X6: A,K4: A] :
( ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X6 @ T2 ) )
= ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D5 ) ) @ T2 ) ) ) ) ) ).
% inf_period(3)
thf(fact_1711_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_1712_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_1713_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_1714_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_1715_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_1716_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_1717_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_1718_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_1719_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1720_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1721_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1722_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1723_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1724_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1725_bezout__add__strong__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ? [D6: nat,X4: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D6 @ A2 )
& ( dvd_dvd @ nat @ D6 @ B2 )
& ( ( times_times @ nat @ A2 @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D6 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1726_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_1727_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_1728_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_1729_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_1730_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_1731_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_1732_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_1733_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_1734_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A] :
( ! [Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less @ A @ Z3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X3 ) @ Y3 ) ) )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% field_le_mult_one_interval
thf(fact_1735_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_1736_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_1737_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_1738_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_1739_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_1740_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_1741_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_1742_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,X3: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ ( times_times @ A @ Z @ Y3 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y3 ) @ Z ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_1743_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,Z: A,X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ Y3 ) @ X3 )
=> ( ord_less_eq @ A @ Z @ ( divide_divide @ A @ X3 @ Y3 ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_1744_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_1745_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X3: A,A2: A,Y3: A,U: A,V2: A] :
( ( ord_less_eq @ A @ X3 @ A2 )
=> ( ( ord_less_eq @ A @ Y3 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U @ V2 )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X3 ) @ ( times_times @ A @ V2 @ Y3 ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1746_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_1747_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ 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 @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_1748_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,Z: A,X3: A,W: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y3 ) @ ( divide_divide @ A @ W @ Z ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z ) @ ( times_times @ A @ W @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_1749_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_1750_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,Z: A,X3: A,W: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y3 ) @ ( divide_divide @ A @ W @ Z ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z ) @ ( times_times @ A @ W @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_1751_mult__2,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z )
= ( plus_plus @ A @ Z @ Z ) ) ) ).
% mult_2
thf(fact_1752_mult__2__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z: A] :
( ( times_times @ A @ Z @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ Z @ Z ) ) ) ).
% mult_2_right
thf(fact_1753_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_1754_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_1755_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_1756_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ~ ( ( A2
!= ( zero_zero @ A ) )
=> ! [B5: A] :
( ( B5
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B5 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
= B5 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B5 )
= A2 )
=> ( ( ( times_times @ A @ A2 @ B5 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C2 @ A2 )
!= ( times_times @ A @ C2 @ B5 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_1757_evenE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ~ ! [B5: A] :
( A2
!= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) ) ) ) ).
% evenE
thf(fact_1758_power4__eq__xxxx,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X3: A] :
( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( times_times @ A @ ( times_times @ A @ X3 @ X3 ) @ X3 ) @ X3 ) ) ) ).
% power4_eq_xxxx
thf(fact_1759_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_1760_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_1761_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_1762_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_1763_frac__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_frac @ A )
= ( ^ [X: A] : ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ) ) ).
% frac_def
thf(fact_1764_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_1765_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_1766_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_1767_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_1768_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_1769_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_1770_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1771_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1772_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_1773_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_1774_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_1775_dvd__minus__add,axiom,
! [Q3: nat,N: nat,R2: nat,M: nat] :
( ( ord_less_eq @ nat @ Q3 @ N )
=> ( ( ord_less_eq @ nat @ Q3 @ ( times_times @ nat @ R2 @ M ) )
=> ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ Q3 ) )
= ( dvd_dvd @ nat @ M @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( times_times @ nat @ R2 @ M ) @ Q3 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_1776_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X3: A,A2: A,Y3: A,U: A,V2: A] :
( ( ord_less @ A @ X3 @ A2 )
=> ( ( ord_less @ A @ Y3 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U @ V2 )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X3 ) @ ( times_times @ A @ V2 @ Y3 ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1777_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ 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 @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_1778_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ 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 @ W ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_1779_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V2: A,R2: A,S: A] :
( ( ord_less_eq @ A @ U @ V2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R2 )
=> ( ( ord_less_eq @ A @ R2 @ S )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R2 @ ( minus_minus @ A @ V2 @ U ) ) @ S ) ) @ V2 ) ) ) ) ) ).
% scaling_mono
thf(fact_1780_frac__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ( archimedean_frac @ A @ X3 )
= X3 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
& ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ) ).
% frac_eq
thf(fact_1781_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_1782_frac__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y3: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X3 @ Y3 ) )
= ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y3 ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X3 @ Y3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% frac_add
thf(fact_1783_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_1784_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P5: A,M3: nat] :
( if @ A
@ ( M3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P5 @ ( power_power @ A @ P5 @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1785_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_1786_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 ) ) )
| ? [Q5: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q5 ) @ M )
& ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q5 ) ) )
& ( P @ Q5 ) ) ) ) ).
% split_div'
thf(fact_1787_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ~ ! [B5: A] :
( A2
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_1788_power2__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X3: A,Y3: A] :
( ( power_power @ A @ ( plus_plus @ A @ X3 @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) @ Y3 ) ) ) ) ).
% power2_sum
thf(fact_1789_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_1790_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_1791_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P6: 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 @ P6 @ Q3 ) ) ) @ Q3 ) @ P6 ) ) ) ).
% floor_divide_lower
thf(fact_1792_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P6: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less_eq @ A @ P6 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P6 @ Q3 ) ) ) @ Q3 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_1793_vebt__buildup_Ocases,axiom,
! [X3: nat] :
( ( X3
!= ( zero_zero @ nat ) )
=> ( ( X3
!= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ! [Va: nat] :
( X3
!= ( suc @ ( suc @ Va ) ) ) ) ) ).
% vebt_buildup.cases
thf(fact_1794_option_Osize_I4_J,axiom,
! [A: $tType,X2: A] :
( ( size_size @ ( option @ A ) @ ( some @ A @ X2 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(4)
thf(fact_1795_sum__squares__bound,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) @ Y3 ) @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_squares_bound
thf(fact_1796_power2__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X3: A,Y3: A] :
( ( power_power @ A @ ( minus_minus @ A @ X3 @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) @ Y3 ) ) ) ) ).
% power2_diff
thf(fact_1797_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_1798_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_1799_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P6: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less @ A @ P6 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P6 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) ) ) ) ).
% floor_divide_upper
thf(fact_1800_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P6: 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 @ P6 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) @ P6 ) ) ) ).
% ceiling_divide_lower
thf(fact_1801_floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y3: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y3 ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y3 ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_add
thf(fact_1802_num_Osize_I5_J,axiom,
! [X2: num] :
( ( size_size @ num @ ( bit0 @ X2 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(5)
thf(fact_1803_arith__geo__mean,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,X3: A,Y3: A] :
( ( ( power_power @ A @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ X3 @ Y3 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ U @ ( divide_divide @ A @ ( plus_plus @ A @ X3 @ Y3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_1804_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_1805_of__real__of__int__eq,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [Z: int] :
( ( real_Vector_of_real @ A @ ( ring_1_of_int @ real @ Z ) )
= ( ring_1_of_int @ A @ Z ) ) ) ).
% of_real_of_int_eq
thf(fact_1806_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_1807_of__real__power,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X3: real,N: nat] :
( ( real_Vector_of_real @ A @ ( power_power @ real @ X3 @ N ) )
= ( power_power @ A @ ( real_Vector_of_real @ A @ X3 ) @ N ) ) ) ).
% of_real_power
thf(fact_1808_of__real__numeral,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [W: num] :
( ( real_Vector_of_real @ A @ ( numeral_numeral @ real @ W ) )
= ( numeral_numeral @ A @ W ) ) ) ).
% of_real_numeral
thf(fact_1809_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_1810_of__real__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X3: real] :
( ( ( real_Vector_of_real @ A @ X3 )
= ( zero_zero @ A ) )
= ( X3
= ( zero_zero @ real ) ) ) ) ).
% of_real_eq_0_iff
thf(fact_1811_mul__def,axiom,
( vEBT_VEBT_mul
= ( vEBT_V2048590022279873568_shift @ nat @ ( times_times @ nat ) ) ) ).
% mul_def
thf(fact_1812_mul__shift,axiom,
! [X3: nat,Y3: nat,Z: nat] :
( ( ( times_times @ nat @ X3 @ Y3 )
= Z )
= ( ( vEBT_VEBT_mul @ ( some @ nat @ X3 ) @ ( some @ nat @ Y3 ) )
= ( some @ nat @ Z ) ) ) ).
% mul_shift
thf(fact_1813_real__divide__square__eq,axiom,
! [R2: real,A2: real] :
( ( divide_divide @ real @ ( times_times @ real @ R2 @ A2 ) @ ( times_times @ real @ R2 @ R2 ) )
= ( divide_divide @ real @ A2 @ R2 ) ) ).
% real_divide_square_eq
thf(fact_1814_not__real__square__gt__zero,axiom,
! [X3: real] :
( ( ~ ( ord_less @ real @ ( zero_zero @ real ) @ ( times_times @ real @ X3 @ X3 ) ) )
= ( X3
= ( zero_zero @ real ) ) ) ).
% not_real_square_gt_zero
thf(fact_1815_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_1816_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times @ num @ M @ one2 )
= M ) ).
% semiring_norm(11)
thf(fact_1817_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times @ num @ one2 @ N )
= N ) ).
% semiring_norm(12)
thf(fact_1818_num__double,axiom,
! [N: num] :
( ( times_times @ num @ ( bit0 @ one2 ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1819_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_1820_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [X3: real,Y3: real,Xa: real,Ya: real] :
( ( power_power @ real @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( 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 @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( 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_1821_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times @ int @ ( zero_zero @ int ) @ L )
= ( zero_zero @ int ) ) ).
% times_int_code(2)
thf(fact_1822_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times @ int @ K @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% times_int_code(1)
thf(fact_1823_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W )
= ( plus_plus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1824_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times @ int @ W @ ( plus_plus @ int @ Z1 @ Z22 ) )
= ( plus_plus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1825_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times @ int @ ( minus_minus @ int @ Z1 @ Z22 ) @ W )
= ( minus_minus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1826_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times @ int @ W @ ( minus_minus @ int @ Z1 @ Z22 ) )
= ( minus_minus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1827_real__sqrt__mult,axiom,
! [X3: real,Y3: real] :
( ( sqrt @ ( times_times @ real @ X3 @ Y3 ) )
= ( times_times @ real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).
% real_sqrt_mult
thf(fact_1828_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_1829_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_1830_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_1831_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_1832_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_1833_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_1834_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_1835_zdvd__period,axiom,
! [A2: int,D3: int,X3: int,T2: int,C2: int] :
( ( dvd_dvd @ int @ A2 @ D3 )
=> ( ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ X3 @ T2 ) )
= ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ ( plus_plus @ int @ X3 @ ( times_times @ int @ C2 @ D3 ) ) @ T2 ) ) ) ) ).
% zdvd_period
thf(fact_1836_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_1837_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_1838_reals__Archimedean3,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ! [Y5: real] :
? [N2: nat] : ( ord_less @ real @ Y5 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X3 ) ) ) ).
% reals_Archimedean3
thf(fact_1839_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_1840_powr__mult,axiom,
! [X3: real,Y3: real,A2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( powr @ real @ ( times_times @ real @ X3 @ Y3 ) @ A2 )
= ( times_times @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ Y3 @ A2 ) ) ) ) ) ).
% powr_mult
thf(fact_1841_minusinfinity,axiom,
! [D3: int,P1: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X4: int,K2: int] :
( ( P1 @ X4 )
= ( P1 @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K2 @ D3 ) ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less @ int @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P1 @ X4 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1842_plusinfinity,axiom,
! [D3: int,P4: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X4: int,K2: int] :
( ( P4 @ X4 )
= ( P4 @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K2 @ D3 ) ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less @ int @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [X_12: int] : ( P4 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1843_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_1844_le__real__sqrt__sumsq,axiom,
! [X3: real,Y3: real] : ( ord_less_eq @ real @ X3 @ ( sqrt @ ( plus_plus @ real @ ( times_times @ real @ X3 @ X3 ) @ ( times_times @ real @ Y3 @ Y3 ) ) ) ) ).
% le_real_sqrt_sumsq
thf(fact_1845_log__powr,axiom,
! [X3: real,B2: real,Y3: real] :
( ( X3
!= ( zero_zero @ real ) )
=> ( ( log @ B2 @ ( powr @ real @ X3 @ Y3 ) )
= ( times_times @ real @ Y3 @ ( log @ B2 @ X3 ) ) ) ) ).
% log_powr
thf(fact_1846_ln__powr,axiom,
! [X3: real,Y3: real] :
( ( X3
!= ( zero_zero @ real ) )
=> ( ( ln_ln @ real @ ( powr @ real @ X3 @ Y3 ) )
= ( times_times @ real @ Y3 @ ( ln_ln @ real @ X3 ) ) ) ) ).
% ln_powr
thf(fact_1847_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_1848_incr__mult__lemma,axiom,
! [D3: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus @ int @ X4 @ D3 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( plus_plus @ int @ X6 @ ( times_times @ int @ K @ D3 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1849_q__pos__lemma,axiom,
! [B3: int,Q6: int,R4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q6 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q6 ) ) ) ) ).
% q_pos_lemma
thf(fact_1850_zdiv__mono2__lemma,axiom,
! [B2: int,Q3: int,R2: int,B3: int,Q6: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q6 ) @ R4 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q6 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ B3 @ B2 )
=> ( ord_less_eq @ int @ Q3 @ Q6 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_1851_zdiv__mono2__neg__lemma,axiom,
! [B2: int,Q3: int,R2: int,B3: int,Q6: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q6 ) @ R4 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q6 ) @ R4 ) @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ B3 @ B2 )
=> ( ord_less_eq @ int @ Q6 @ Q3 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_1852_unique__quotient__lemma,axiom,
! [B2: int,Q6: int,R4: int,Q3: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ R4 @ B2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ord_less_eq @ int @ Q6 @ Q3 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_1853_unique__quotient__lemma__neg,axiom,
! [B2: int,Q6: int,R4: int,Q3: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R2 )
=> ( ( ord_less @ int @ B2 @ R4 )
=> ( ord_less_eq @ int @ Q3 @ Q6 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_1854_nat__mult__distrib,axiom,
! [Z: int,Z5: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( nat2 @ ( times_times @ int @ Z @ Z5 ) )
= ( times_times @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z5 ) ) ) ) ).
% nat_mult_distrib
thf(fact_1855_decr__mult__lemma,axiom,
! [D3: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus @ int @ X4 @ D3 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( minus_minus @ int @ X6 @ ( times_times @ int @ K @ D3 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1856_ln__mult,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ln_ln @ real @ ( times_times @ real @ X3 @ Y3 ) )
= ( plus_plus @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y3 ) ) ) ) ) ).
% ln_mult
thf(fact_1857_four__x__squared,axiom,
! [X3: real] :
( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% four_x_squared
thf(fact_1858_real__archimedian__rdiv__eq__0,axiom,
! [X3: real,C2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
=> ( ! [M4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M4 )
=> ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M4 ) @ X3 ) @ C2 ) )
=> ( X3
= ( zero_zero @ real ) ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1859_log__mult,axiom,
! [A2: real,X3: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( log @ A2 @ ( times_times @ real @ X3 @ Y3 ) )
= ( plus_plus @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ Y3 ) ) ) ) ) ) ) ).
% log_mult
thf(fact_1860_powr__mult__base,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( times_times @ real @ X3 @ ( powr @ real @ X3 @ Y3 ) )
= ( powr @ real @ X3 @ ( plus_plus @ real @ ( one_one @ real ) @ Y3 ) ) ) ) ).
% powr_mult_base
thf(fact_1861_log__nat__power,axiom,
! [X3: real,B2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( log @ B2 @ ( power_power @ real @ X3 @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ X3 ) ) ) ) ).
% log_nat_power
thf(fact_1862_int__div__pos__eq,axiom,
! [A2: int,B2: int,Q3: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ( divide_divide @ int @ A2 @ B2 )
= Q3 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1863_int__div__neg__eq,axiom,
! [A2: int,B2: int,Q3: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R2 )
=> ( ( divide_divide @ int @ A2 @ B2 )
= Q3 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1864_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_1865_ln__realpow,axiom,
! [X3: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ln_ln @ real @ ( power_power @ real @ X3 @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( ln_ln @ real @ X3 ) ) ) ) ).
% ln_realpow
thf(fact_1866_linear__plus__1__le__power,axiom,
! [X3: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X3 ) @ ( one_one @ real ) ) @ ( power_power @ real @ ( plus_plus @ real @ X3 @ ( one_one @ real ) ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_1867_ln__powr__bound2,axiom,
! [X3: real,A2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( powr @ real @ ( ln_ln @ real @ X3 ) @ A2 ) @ ( times_times @ real @ ( powr @ real @ A2 @ A2 ) @ X3 ) ) ) ) ).
% ln_powr_bound2
thf(fact_1868_log__eq__div__ln__mult__log,axiom,
! [A2: real,B2: real,X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( log @ A2 @ X3 )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( ln_ln @ real @ A2 ) ) @ ( log @ B2 @ X3 ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_1869_log__add__eq__powr,axiom,
! [B2: real,X3: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( plus_plus @ real @ ( log @ B2 @ X3 ) @ Y3 )
= ( log @ B2 @ ( times_times @ real @ X3 @ ( powr @ real @ B2 @ Y3 ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_1870_add__log__eq__powr,axiom,
! [B2: real,X3: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( plus_plus @ real @ Y3 @ ( log @ B2 @ X3 ) )
= ( log @ B2 @ ( times_times @ real @ ( powr @ real @ B2 @ Y3 ) @ X3 ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_1871_L2__set__mult__ineq__lemma,axiom,
! [A2: real,C2: real,B2: real,D3: 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 @ D3 ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ real @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ C2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_1872_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X3: real,Y3: real,Xa: real,Ya: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( 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_1873_arith__geo__mean__sqrt,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ord_less_eq @ real @ ( sqrt @ ( times_times @ real @ X3 @ Y3 ) ) @ ( divide_divide @ real @ ( plus_plus @ real @ X3 @ Y3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arith_geo_mean_sqrt
thf(fact_1874_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_1875_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_1876_Bernoulli__inequality__even,axiom,
! [N: nat,X3: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X3 ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) @ N ) ) ) ).
% Bernoulli_inequality_even
thf(fact_1877_nonzero__of__real__divide,axiom,
! [A: $tType] :
( ( real_V7773925162809079976_field @ A )
=> ! [Y3: real,X3: real] :
( ( Y3
!= ( zero_zero @ real ) )
=> ( ( real_Vector_of_real @ A @ ( divide_divide @ real @ X3 @ Y3 ) )
= ( divide_divide @ A @ ( real_Vector_of_real @ A @ X3 ) @ ( real_Vector_of_real @ A @ Y3 ) ) ) ) ) ).
% nonzero_of_real_divide
thf(fact_1878_lemma__termdiff3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [H: A,Z: A,K5: real,N: nat] :
( ( H
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K5 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ Z @ H ) ) @ K5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H ) @ N ) @ ( power_power @ A @ Z @ N ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) @ ( times_times @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( power_power @ real @ K5 @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_1879_low__inv,axiom,
! [X3: nat,N: nat,Y3: nat] :
( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ Y3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ X3 ) @ N )
= X3 ) ) ).
% low_inv
thf(fact_1880_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X3: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X3 @ Z ) @ ( times_times @ A @ Y3 @ Z ) )
= ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1881_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X3: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ X3 ) @ ( times_times @ A @ Z @ Y3 ) )
= ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1882_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_1883_ln__one__minus__pos__lower__bound,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( uminus_uminus @ real @ X3 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X3 ) ) ) ) ) ).
% ln_one_minus_pos_lower_bound
thf(fact_1884_abs__ln__one__plus__x__minus__x__bound,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) ) @ X3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound
thf(fact_1885_bit__split__inv,axiom,
! [X3: nat,D3: nat] :
( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X3 @ D3 ) @ ( vEBT_VEBT_low @ X3 @ D3 ) @ D3 )
= X3 ) ).
% bit_split_inv
thf(fact_1886_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_1887_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_1888_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_1889_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_1890_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_1891_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_1892_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_1893_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_1894_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_1895_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_1896_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_1897_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_1898_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_1899_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_1900_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_1901_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_1902_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_1903_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_1904_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_1905_abs__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_0
thf(fact_1906_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_1907_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_1908_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_1909_of__int__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ Z ) )
= ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_minus
thf(fact_1910_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_1911_of__int__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: int] :
( ( ring_1_of_int @ A @ ( abs_abs @ int @ X3 ) )
= ( abs_abs @ A @ ( ring_1_of_int @ A @ X3 ) ) ) ) ).
% of_int_abs
thf(fact_1912_triangle__0,axiom,
( ( nat_triangle @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% triangle_0
thf(fact_1913_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_1914_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_1915_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_1916_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_1917_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_1918_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_1919_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_1920_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_1921_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_1922_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_1923_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_1924_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_1925_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_1926_mult__minus1__right,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: A] :
( ( times_times @ A @ Z @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ Z ) ) ) ).
% mult_minus1_right
thf(fact_1927_mult__minus1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ Z )
= ( uminus_uminus @ A @ Z ) ) ) ).
% mult_minus1
thf(fact_1928_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_1929_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_1930_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_1931_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_1932_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_1933_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_1934_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_1935_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_1936_norm__eq__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A] :
( ( ( real_V7770717601297561774m_norm @ A @ X3 )
= ( zero_zero @ real ) )
= ( X3
= ( zero_zero @ A ) ) ) ) ).
% norm_eq_zero
thf(fact_1937_norm__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( zero_zero @ A ) )
= ( zero_zero @ real ) ) ) ).
% norm_zero
thf(fact_1938_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_1939_norm__numeral,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( numeral_numeral @ A @ W ) )
= ( numeral_numeral @ real @ W ) ) ) ).
% norm_numeral
thf(fact_1940_bit_Odisj__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_left
thf(fact_1941_bit_Odisj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se1065995026697491101ons_or @ A @ X3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_right
thf(fact_1942_real__add__minus__iff,axiom,
! [X3: real,A2: real] :
( ( ( plus_plus @ real @ X3 @ ( uminus_uminus @ real @ A2 ) )
= ( zero_zero @ real ) )
= ( X3 = A2 ) ) ).
% real_add_minus_iff
thf(fact_1943_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_1944_floor__uminus__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z ) ) )
= ( uminus_uminus @ int @ Z ) ) ) ).
% floor_uminus_of_int
thf(fact_1945_real__sqrt__abs2,axiom,
! [X3: real] :
( ( sqrt @ ( times_times @ real @ X3 @ X3 ) )
= ( abs_abs @ real @ X3 ) ) ).
% real_sqrt_abs2
thf(fact_1946_real__sqrt__mult__self,axiom,
! [A2: real] :
( ( times_times @ real @ ( sqrt @ A2 ) @ ( sqrt @ A2 ) )
= ( abs_abs @ real @ A2 ) ) ).
% real_sqrt_mult_self
thf(fact_1947_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_1948_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_1949_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_1950_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_1951_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_1952_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_1953_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_1954_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_1955_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_1956_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_1957_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_1958_zero__less__norm__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) )
= ( X3
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_norm_iff
thf(fact_1959_norm__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( zero_zero @ real ) )
= ( X3
= ( zero_zero @ A ) ) ) ) ).
% norm_le_zero_iff
thf(fact_1960_norm__neg__numeral,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( numeral_numeral @ real @ W ) ) ) ).
% norm_neg_numeral
thf(fact_1961_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_1962_semiring__norm_I168_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W: num,Y3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y3 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W ) ) ) @ Y3 ) ) ) ).
% semiring_norm(168)
thf(fact_1963_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_1964_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_1965_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_1966_semiring__norm_I172_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V2: num,W: num,Y3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y3 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W ) ) @ Y3 ) ) ) ).
% semiring_norm(172)
thf(fact_1967_semiring__norm_I171_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V2: num,W: num,Y3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y3 ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W ) ) ) @ Y3 ) ) ) ).
% semiring_norm(171)
thf(fact_1968_semiring__norm_I170_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V2: num,W: num,Y3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Y3 ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W ) ) ) @ Y3 ) ) ) ).
% semiring_norm(170)
thf(fact_1969_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_1970_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_1971_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_1972_artanh__minus__real,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( ( artanh @ real @ ( uminus_uminus @ real @ X3 ) )
= ( uminus_uminus @ real @ ( artanh @ real @ X3 ) ) ) ) ).
% artanh_minus_real
thf(fact_1973_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_1974_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_1975_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_1976_norm__of__int,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [Z: int] :
( ( real_V7770717601297561774m_norm @ A @ ( ring_1_of_int @ A @ Z ) )
= ( abs_abs @ real @ ( ring_1_of_int @ real @ Z ) ) ) ) ).
% norm_of_int
thf(fact_1977_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_1978_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_1979_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W: num] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_1980_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B2 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_1981_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W: num,A2: A] :
( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= A2 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_1982_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,W: num] :
( ( A2
= ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= B2 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_1983_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B2 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_1984_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W: num] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_1985_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_1986_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_1987_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_1988_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_1989_norm__mult__numeral1,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [W: num,A2: A] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A2 ) )
= ( times_times @ real @ ( numeral_numeral @ real @ W ) @ ( real_V7770717601297561774m_norm @ A @ A2 ) ) ) ) ).
% norm_mult_numeral1
thf(fact_1990_norm__mult__numeral2,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: A,W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) )
= ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( numeral_numeral @ real @ W ) ) ) ) ).
% norm_mult_numeral2
thf(fact_1991_norm__divide__numeral,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: A,W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ W ) ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( numeral_numeral @ real @ W ) ) ) ) ).
% norm_divide_numeral
thf(fact_1992_of__real__neg__numeral,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [W: num] :
( ( real_Vector_of_real @ A @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ W ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ).
% of_real_neg_numeral
thf(fact_1993_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_1994_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_1995_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_1996_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_1997_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_1998_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_1999_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_2000_power__even__abs__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: num,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
=> ( ( power_power @ A @ ( abs_abs @ A @ A2 ) @ ( numeral_numeral @ nat @ W ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).
% power_even_abs_numeral
thf(fact_2001_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_2002_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_2003_neg__numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X3 ) ) ) ).
% neg_numeral_le_floor
thf(fact_2004_floor__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_2005_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_2006_ceiling__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_zero
thf(fact_2007_zero__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X3 ) ) ) ).
% zero_le_ceiling
thf(fact_2008_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X3 ) ) ) ).
% neg_numeral_less_ceiling
thf(fact_2009_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X3: num,N: nat,Y3: int] :
( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N )
= ( ring_1_of_int @ A @ Y3 ) )
= ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N )
= Y3 ) ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_2010_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y3: int,X3: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y3 )
= ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N ) )
= ( Y3
= ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N ) ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_2011_real__sqrt__abs,axiom,
! [X3: real] :
( ( sqrt @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( abs_abs @ real @ X3 ) ) ).
% real_sqrt_abs
thf(fact_2012_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_2013_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_2014_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_2015_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_2016_norm__of__real__addn,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X3: real,B2: num] :
( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X3 ) @ ( numeral_numeral @ A @ B2 ) ) )
= ( abs_abs @ real @ ( plus_plus @ real @ X3 @ ( numeral_numeral @ real @ B2 ) ) ) ) ) ).
% norm_of_real_addn
thf(fact_2017_neg__numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% neg_numeral_less_floor
thf(fact_2018_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_2019_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X3 @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_2020_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_2021_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X3: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_2022_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: num,N: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N ) @ A2 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_2023_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: num,N: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N ) @ A2 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_2024_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X3: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_2025_square__powr__half,axiom,
! [X3: real] :
( ( powr @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( abs_abs @ real @ X3 ) ) ).
% square_powr_half
thf(fact_2026_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_2027_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_2028_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_2029_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_2030_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_2031_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_2032_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_2033_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_2034_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_2035_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_2036_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_2037_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_2038_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_2039_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_2040_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_2041_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_2042_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_2043_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_2044_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_2045_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_2046_ceiling__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X: A] : ( uminus_uminus @ int @ ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ X ) ) ) ) ) ) ).
% ceiling_def
thf(fact_2047_floor__minus,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ X3 ) )
= ( uminus_uminus @ int @ ( archimedean_ceiling @ A @ X3 ) ) ) ) ).
% floor_minus
thf(fact_2048_ceiling__minus,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ X3 ) )
= ( uminus_uminus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) ) ) ) ).
% ceiling_minus
thf(fact_2049_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_2050_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_2051_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_2052_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_2053_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_2054_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_2055_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_2056_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_2057_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_2058_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,K: A,A2: A] :
( ( A4
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( uminus_uminus @ A @ A4 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_2059_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_2060_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_2061_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_2062_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_2063_real__minus__mult__self__le,axiom,
! [U: real,X3: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( times_times @ real @ U @ U ) ) @ ( times_times @ real @ X3 @ X3 ) ) ).
% real_minus_mult_self_le
thf(fact_2064_real__sqrt__minus,axiom,
! [X3: real] :
( ( sqrt @ ( uminus_uminus @ real @ X3 ) )
= ( uminus_uminus @ real @ ( sqrt @ X3 ) ) ) ).
% real_sqrt_minus
thf(fact_2065_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_2066_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_2067_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_2068_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_2069_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_2070_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_2071_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C2: A,B2: A,D3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ C2 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D3 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C2 @ D3 ) ) ) ) ) ).
% abs_mult_less
thf(fact_2072_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_2073_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_2074_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_2075_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_2076_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_2077_norm__mult__less,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X3: A,R2: real,Y3: A,S: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ R2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y3 ) @ S )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X3 @ Y3 ) ) @ ( times_times @ real @ R2 @ S ) ) ) ) ) ).
% norm_mult_less
thf(fact_2078_norm__mult__ineq,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X3 @ Y3 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ Y3 ) ) ) ) ).
% norm_mult_ineq
thf(fact_2079_norm__not__less__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A] :
~ ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( zero_zero @ real ) ) ) ).
% norm_not_less_zero
thf(fact_2080_norm__ge__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) ) ) ).
% norm_ge_zero
thf(fact_2081_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_2082_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_2083_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_2084_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_2085_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_2086_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_2087_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_2088_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_2089_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_2090_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_2091_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_2092_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_2093_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_2094_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_2095_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_2096_numeral__times__minus__swap,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [W: num,X3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ ( uminus_uminus @ A @ X3 ) )
= ( times_times @ A @ X3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_2097_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_2098_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_2099_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_2100_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_2101_norm__power,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X3: A,N: nat] :
( ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X3 @ N ) )
= ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ N ) ) ) ).
% norm_power
thf(fact_2102_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B6: A,K: A,B2: A,A2: A] :
( ( B6
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( minus_minus @ A @ A2 @ B6 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_2103_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_2104_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_2105_minus__real__def,axiom,
( ( minus_minus @ real )
= ( ^ [X: real,Y: real] : ( plus_plus @ real @ X @ ( uminus_uminus @ real @ Y ) ) ) ) ).
% minus_real_def
thf(fact_2106_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_2107_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X3: A] :
( ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ E ) )
=> ( X3
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_2108_abs__mult__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( times_times @ A @ ( abs_abs @ A @ Y3 ) @ X3 )
= ( abs_abs @ A @ ( times_times @ A @ Y3 @ X3 ) ) ) ) ) ).
% abs_mult_pos
thf(fact_2109_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_2110_abs__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( divide_divide @ A @ ( abs_abs @ A @ X3 ) @ Y3 )
= ( abs_abs @ A @ ( divide_divide @ A @ X3 @ Y3 ) ) ) ) ) ).
% abs_div_pos
thf(fact_2111_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_2112_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,A2: A,R2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ R2 )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ R2 ) @ X3 )
& ( ord_less_eq @ A @ X3 @ ( plus_plus @ A @ A2 @ R2 ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_2113_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D3 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_2114_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_2115_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,A2: A,R2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ R2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ A2 @ R2 ) @ X3 )
& ( ord_less @ A @ X3 @ ( plus_plus @ A @ A2 @ R2 ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_2116_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_2117_power__eq__imp__eq__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W: A,N: nat,Z: A] :
( ( ( power_power @ A @ W @ N )
= ( power_power @ A @ Z @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( real_V7770717601297561774m_norm @ A @ W )
= ( real_V7770717601297561774m_norm @ A @ Z ) ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_2118_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_2119_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_2120_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_2121_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_2122_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_2123_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_2124_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_2125_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_2126_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_2127_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_2128_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_2129_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_2130_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_2131_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_2132_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_2133_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_2134_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_2135_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_2136_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_2137_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_2138_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_2139_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_2140_norm__triangle__lt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,Y3: A,E2: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ Y3 ) ) @ E2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y3 ) ) @ E2 ) ) ) ).
% norm_triangle_lt
thf(fact_2141_norm__add__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,R2: real,Y3: A,S: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ R2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y3 ) @ S )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y3 ) ) @ ( plus_plus @ real @ R2 @ S ) ) ) ) ) ).
% norm_add_less
thf(fact_2142_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_2143_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_2144_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_2145_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_2146_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_2147_norm__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,Y3: A,E2: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ Y3 ) ) @ E2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y3 ) ) @ E2 ) ) ) ).
% norm_triangle_le
thf(fact_2148_norm__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y3 ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ Y3 ) ) ) ) ).
% norm_triangle_ineq
thf(fact_2149_norm__triangle__mono,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,R2: real,B2: A,S: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ R2 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ S )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ real @ R2 @ S ) ) ) ) ) ).
% norm_triangle_mono
thf(fact_2150_norm__power__ineq,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X3: A,N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X3 @ N ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ N ) ) ) ).
% norm_power_ineq
thf(fact_2151_norm__diff__triangle__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,Y3: A,E1: real,Z: A,E22: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ Y3 ) ) @ E1 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y3 @ Z ) ) @ E22 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ Z ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_2152_norm__triangle__sub,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ Y3 ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ Y3 ) ) ) ) ) ).
% norm_triangle_sub
thf(fact_2153_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_2154_norm__diff__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,Y3: A,E1: real,Z: A,E22: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ Y3 ) ) @ E1 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y3 @ Z ) ) @ E22 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ Z ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_2155_norm__triangle__le__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A,Y3: A,E2: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ Y3 ) ) @ E2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ Y3 ) ) @ E2 ) ) ) ).
% norm_triangle_le_diff
thf(fact_2156_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_2157_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_2158_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_2159_power__minus__Bit0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: A,K: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) )
= ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ) ) ).
% power_minus_Bit0
thf(fact_2160_lemma__interval__lt,axiom,
! [A2: real,X3: real,B2: real] :
( ( ord_less @ real @ A2 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [Y5: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y5 ) ) @ D6 )
=> ( ( ord_less @ real @ A2 @ Y5 )
& ( ord_less @ real @ Y5 @ B2 ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_2161_sin__bound__lemma,axiom,
! [X3: real,Y3: real,U: real,V2: real] :
( ( X3 = Y3 )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ U ) @ V2 )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( plus_plus @ real @ X3 @ U ) @ Y3 ) ) @ V2 ) ) ) ).
% sin_bound_lemma
thf(fact_2162_real__0__less__add__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X3 @ Y3 ) )
= ( ord_less @ real @ ( uminus_uminus @ real @ X3 ) @ Y3 ) ) ).
% real_0_less_add_iff
thf(fact_2163_real__add__less__0__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( plus_plus @ real @ X3 @ Y3 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ Y3 @ ( uminus_uminus @ real @ X3 ) ) ) ).
% real_add_less_0_iff
thf(fact_2164_real__0__le__add__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X3 @ Y3 ) )
= ( ord_less_eq @ real @ ( uminus_uminus @ real @ X3 ) @ Y3 ) ) ).
% real_0_le_add_iff
thf(fact_2165_real__add__le__0__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ X3 @ Y3 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ Y3 @ ( uminus_uminus @ real @ X3 ) ) ) ).
% real_add_le_0_iff
thf(fact_2166_tanh__real__gt__neg1,axiom,
! [X3: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( tanh @ real @ X3 ) ) ).
% tanh_real_gt_neg1
thf(fact_2167_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X3 ) ) ) ) ).
% abs_add_one_gt_zero
thf(fact_2168_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X3 )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ) ).
% of_int_leD
thf(fact_2169_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X3 )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X3 ) ) ) ) ).
% of_int_lessD
thf(fact_2170_power__eq__1__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W: A,N: nat] :
( ( ( power_power @ A @ W @ N )
= ( one_one @ A ) )
=> ( ( ( real_V7770717601297561774m_norm @ A @ W )
= ( one_one @ real ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% power_eq_1_iff
thf(fact_2171_norm__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D3 ) ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_2172_norm__less__p1,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X3: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ ( real_V7770717601297561774m_norm @ A @ X3 ) ) @ ( one_one @ A ) ) ) ) ) ).
% norm_less_p1
thf(fact_2173_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_2174_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_2175_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_2176_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_2177_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_2178_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_2179_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ( divide_divide @ A @ B2 @ C2 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_2180_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_2181_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X3: A,Y3: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X3 @ Z ) ) @ Y3 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X3 ) @ ( times_times @ A @ Y3 @ Z ) ) @ Z ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_2182_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z ) ) @ B2 )
= B2 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z ) ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_2183_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X3: A,Y3: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X3 @ Z ) ) @ Y3 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X3 ) @ ( times_times @ A @ Y3 @ Z ) ) @ Z ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_2184_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ A2 @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_2185_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z ) ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_2186_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_2187_power2__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [X3: A,Y3: A] :
( ( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( X3 = Y3 )
| ( X3
= ( uminus_uminus @ A @ Y3 ) ) ) ) ) ).
% power2_eq_iff
thf(fact_2188_lemma__interval,axiom,
! [A2: real,X3: real,B2: real] :
( ( ord_less @ real @ A2 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [Y5: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y5 ) ) @ D6 )
=> ( ( ord_less_eq @ real @ A2 @ Y5 )
& ( ord_less_eq @ real @ Y5 @ B2 ) ) ) ) ) ) ).
% lemma_interval
thf(fact_2189_round__diff__minimal,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: A,M: int] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ Z @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ Z ) ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ Z @ ( ring_1_of_int @ A @ M ) ) ) ) ) ).
% round_diff_minimal
thf(fact_2190_abs__le__square__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ ( abs_abs @ A @ Y3 ) )
= ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_2191_abs__square__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( abs_abs @ A @ X3 )
= ( one_one @ A ) ) ) ) ).
% abs_square_eq_1
thf(fact_2192_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_2193_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_2194_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_2195_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_2196_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_2197_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_2198_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_2199_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ 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 @ W ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_2200_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ 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 @ W ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_2201_lemma__NBseq__def2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X8: A > B] :
( ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N3 ) ) @ K6 ) ) )
= ( ? [N6: nat] :
! [N3: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) ) ) ) ).
% lemma_NBseq_def2
thf(fact_2202_lemma__NBseq__def,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X8: A > B] :
( ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N3 ) ) @ K6 ) ) )
= ( ? [N6: nat] :
! [N3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) ) ) ) ).
% lemma_NBseq_def
thf(fact_2203_norm__power__diff,axiom,
! [A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z: A,W: A,M: nat] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ W ) @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( power_power @ A @ Z @ M ) @ ( power_power @ A @ W @ M ) ) ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Z @ W ) ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_2204_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_2205_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_2206_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_2207_realpow__square__minus__le,axiom,
! [U: real,X3: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( power_power @ real @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% realpow_square_minus_le
thf(fact_2208_powr__neg__one,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( powr @ real @ X3 @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ X3 ) ) ) ).
% powr_neg_one
thf(fact_2209_Bernoulli__inequality,axiom,
! [X3: real,N: nat] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X3 ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_2210_ln__add__one__self__le__self2,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) ) @ X3 ) ) ).
% ln_add_one_self_le_self2
thf(fact_2211_power2__le__iff__abs__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ Y3 ) ) ) ) ).
% power2_le_iff_abs_le
thf(fact_2212_abs__square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ ( one_one @ A ) ) ) ) ).
% abs_square_le_1
thf(fact_2213_abs__square__less__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less @ A @ ( abs_abs @ A @ X3 ) @ ( one_one @ A ) ) ) ) ).
% abs_square_less_1
thf(fact_2214_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_2215_square__norm__one,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X3: A] :
( ( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ X3 )
= ( one_one @ real ) ) ) ) ).
% square_norm_one
thf(fact_2216_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( 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 @ W ) ) @ 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 @ W ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_2217_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ 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 @ W ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_2218_sqrt__ge__absD,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( sqrt @ Y3 ) )
=> ( ord_less_eq @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y3 ) ) ).
% sqrt_ge_absD
thf(fact_2219_square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% square_le_1
thf(fact_2220_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) ) @ X3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_2221_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_2222_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_2223_ln__one__minus__pos__upper__bound,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X3 ) ) @ ( uminus_uminus @ real @ X3 ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_2224_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X3: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X3 @ ( 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 @ X3 @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_2225_log__minus__eq__powr,axiom,
! [B2: real,X3: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( minus_minus @ real @ ( log @ B2 @ X3 ) @ Y3 )
= ( log @ B2 @ ( times_times @ real @ X3 @ ( powr @ real @ B2 @ ( uminus_uminus @ real @ Y3 ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_2226_real__sqrt__ge__abs1,axiom,
! [X3: real,Y3: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs1
thf(fact_2227_real__sqrt__ge__abs2,axiom,
! [Y3: real,X3: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ Y3 ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs2
thf(fact_2228_sqrt__sum__squares__le__sum__abs,axiom,
! [X3: real,Y3: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( abs_abs @ real @ X3 ) @ ( abs_abs @ real @ Y3 ) ) ) ).
% sqrt_sum_squares_le_sum_abs
thf(fact_2229_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_2230_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_2231_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_2232_powr__neg__numeral,axiom,
! [X3: real,N: num] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( powr @ real @ X3 @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ N ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% powr_neg_numeral
thf(fact_2233_of__int__round__abs__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) @ X3 ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_int_round_abs_le
thf(fact_2234_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,N: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ ( ring_1_of_int @ A @ N ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X3 )
= N ) ) ) ).
% round_unique'
thf(fact_2235_cos__x__y__le__one,axiom,
! [X3: real,Y3: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( divide_divide @ real @ X3 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( one_one @ real ) ) ).
% cos_x_y_le_one
thf(fact_2236_real__sqrt__sum__squares__less,axiom,
! [X3: real,U: real,Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y3 ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ).
% real_sqrt_sum_squares_less
thf(fact_2237_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) ) @ X3 ) ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_2238_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X3: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less @ A @ ( times_times @ A @ X3 @ Z ) @ ( times_times @ A @ Y3 @ Z ) )
= ( ord_less @ A @ X3 @ Y3 ) ) ) ) ).
% mult_less_iff1
thf(fact_2239_abs__sqrt__wlog,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A > A > $o,X3: A] :
( ! [X4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 )
=> ( P @ X4 @ ( power_power @ A @ X4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ ( abs_abs @ A @ X3 ) @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_2240__C5_Ohyps_C_I9_J,axiom,
( ( mi != ma )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( ( ( vEBT_VEBT_high @ ma @ na )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I2 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
& ! [X6: nat] :
( ( ( ( vEBT_VEBT_high @ X6 @ na )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I2 ) @ ( vEBT_VEBT_low @ X6 @ na ) ) )
=> ( ( ord_less @ nat @ mi @ X6 )
& ( ord_less_eq @ nat @ X6 @ ma ) ) ) ) ) ) ).
% "5.hyps"(9)
thf(fact_2241_compl__less__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y3 ) )
= ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% compl_less_compl_iff
thf(fact_2242_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% compl_le_compl_iff
thf(fact_2243_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X: nat,N3: nat] : ( modulo_modulo @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% low_def
thf(fact_2244_arctan__double,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ X3 ) )
= ( arctan @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arctan_double
thf(fact_2245_pochhammer__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z ) @ ( 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 @ Z @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N ) ) ) ) ).
% pochhammer_double
thf(fact_2246_mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_0
thf(fact_2247_mod__by__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% mod_by_0
thf(fact_2248_mod__self,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_self
thf(fact_2249_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_2250_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( modulo_modulo @ nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_2251_arctan__eq__zero__iff,axiom,
! [X3: real] :
( ( ( arctan @ X3 )
= ( zero_zero @ real ) )
= ( X3
= ( zero_zero @ real ) ) ) ).
% arctan_eq_zero_iff
thf(fact_2252_arctan__zero__zero,axiom,
( ( arctan @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% arctan_zero_zero
thf(fact_2253_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_2254_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_2255_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_2256_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_2257_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_2258_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_2259_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_2260_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_2261_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_2262_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_2263_pochhammer__Suc0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% pochhammer_Suc0
thf(fact_2264_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_2265_zero__less__arctan__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( arctan @ X3 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ).
% zero_less_arctan_iff
thf(fact_2266_arctan__less__zero__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( arctan @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ).
% arctan_less_zero_iff
thf(fact_2267_zero__le__arctan__iff,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arctan @ X3 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) ) ).
% zero_le_arctan_iff
thf(fact_2268_arctan__le__zero__iff,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( arctan @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) ) ) ).
% arctan_le_zero_iff
thf(fact_2269_zdvd1__eq,axiom,
! [X3: int] :
( ( dvd_dvd @ int @ X3 @ ( one_one @ int ) )
= ( ( abs_abs @ int @ X3 )
= ( one_one @ int ) ) ) ).
% zdvd1_eq
thf(fact_2270__C5_Ohyps_C_I5_J,axiom,
! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I2 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ summary @ I2 ) ) ) ).
% "5.hyps"(5)
thf(fact_2271_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_2272_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_2273_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( zero_zero @ nat ) ) ).
% nat_neg_numeral
thf(fact_2274_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ nat ) ) ).
% nat_zminus_int
thf(fact_2275_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less @ int @ ( abs_abs @ int @ Z ) @ ( one_one @ int ) )
= ( Z
= ( zero_zero @ int ) ) ) ).
% zabs_less_one_iff
thf(fact_2276_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_2277_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_2278_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_2279_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_2280_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_2281_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_2282_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_2283_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_2284_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_2285_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_2286_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_2287_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_2288_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_2289_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_2290_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_2291_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_2292_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_2293_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_2294__C5_Ohyps_C_I6_J,axiom,
( ( mi = ma )
=> ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ treeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) ) ).
% "5.hyps"(6)
thf(fact_2295_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_2296_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_2297_pochhammer__of__int,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X3: int,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( ring_1_of_int @ A @ X3 ) @ N )
= ( ring_1_of_int @ A @ ( comm_s3205402744901411588hammer @ int @ X3 @ N ) ) ) ) ).
% pochhammer_of_int
thf(fact_2298_pochhammer__of__nat,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X3: nat,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( semiring_1_of_nat @ A @ X3 ) @ N )
= ( semiring_1_of_nat @ A @ ( comm_s3205402744901411588hammer @ nat @ X3 @ N ) ) ) ) ).
% pochhammer_of_nat
thf(fact_2299_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_2300_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_2301_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_2302_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_2303_uminus__int__code_I1_J,axiom,
( ( uminus_uminus @ int @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% uminus_int_code(1)
thf(fact_2304_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_2305_arctan__less__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) )
= ( ord_less @ real @ X3 @ Y3 ) ) ).
% arctan_less_iff
thf(fact_2306_arctan__monotone,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ X3 @ Y3 )
=> ( ord_less @ real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) ) ) ).
% arctan_monotone
thf(fact_2307_arctan__le__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) )
= ( ord_less_eq @ real @ X3 @ Y3 ) ) ).
% arctan_le_iff
thf(fact_2308_arctan__monotone_H,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ X3 @ Y3 )
=> ( ord_less_eq @ real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) ) ) ).
% arctan_monotone'
thf(fact_2309_int__cases2,axiom,
! [Z: int] :
( ! [N2: nat] :
( Z
!= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ! [N2: nat] :
( Z
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_2310_uminus__dvd__conv_I2_J,axiom,
( ( dvd_dvd @ int )
= ( ^ [D4: int,T3: int] : ( dvd_dvd @ int @ D4 @ ( uminus_uminus @ int @ T3 ) ) ) ) ).
% uminus_dvd_conv(2)
thf(fact_2311_uminus__dvd__conv_I1_J,axiom,
( ( dvd_dvd @ int )
= ( ^ [D4: int] : ( dvd_dvd @ int @ ( uminus_uminus @ int @ D4 ) ) ) ) ).
% uminus_dvd_conv(1)
thf(fact_2312_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_2313_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_2314_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_2315_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_2316_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_2317_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_2318_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_2319_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A5: A,B4: A] :
( ( modulo_modulo @ A @ B4 @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_2320_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_2321_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_2322_mod__induct,axiom,
! [P: nat > $o,N: nat,P6: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less @ nat @ N @ P6 )
=> ( ( ord_less @ nat @ M @ P6 )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ P6 )
=> ( ( P @ N2 )
=> ( P @ ( modulo_modulo @ nat @ ( suc @ N2 ) @ P6 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_2323_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_2324_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_2325_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_2326_mod__eq__0D,axiom,
! [M: nat,D3: nat] :
( ( ( modulo_modulo @ nat @ M @ D3 )
= ( zero_zero @ nat ) )
=> ? [Q4: nat] :
( M
= ( times_times @ nat @ D3 @ Q4 ) ) ) ).
% mod_eq_0D
thf(fact_2327_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_2328_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_2329_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_2330_pochhammer__pos,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X3 @ N ) ) ) ) ).
% pochhammer_pos
thf(fact_2331_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_2332_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_2333_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_2334_int__cases,axiom,
! [Z: int] :
( ! [N2: nat] :
( Z
!= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ! [N2: nat] :
( Z
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_2335_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N2: nat] : ( P @ ( semiring_1_of_nat @ int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_2336_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_2337_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_2338_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_2339_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_2340_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_2341_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_2342_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_2343_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_2344_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_2345_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_2346_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_2347_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_2348_div__less__mono,axiom,
! [A4: nat,B6: nat,N: nat] :
( ( ord_less @ nat @ A4 @ B6 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( modulo_modulo @ nat @ A4 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B6 @ N )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A4 @ N ) @ ( divide_divide @ nat @ B6 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_2349_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_2350_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_2351_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_2352_nat__mod__eq__lemma,axiom,
! [X3: nat,N: nat,Y3: nat] :
( ( ( modulo_modulo @ nat @ X3 @ N )
= ( modulo_modulo @ nat @ Y3 @ N ) )
=> ( ( ord_less_eq @ nat @ Y3 @ X3 )
=> ? [Q4: nat] :
( X3
= ( plus_plus @ nat @ Y3 @ ( times_times @ nat @ N @ Q4 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_2353_div__mod__decomp,axiom,
! [A4: nat,N: nat] :
( A4
= ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A4 @ N ) @ N ) @ ( modulo_modulo @ nat @ A4 @ N ) ) ) ).
% div_mod_decomp
thf(fact_2354_pochhammer__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X3 @ N ) ) ) ) ).
% pochhammer_nonneg
thf(fact_2355_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_2356_dvd__imp__le__int,axiom,
! [I: int,D3: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ D3 @ I )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ D3 ) @ ( abs_abs @ int @ I ) ) ) ) ).
% dvd_imp_le_int
thf(fact_2357_nat__abs__mult__distrib,axiom,
! [W: int,Z: int] :
( ( nat2 @ ( abs_abs @ int @ ( times_times @ int @ W @ Z ) ) )
= ( times_times @ nat @ ( nat2 @ ( abs_abs @ int @ W ) ) @ ( nat2 @ ( abs_abs @ int @ Z ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_2358_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_2359_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_2360_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_2361_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_2362_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_2363_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_2364_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_2365_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_2366_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_2367_mod__nat__eqI,axiom,
! [R2: nat,N: nat,M: nat] :
( ( ord_less @ nat @ R2 @ N )
=> ( ( ord_less_eq @ nat @ R2 @ M )
=> ( ( dvd_dvd @ nat @ N @ ( minus_minus @ nat @ M @ R2 ) )
=> ( ( modulo_modulo @ nat @ M @ N )
= R2 ) ) ) ) ).
% mod_nat_eqI
thf(fact_2368_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_2369_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_2370_real__of__nat__div__aux,axiom,
! [X3: nat,D3: nat] :
( ( divide_divide @ real @ ( semiring_1_of_nat @ real @ X3 ) @ ( semiring_1_of_nat @ real @ D3 ) )
= ( plus_plus @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ X3 @ D3 ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ ( modulo_modulo @ nat @ X3 @ D3 ) ) @ ( semiring_1_of_nat @ real @ D3 ) ) ) ) ).
% real_of_nat_div_aux
thf(fact_2371_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_2372_pochhammer__rec_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z @ ( suc @ N ) )
= ( times_times @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ Z @ N ) ) ) ) ).
% pochhammer_rec'
thf(fact_2373_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_2374_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_2375_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_2376_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_2377_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_2378_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z: A,N: nat,M: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z @ ( plus_plus @ nat @ N @ M ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ N ) ) @ M ) ) ) ) ).
% pochhammer_product'
thf(fact_2379_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_2380_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_2381_negD,axiom,
! [X3: int] :
( ( ord_less @ int @ X3 @ ( zero_zero @ int ) )
=> ? [N2: nat] :
( X3
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_2382_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_2383_verit__less__mono__div__int2,axiom,
! [A4: int,B6: int,N: int] :
( ( ord_less_eq @ int @ A4 @ B6 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ B6 @ N ) @ ( divide_divide @ int @ A4 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_2384_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_2385_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_2386_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_2387_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_2388_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_2389_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_2390_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_2391_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_2392_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_2393_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_2394_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_2395_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_2396_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N: nat,Z: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ Z @ N )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ M ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ M ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_2397_nat__mult__distrib__neg,axiom,
! [Z: int,Z5: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z @ Z5 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z5 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_2398_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_2399_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_2400_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_2401_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_2402_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_2403_verit__le__mono__div,axiom,
! [A4: nat,B6: nat,N: nat] :
( ( ord_less @ nat @ A4 @ B6 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A4 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B6 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B6 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_2404_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F2: nat > int,K: int] :
( ! [I3: nat] :
( ( ( ord_less_eq @ nat @ M @ I3 )
& ( ord_less @ nat @ I3 @ N ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ int @ ( F2 @ M ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ M @ I3 )
& ( ord_less_eq @ nat @ I3 @ N )
& ( ( F2 @ I3 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_2405_decr__lemma,axiom,
! [D3: int,X3: int,Z: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ord_less @ int @ ( minus_minus @ int @ X3 @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X3 @ Z ) ) @ ( one_one @ int ) ) @ D3 ) ) @ Z ) ) ).
% decr_lemma
thf(fact_2406_incr__lemma,axiom,
! [D3: int,Z: int,X3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ord_less @ int @ Z @ ( plus_plus @ int @ X3 @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X3 @ Z ) ) @ ( one_one @ int ) ) @ D3 ) ) ) ) ).
% incr_lemma
thf(fact_2407_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [R2: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ R2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R2 ) @ K ) )
= ( times_times @ A @ R2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R2 ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_absorb_comp
thf(fact_2408_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_2409_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_2410_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_2411_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_2412_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_2413_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ X3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y3 ) ) ) ).
% compl_le_swap2
thf(fact_2414_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ ( uminus_uminus @ A @ X3 ) )
=> ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% compl_le_swap1
thf(fact_2415_compl__mono,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% compl_mono
thf(fact_2416_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less @ A @ Y3 @ ( uminus_uminus @ A @ X3 ) )
=> ( ord_less @ A @ X3 @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% compl_less_swap1
thf(fact_2417_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y3 ) @ X3 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X3 ) @ Y3 ) ) ) ).
% compl_less_swap2
thf(fact_2418_nat__ivt__aux,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
& ( ( F2 @ I3 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_2419_complex__mod__minus__le__complex__mod,axiom,
! [X3: complex] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( real_V7770717601297561774m_norm @ complex @ X3 ) ) @ ( real_V7770717601297561774m_norm @ complex @ X3 ) ) ).
% complex_mod_minus_le_complex_mod
thf(fact_2420_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_2421_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_2422_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_2423_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_2424_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: A,X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ( modulo_modulo @ A @ X3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ A @ X3 @ M ) )
| ( ( modulo_modulo @ A @ X3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X3 @ M ) @ M ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_2425_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_2426_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_2427_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_2428_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_2429_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_2430_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_2431_nat0__intermed__int__val,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) @ ( F2 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
& ( ( F2 @ I3 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_2432_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_2433_arctan__add,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y3 ) @ ( one_one @ real ) )
=> ( ( plus_plus @ real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) )
= ( arctan @ ( divide_divide @ real @ ( plus_plus @ real @ X3 @ Y3 ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( times_times @ real @ X3 @ Y3 ) ) ) ) ) ) ) ).
% arctan_add
thf(fact_2434_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_2435_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_2436_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_2437_powr__int,axiom,
! [X3: real,I: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ I ) )
= ( power_power @ real @ X3 @ ( nat2 @ I ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ I ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( nat2 @ ( uminus_uminus @ int @ I ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_2438_arctan__half,axiom,
( arctan
= ( ^ [X: real] : ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ ( divide_divide @ real @ X @ ( plus_plus @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% arctan_half
thf(fact_2439_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N3: nat,TreeList: list @ vEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ N3 ) ) @ ( vEBT_VEBT_low @ X @ N3 ) ) ) ) ).
% in_children_def
thf(fact_2440_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_2441_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_2442_nth__equalityI,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I3 )
= ( nth @ A @ Ys @ I3 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_2443_inthall,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o,N: nat] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X4 ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ N ) ) ) ) ).
% inthall
thf(fact_2444_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_2445_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_2446_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_2447_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_2448_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_2449_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_2450_zmod__numeral__Bit0,axiom,
! [V2: num,W: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit0 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W ) ) ) ) ).
% zmod_numeral_Bit0
thf(fact_2451_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_2452_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_2453_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_2454__C5_Ohyps_C_I1_J,axiom,
vEBT_invar_vebt @ summary @ m ).
% "5.hyps"(1)
thf(fact_2455_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_2456_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_2457_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_2458_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_2459_length__pos__if__in__set,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_2460_all__set__conv__all__nth,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( P @ X ) ) )
= ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_2461_all__nth__imp__all__set,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o,X3: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ I3 ) ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_2462_in__set__conv__nth,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ( nth @ A @ Xs2 @ I4 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_2463_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs2: list @ A,P: A > $o] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X4 ) )
=> ( P @ ( nth @ A @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_2464_nth__mem,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( member @ A @ ( nth @ A @ Xs2 @ N ) @ ( set2 @ A @ Xs2 ) ) ) ).
% nth_mem
thf(fact_2465_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_2466_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_2467_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_2468_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_2469_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_2470_zmod__eq__0__iff,axiom,
! [M: int,D3: int] :
( ( ( modulo_modulo @ int @ M @ D3 )
= ( zero_zero @ int ) )
= ( ? [Q5: int] :
( M
= ( times_times @ int @ D3 @ Q5 ) ) ) ) ).
% zmod_eq_0_iff
thf(fact_2471_zmod__eq__0D,axiom,
! [M: int,D3: int] :
( ( ( modulo_modulo @ int @ M @ D3 )
= ( zero_zero @ int ) )
=> ? [Q4: int] :
( M
= ( times_times @ int @ D3 @ Q4 ) ) ) ).
% zmod_eq_0D
thf(fact_2472_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_2473_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_2474_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_2475_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_2476_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_2477_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_2478_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_2479_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_2480_zdiv__mono__strict,axiom,
! [A4: int,B6: int,N: int] :
( ( ord_less @ int @ A4 @ B6 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ( modulo_modulo @ int @ A4 @ N )
= ( zero_zero @ int ) )
=> ( ( ( modulo_modulo @ int @ B6 @ N )
= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( divide_divide @ int @ A4 @ N ) @ ( divide_divide @ int @ B6 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_2481_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_2482_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_2483_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_2484_div__mod__decomp__int,axiom,
! [A4: int,N: int] :
( A4
= ( plus_plus @ int @ ( times_times @ int @ ( divide_divide @ int @ A4 @ N ) @ N ) @ ( modulo_modulo @ int @ A4 @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_2485_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_2486_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_2487_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_2488_nat__mod__distrib,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( nat2 @ ( modulo_modulo @ int @ X3 @ Y3 ) )
= ( modulo_modulo @ nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_2489_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_2490_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_2491_real__of__int__div__aux,axiom,
! [X3: int,D3: int] :
( ( divide_divide @ real @ ( ring_1_of_int @ real @ X3 ) @ ( ring_1_of_int @ real @ D3 ) )
= ( plus_plus @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ X3 @ D3 ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( modulo_modulo @ int @ X3 @ D3 ) ) @ ( ring_1_of_int @ real @ D3 ) ) ) ) ).
% real_of_int_div_aux
thf(fact_2492_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_2493_binomial__antimono,axiom,
! [K: nat,K7: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ K )
=> ( ( ord_less_eq @ nat @ K7 @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K7 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_antimono
thf(fact_2494_binomial__mono,axiom,
! [K: nat,K7: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K7 ) ) ) ) ).
% binomial_mono
thf(fact_2495_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_2496_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_2497_int__mod__neg__eq,axiom,
! [A2: int,B2: int,Q3: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R2 )
=> ( ( modulo_modulo @ int @ A2 @ B2 )
= R2 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_2498_int__mod__pos__eq,axiom,
! [A2: int,B2: int,Q3: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ( modulo_modulo @ int @ A2 @ B2 )
= R2 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_2499_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_2500_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_2501_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_2502_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_2503_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_2504_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_2505_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_2506_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_2507_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_2508_binomial__strict__mono,axiom,
! [K: nat,K7: nat,N: nat] :
( ( ord_less @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K7 ) ) ) ) ).
% binomial_strict_mono
thf(fact_2509_binomial__strict__antimono,axiom,
! [K: nat,K7: nat,N: nat] :
( ( ord_less @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ N @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) )
=> ( ( ord_less_eq @ nat @ K7 @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K7 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_2510_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_2511_verit__le__mono__div__int,axiom,
! [A4: int,B6: int,N: int] :
( ( ord_less @ int @ A4 @ B6 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A4 @ N )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B6 @ N )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B6 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_2512_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_2513_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_2514_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_2515_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_2516_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_2517_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_2518_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_2519_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs2: list @ A] :
( ! [Xs3: list @ A] :
( ! [Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_2520_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_2521_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_2522_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_2523_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_2524_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_2525_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_2526_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_2527_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_2528_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y6: list @ A,Z2: list @ A] : Y6 = Z2 )
= ( ^ [Xs: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I4 )
= ( nth @ A @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_2529_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ? [X9: A] : ( P @ I4 @ X9 ) ) )
= ( ? [Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= K )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ( P @ I4 @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_2530_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_2531_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_2532_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_2533_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_2534_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) )
= ( zero_zero @ nat ) ) ).
% binomial_0_Suc
thf(fact_2535_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ ( zero_zero @ nat ) ) )
= N ) ).
% binomial_1
thf(fact_2536_valid__0__not,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_0_not
thf(fact_2537_valid__tree__deg__neq__0,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_tree_deg_neq_0
thf(fact_2538_deg__not__0,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% deg_not_0
thf(fact_2539_set__n__deg__not__0,axiom,
! [TreeList2: list @ vEBT_VEBT,N: nat,M: nat] :
( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ord_less_eq @ nat @ ( one_one @ nat ) @ N ) ) ) ).
% set_n_deg_not_0
thf(fact_2540_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( ( binomial @ N @ K )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_2541_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_2542_binomial__le__pow,axiom,
! [R2: nat,N: nat] :
( ( ord_less_eq @ nat @ R2 @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ R2 ) @ ( power_power @ nat @ N @ R2 ) ) ) ).
% binomial_le_pow
thf(fact_2543_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_2544_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_2545_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_2546_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_2547_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_2548_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_2549_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_2550_valid__insert__both__member__options__pres,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Y3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ Y3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X3 )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ Y3 ) @ X3 ) ) ) ) ) ).
% valid_insert_both_member_options_pres
thf(fact_2551_valid__insert__both__member__options__add,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ X3 ) @ X3 ) ) ) ).
% valid_insert_both_member_options_add
thf(fact_2552_height__double__log__univ__size,axiom,
! [U: real,Deg: nat,T2: vEBT_VEBT] :
( ( U
= ( 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 ) ) @ U ) ) ) ) ) ) ).
% height_double_log_univ_size
thf(fact_2553_heigt__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 ) ) ) ) ) ).
% heigt_uplog_rel
thf(fact_2554_helpypredd,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Y3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X3 )
= ( some @ nat @ Y3 ) )
=> ( ord_less @ nat @ Y3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% helpypredd
thf(fact_2555_helpyd,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Y3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X3 )
= ( some @ nat @ Y3 ) )
=> ( ord_less @ nat @ Y3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% helpyd
thf(fact_2556_two__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 ) ) ) ).
% two_powr_height_bound_deg
thf(fact_2557_both__member__options__ding,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X3 ) ) ) ) ).
% both_member_options_ding
thf(fact_2558_post__member__pre__member,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Y3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ Y3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T2 @ X3 ) @ Y3 )
=> ( ( vEBT_vebt_member @ T2 @ Y3 )
| ( X3 = Y3 ) ) ) ) ) ) ).
% post_member_pre_member
thf(fact_2559_buildup__gives__valid,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% buildup_gives_valid
thf(fact_2560_member__bound,axiom,
! [Tree: vEBT_VEBT,X3: nat,N: nat] :
( ( vEBT_vebt_member @ Tree @ X3 )
=> ( ( vEBT_invar_vebt @ Tree @ N )
=> ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% member_bound
thf(fact_2561_misiz,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 ) ) ) ) ).
% misiz
thf(fact_2562_height__compose__child,axiom,
! [T2: vEBT_VEBT,TreeList2: list @ vEBT_VEBT,Info: option @ ( product_prod @ nat @ nat ),Deg: nat,Summary: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ T2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% height_compose_child
thf(fact_2563_deg__deg__n,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
=> ( Deg = N ) ) ).
% deg_deg_n
thf(fact_2564_deg__SUcn__Node,axiom,
! [Tree: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
=> ? [Info2: option @ ( product_prod @ nat @ nat ),TreeList3: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( Tree
= ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList3 @ S3 ) ) ) ).
% deg_SUcn_Node
thf(fact_2565_valid__member__both__member__options,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X3 )
=> ( vEBT_vebt_member @ T2 @ X3 ) ) ) ).
% valid_member_both_member_options
thf(fact_2566_both__member__options__equiv__member,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X3 )
= ( vEBT_vebt_member @ T2 @ X3 ) ) ) ).
% both_member_options_equiv_member
thf(fact_2567_mint__member,axiom,
! [T2: vEBT_VEBT,N: nat,Maxi: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ Maxi ) )
=> ( vEBT_vebt_member @ T2 @ Maxi ) ) ) ).
% mint_member
thf(fact_2568_height__compose__summary,axiom,
! [Summary: vEBT_VEBT,Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ Summary ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) ) ) ).
% height_compose_summary
thf(fact_2569_mint__corr__help,axiom,
! [T2: vEBT_VEBT,N: nat,Mini: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ Mini ) )
=> ( ( vEBT_vebt_member @ T2 @ X3 )
=> ( ord_less_eq @ nat @ Mini @ X3 ) ) ) ) ).
% mint_corr_help
thf(fact_2570_member__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_vebt_member @ T2 @ X3 )
= ( member @ nat @ X3 @ ( vEBT_set_vebt @ T2 ) ) ) ) ).
% member_correct
thf(fact_2571_mint__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X3 ) )
=> ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 ) ) ) ).
% mint_corr
thf(fact_2572_mint__sound,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 )
=> ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X3 ) ) ) ) ).
% mint_sound
thf(fact_2573_both__member__options__from__chilf__to__complete__tree,axiom,
! [X3: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_2574_member__inv,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
& ( ( X3 = Mi )
| ( X3 = Ma )
| ( ( ord_less @ nat @ X3 @ Ma )
& ( ord_less @ nat @ Mi @ X3 )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% member_inv
thf(fact_2575_mintlistlength,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( Mi != Ma )
=> ( ( ord_less @ nat @ Mi @ Ma )
& ? [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 ) ) ) ) ) ) ) ) ) ) ).
% mintlistlength
thf(fact_2576_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
| ( X3 = Mi )
| ( X3 = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_2577_set__vebt__set__vebt_H__valid,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_set_vebt @ T2 )
= ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).
% set_vebt_set_vebt'_valid
thf(fact_2578_mi__eq__ma__no__ch,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
=> ( ( Mi = Ma )
=> ( ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) )
& ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% mi_eq_ma_no_ch
thf(fact_2579_insert__simp__mima,axiom,
! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X3 = Mi )
| ( X3 = Ma ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% insert_simp_mima
thf(fact_2580_mi__ma__2__deg,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
& ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) ) ) ).
% mi_ma_2_deg
thf(fact_2581_pred__max,axiom,
! [Deg: nat,Ma: nat,X3: nat,Mi: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less @ nat @ Ma @ X3 )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( some @ nat @ Ma ) ) ) ) ).
% pred_max
thf(fact_2582_succ__min,axiom,
! [Deg: nat,X3: nat,Mi: nat,Ma: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less @ nat @ X3 @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( some @ nat @ Mi ) ) ) ) ).
% succ_min
thf(fact_2583_succ__member,axiom,
! [T2: vEBT_VEBT,X3: nat,Y3: nat] :
( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 @ Y3 )
= ( ( vEBT_vebt_member @ T2 @ Y3 )
& ( ord_less @ nat @ X3 @ Y3 )
& ! [Z6: nat] :
( ( ( vEBT_vebt_member @ T2 @ Z6 )
& ( ord_less @ nat @ X3 @ Z6 ) )
=> ( ord_less_eq @ nat @ Y3 @ Z6 ) ) ) ) ).
% succ_member
thf(fact_2584_pred__member,axiom,
! [T2: vEBT_VEBT,X3: nat,Y3: nat] :
( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 @ Y3 )
= ( ( vEBT_vebt_member @ T2 @ Y3 )
& ( ord_less @ nat @ Y3 @ X3 )
& ! [Z6: nat] :
( ( ( vEBT_vebt_member @ T2 @ Z6 )
& ( ord_less @ nat @ Z6 @ X3 ) )
=> ( ord_less_eq @ nat @ Z6 @ Y3 ) ) ) ) ).
% pred_member
thf(fact_2585_pred__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Px: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X3 )
= ( some @ nat @ Px ) )
= ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 @ Px ) ) ) ).
% pred_corr
thf(fact_2586_succ__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X3 )
= ( some @ nat @ Sx ) )
= ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 @ Sx ) ) ) ).
% succ_corr
thf(fact_2587_pred__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X3 )
= ( some @ nat @ Sx ) )
= ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T2 ) @ X3 @ Sx ) ) ) ).
% pred_correct
thf(fact_2588_succ__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X3 )
= ( some @ nat @ Sx ) )
= ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T2 ) @ X3 @ Sx ) ) ) ).
% succ_correct
thf(fact_2589_vebt__mint_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( some @ nat @ Mi ) ) ).
% vebt_mint.simps(3)
thf(fact_2590_invar__vebt_Ointros_I4_J,axiom,
! [TreeList2: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_2591_invar__vebt_Ointros_I5_J,axiom,
! [TreeList2: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_2592_nested__mint,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: 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 @ TreeList2 @ 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide @ nat @ Va2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ).
% nested_mint
thf(fact_2593_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R2: A,Q3: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q3 @ R2 ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R2 @ ( numeral_numeral @ A @ L ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q3 @ R2 ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ R2 ) ) ) ) ) ).
% divmod_step_eq
thf(fact_2594_pred__list__to__short,axiom,
! [Deg: nat,X3: nat,Ma: nat,TreeList2: list @ vEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X3 @ Ma )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( none @ nat ) ) ) ) ) ).
% pred_list_to_short
thf(fact_2595_succ__list__to__short,axiom,
! [Deg: nat,Mi: nat,X3: nat,TreeList2: list @ vEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X3 )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( none @ nat ) ) ) ) ) ).
% succ_list_to_short
thf(fact_2596_height__node,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ord_less_eq @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% height_node
thf(fact_2597_inrange,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ).
% inrange
thf(fact_2598_geqmaxNone,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( ord_less_eq @ nat @ Ma @ X3 )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( none @ nat ) ) ) ) ).
% geqmaxNone
thf(fact_2599_not__None__eq,axiom,
! [A: $tType,X3: option @ A] :
( ( X3
!= ( none @ A ) )
= ( ? [Y: A] :
( X3
= ( some @ A @ Y ) ) ) ) ).
% not_None_eq
thf(fact_2600_not__Some__eq,axiom,
! [A: $tType,X3: option @ A] :
( ( ! [Y: A] :
( X3
!= ( some @ A @ Y ) ) )
= ( X3
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_2601_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the2 @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_2602_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_2603_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] :
( ! [Uu: A > A > $o,Uv: option @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv ) ) )
=> ( ! [Uw: A > A > $o,V4: A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V4 ) @ ( none @ A ) ) ) )
=> ~ ! [F5: A > A > $o,X4: A,Y4: A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F5 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ X4 ) @ ( some @ A @ Y4 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.cases
thf(fact_2604_VEBT__internal_Ooption__shift_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] :
( ! [Uu: A > A > A,Uv: option @ A] :
( X3
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv ) ) )
=> ( ! [Uw: A > A > A,V4: A] :
( X3
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V4 ) @ ( none @ A ) ) ) )
=> ~ ! [F5: A > A > A,A6: A,B5: A] :
( X3
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F5 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ A6 ) @ ( some @ A @ B5 ) ) ) ) ) ) ).
% VEBT_internal.option_shift.cases
thf(fact_2605_option_Oexpand,axiom,
! [A: $tType,Option: option @ A,Option2: option @ A] :
( ( ( Option
= ( none @ A ) )
= ( Option2
= ( none @ A ) ) )
=> ( ( ( Option
!= ( none @ A ) )
=> ( ( Option2
!= ( none @ A ) )
=> ( ( the2 @ A @ Option )
= ( the2 @ A @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_2606_option_Odistinct_I1_J,axiom,
! [A: $tType,X2: A] :
( ( none @ A )
!= ( some @ A @ X2 ) ) ).
% option.distinct(1)
thf(fact_2607_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X2: A] :
( ( Option
= ( some @ A @ X2 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_2608_option_Oexhaust,axiom,
! [A: $tType,Y3: option @ A] :
( ( Y3
!= ( none @ A ) )
=> ~ ! [X22: A] :
( Y3
!= ( some @ A @ X22 ) ) ) ).
% option.exhaust
thf(fact_2609_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
? [X5: option @ A] : ( P2 @ X5 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
| ? [X: A] : ( P3 @ ( some @ A @ X ) ) ) ) ) ).
% split_option_ex
thf(fact_2610_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
! [X5: option @ A] : ( P2 @ X5 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
& ! [X: A] : ( P3 @ ( some @ A @ X ) ) ) ) ) ).
% split_option_all
thf(fact_2611_combine__options__cases,axiom,
! [A: $tType,B: $tType,X3: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y3: option @ B] :
( ( ( X3
= ( none @ A ) )
=> ( P @ X3 @ Y3 ) )
=> ( ( ( Y3
= ( none @ B ) )
=> ( P @ X3 @ Y3 ) )
=> ( ! [A6: A,B5: B] :
( ( X3
= ( some @ A @ A6 ) )
=> ( ( Y3
= ( some @ B @ B5 ) )
=> ( P @ X3 @ Y3 ) ) )
=> ( P @ X3 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_2612_option_Osel,axiom,
! [A: $tType,X2: A] :
( ( the2 @ A @ ( some @ A @ X2 ) )
= X2 ) ).
% option.sel
thf(fact_2613_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
! [A: $tType,Uu2: A > A > A,Uv2: option @ A] :
( ( vEBT_V2048590022279873568_shift @ A @ Uu2 @ ( none @ A ) @ Uv2 )
= ( none @ A ) ) ).
% VEBT_internal.option_shift.simps(1)
thf(fact_2614_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
=> ( P @ M3 ) ) )
= ( ! [X: nat] :
( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X ) ) ) ) ).
% all_nat_less
thf(fact_2615_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
& ( P @ M3 ) ) )
= ( ? [X: nat] :
( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X ) ) ) ) ).
% ex_nat_less
thf(fact_2616_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [A: $tType,Uw2: A > A > A,V2: A] :
( ( vEBT_V2048590022279873568_shift @ A @ Uw2 @ ( some @ A @ V2 ) @ ( none @ A ) )
= ( none @ A ) ) ).
% VEBT_internal.option_shift.simps(2)
thf(fact_2617_VEBT__internal_Ooption__shift_Oelims,axiom,
! [A: $tType,X3: A > A > A,Xa: option @ A,Xb: option @ A,Y3: option @ A] :
( ( ( vEBT_V2048590022279873568_shift @ A @ X3 @ Xa @ Xb )
= Y3 )
=> ( ( ( Xa
= ( none @ A ) )
=> ( Y3
!= ( none @ A ) ) )
=> ( ( ? [V4: A] :
( Xa
= ( some @ A @ V4 ) )
=> ( ( Xb
= ( none @ A ) )
=> ( Y3
!= ( none @ A ) ) ) )
=> ~ ! [A6: A] :
( ( Xa
= ( some @ A @ A6 ) )
=> ! [B5: A] :
( ( Xb
= ( some @ A @ B5 ) )
=> ( Y3
!= ( some @ A @ ( X3 @ A6 @ B5 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
thf(fact_2618_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_2619_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_2620_summaxma,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
=> ( ( Mi != Ma )
=> ( ( the2 @ nat @ ( vEBT_vebt_maxt @ Summary ) )
= ( vEBT_VEBT_high @ Ma @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% summaxma
thf(fact_2621_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_2622_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_2623__C5_Oprems_C,axiom,
vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ x ) ).
% "5.prems"
thf(fact_2624_delt__out__of__range,axiom,
! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X3 @ Mi )
| ( ord_less @ nat @ Ma @ X3 ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% delt_out_of_range
thf(fact_2625_delete__pres__valid,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T2 @ X3 ) @ N ) ) ).
% delete_pres_valid
thf(fact_2626_not__min__Null__member,axiom,
! [T2: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ T2 )
=> ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_1 ) ) ).
% not_min_Null_member
thf(fact_2627_min__Null__member,axiom,
! [T2: vEBT_VEBT,X3: nat] :
( ( vEBT_VEBT_minNull @ T2 )
=> ~ ( vEBT_vebt_member @ T2 @ X3 ) ) ).
% min_Null_member
thf(fact_2628_maxbmo,axiom,
! [T2: vEBT_VEBT,X3: nat] :
( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X3 ) )
=> ( vEBT_V8194947554948674370ptions @ T2 @ X3 ) ) ).
% maxbmo
thf(fact_2629_dele__bmo__cont__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Y3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T2 @ X3 ) @ Y3 )
= ( ( X3 != Y3 )
& ( vEBT_V8194947554948674370ptions @ T2 @ Y3 ) ) ) ) ).
% dele_bmo_cont_corr
thf(fact_2630_minNullmin,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ T2 )
=> ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) ) ) ).
% minNullmin
thf(fact_2631_minminNull,axiom,
! [T2: vEBT_VEBT] :
( ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) )
=> ( vEBT_VEBT_minNull @ T2 ) ) ).
% minminNull
thf(fact_2632_dele__member__cont__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat,Y3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T2 @ X3 ) @ Y3 )
= ( ( X3 != Y3 )
& ( vEBT_vebt_member @ T2 @ Y3 ) ) ) ) ).
% dele_member_cont_corr
thf(fact_2633_maxt__member,axiom,
! [T2: vEBT_VEBT,N: nat,Maxi: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ Maxi ) )
=> ( vEBT_vebt_member @ T2 @ Maxi ) ) ) ).
% maxt_member
thf(fact_2634_maxt__corr__help,axiom,
! [T2: vEBT_VEBT,N: nat,Maxi: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ Maxi ) )
=> ( ( vEBT_vebt_member @ T2 @ X3 )
=> ( ord_less_eq @ nat @ X3 @ Maxi ) ) ) ) ).
% maxt_corr_help
thf(fact_2635_maxt__sound,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 )
=> ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X3 ) ) ) ) ).
% maxt_sound
thf(fact_2636_maxt__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X3 ) )
=> ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 ) ) ) ).
% maxt_corr
thf(fact_2637_del__single__cont,axiom,
! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X3 = Mi )
& ( X3 = Ma ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% del_single_cont
thf(fact_2638_Icc__eq__Icc,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,H: A,L3: A,H3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ L @ H )
= ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) )
= ( ( ( L = L3 )
& ( H = H3 ) )
| ( ~ ( ord_less_eq @ A @ L @ H )
& ~ ( ord_less_eq @ A @ L3 @ H3 ) ) ) ) ) ).
% Icc_eq_Icc
thf(fact_2639_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% atLeastAtMost_iff
thf(fact_2640_vebt__maxt_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
= ( none @ nat ) ) ).
% vebt_maxt.simps(2)
thf(fact_2641_aset_I2_J,axiom,
! [D5: int,A4: set @ int,P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( plus_plus @ int @ X4 @ D5 ) ) ) )
=> ( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X4 )
=> ( Q @ ( plus_plus @ int @ X4 @ D5 ) ) ) )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
=> ( ( P @ ( plus_plus @ int @ X6 @ D5 ) )
| ( Q @ ( plus_plus @ int @ X6 @ D5 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_2642_aset_I1_J,axiom,
! [D5: int,A4: set @ int,P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( plus_plus @ int @ X4 @ D5 ) ) ) )
=> ( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X4 )
=> ( Q @ ( plus_plus @ int @ X4 @ D5 ) ) ) )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
=> ( ( P @ ( plus_plus @ int @ X6 @ D5 ) )
& ( Q @ ( plus_plus @ int @ X6 @ D5 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_2643_bset_I2_J,axiom,
! [D5: int,B6: set @ int,P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( minus_minus @ int @ X4 @ D5 ) ) ) )
=> ( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X4 )
=> ( Q @ ( minus_minus @ int @ X4 @ D5 ) ) ) )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
=> ( ( P @ ( minus_minus @ int @ X6 @ D5 ) )
| ( Q @ ( minus_minus @ int @ X6 @ D5 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_2644_bset_I1_J,axiom,
! [D5: int,B6: set @ int,P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( minus_minus @ int @ X4 @ D5 ) ) ) )
=> ( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X4 )
=> ( Q @ ( minus_minus @ int @ X4 @ D5 ) ) ) )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
=> ( ( P @ ( minus_minus @ int @ X6 @ D5 ) )
& ( Q @ ( minus_minus @ int @ X6 @ D5 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_2645_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M7: nat] :
( ( P @ X3 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq @ nat @ X4 @ M7 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq @ nat @ X6 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_2646_vebt__mint_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( vEBT_vebt_mint @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
= ( none @ nat ) ) ).
% vebt_mint.simps(2)
thf(fact_2647_bset_I9_J,axiom,
! [D3: int,D5: int,B6: set @ int,T2: int] :
( ( dvd_dvd @ int @ D3 @ D5 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X6 @ T2 ) )
=> ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( minus_minus @ int @ X6 @ D5 ) @ T2 ) ) ) ) ) ).
% bset(9)
thf(fact_2648_bset_I10_J,axiom,
! [D3: int,D5: int,B6: set @ int,T2: int] :
( ( dvd_dvd @ int @ D3 @ D5 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X6 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( minus_minus @ int @ X6 @ D5 ) @ T2 ) ) ) ) ) ).
% bset(10)
thf(fact_2649_aset_I9_J,axiom,
! [D3: int,D5: int,A4: set @ int,T2: int] :
( ( dvd_dvd @ int @ D3 @ D5 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X6 @ T2 ) )
=> ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( plus_plus @ int @ X6 @ D5 ) @ T2 ) ) ) ) ) ).
% aset(9)
thf(fact_2650_aset_I10_J,axiom,
! [D3: int,D5: int,A4: set @ int,T2: int] :
( ( dvd_dvd @ int @ D3 @ D5 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X6 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( plus_plus @ int @ X6 @ D5 ) @ T2 ) ) ) ) ) ).
% aset(10)
thf(fact_2651_vebt__maxt_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( some @ nat @ Ma ) ) ).
% vebt_maxt.simps(3)
thf(fact_2652_periodic__finite__ex,axiom,
! [D3: int,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X4: int,K2: int] :
( ( P @ X4 )
= ( P @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K2 @ D3 ) ) ) )
=> ( ( ? [X9: int] : ( P @ X9 ) )
= ( ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ( P @ X ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_2653_aset_I7_J,axiom,
! [D5: int,A4: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X6 )
=> ( ord_less @ int @ T2 @ ( plus_plus @ int @ X6 @ D5 ) ) ) ) ) ).
% aset(7)
thf(fact_2654_aset_I5_J,axiom,
! [D5: int,T2: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ T2 @ A4 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X6 @ T2 )
=> ( ord_less @ int @ ( plus_plus @ int @ X6 @ D5 ) @ T2 ) ) ) ) ) ).
% aset(5)
thf(fact_2655_aset_I4_J,axiom,
! [D5: int,T2: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ T2 @ A4 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X6 != T2 )
=> ( ( plus_plus @ int @ X6 @ D5 )
!= T2 ) ) ) ) ) ).
% aset(4)
thf(fact_2656_aset_I3_J,axiom,
! [D5: int,T2: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A4 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X6 = T2 )
=> ( ( plus_plus @ int @ X6 @ D5 )
= T2 ) ) ) ) ) ).
% aset(3)
thf(fact_2657_bset_I7_J,axiom,
! [D5: int,T2: int,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ T2 @ B6 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X6 )
=> ( ord_less @ int @ T2 @ ( minus_minus @ int @ X6 @ D5 ) ) ) ) ) ) ).
% bset(7)
thf(fact_2658_bset_I5_J,axiom,
! [D5: int,B6: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X6 @ T2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X6 @ D5 ) @ T2 ) ) ) ) ).
% bset(5)
thf(fact_2659_bset_I4_J,axiom,
! [D5: int,T2: int,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ T2 @ B6 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X6 != T2 )
=> ( ( minus_minus @ int @ X6 @ D5 )
!= T2 ) ) ) ) ) ).
% bset(4)
thf(fact_2660_bset_I3_J,axiom,
! [D5: int,T2: int,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B6 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X6 = T2 )
=> ( ( minus_minus @ int @ X6 @ D5 )
= T2 ) ) ) ) ) ).
% bset(3)
thf(fact_2661_bset_I6_J,axiom,
! [D5: int,B6: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X6 @ T2 )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X6 @ D5 ) @ T2 ) ) ) ) ).
% bset(6)
thf(fact_2662_bset_I8_J,axiom,
! [D5: int,T2: int,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B6 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X6 )
=> ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X6 @ D5 ) ) ) ) ) ) ).
% bset(8)
thf(fact_2663_aset_I6_J,axiom,
! [D5: int,T2: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A4 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X6 @ T2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X6 @ D5 ) @ T2 ) ) ) ) ) ).
% aset(6)
thf(fact_2664_aset_I8_J,axiom,
! [D5: int,A4: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X6 )
=> ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X6 @ D5 ) ) ) ) ) ).
% aset(8)
thf(fact_2665_cppi,axiom,
! [D5: int,P: int > $o,P4: int > $o,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less @ int @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( plus_plus @ int @ X4 @ D5 ) ) ) )
=> ( ! [X4: int,K2: int] :
( ( P4 @ X4 )
= ( P4 @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K2 @ D5 ) ) ) )
=> ( ( ? [X9: int] : ( P @ X9 ) )
= ( ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
& ( P4 @ X ) )
| ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
& ? [Y: int] :
( ( member @ int @ Y @ A4 )
& ( P @ ( minus_minus @ int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_2666_cpmi,axiom,
! [D5: int,P: int > $o,P4: int > $o,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less @ int @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X4 )
=> ( P @ ( minus_minus @ int @ X4 @ D5 ) ) ) )
=> ( ! [X4: int,K2: int] :
( ( P4 @ X4 )
= ( P4 @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K2 @ D5 ) ) ) )
=> ( ( ? [X9: int] : ( P @ X9 ) )
= ( ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
& ( P4 @ X ) )
| ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
& ? [Y: int] :
( ( member @ int @ Y @ B6 )
& ( P @ ( plus_plus @ int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_2667_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 )
& ( ( ord_less @ A @ C2 @ A2 )
| ( ord_less @ A @ B2 @ D3 ) ) ) )
& ( ord_less_eq @ A @ C2 @ D3 ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_2668_invar__vebt_Ointros_I2_J,axiom,
! [TreeList2: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_2669_invar__vebt_Ointros_I3_J,axiom,
! [TreeList2: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_2670_is__pred__in__set__def,axiom,
( vEBT_is_pred_in_set
= ( ^ [Xs: set @ nat,X: nat,Y: nat] :
( ( member @ nat @ Y @ Xs )
& ( ord_less @ nat @ Y @ X )
& ! [Z6: nat] :
( ( member @ nat @ Z6 @ Xs )
=> ( ( ord_less @ nat @ Z6 @ X )
=> ( ord_less_eq @ nat @ Z6 @ Y ) ) ) ) ) ) ).
% is_pred_in_set_def
thf(fact_2671_is__succ__in__set__def,axiom,
( vEBT_is_succ_in_set
= ( ^ [Xs: set @ nat,X: nat,Y: nat] :
( ( member @ nat @ Y @ Xs )
& ( ord_less @ nat @ X @ Y )
& ! [Z6: nat] :
( ( member @ nat @ Z6 @ Xs )
=> ( ( ord_less @ nat @ X @ Z6 )
=> ( ord_less_eq @ nat @ Y @ Z6 ) ) ) ) ) ) ).
% is_succ_in_set_def
thf(fact_2672_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_2673_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_2674_del__x__mi__lets__in__not__minNull,axiom,
! [X3: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list @ vEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( X3 = Mi )
& ( ord_less @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList2 @ H @ Newnode ) )
=> ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi_lets_in_not_minNull
thf(fact_2675_del__x__not__mi__newnode__not__nil,axiom,
! [Mi: nat,X3: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list @ vEBT_VEBT,Newlist: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) )
=> ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList2 @ H @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X3 = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_not_mi_newnode_not_nil
thf(fact_2676__C5_OIH_C_I1_J,axiom,
! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ treeList ) )
=> ( ( vEBT_invar_vebt @ X6 @ na )
& ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ X6 @ x ) )
=> ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ X6 @ x ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% "5.IH"(1)
thf(fact_2677_vebt__delete_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,Tr: list @ vEBT_VEBT,Sm: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) ) ).
% vebt_delete.simps(6)
thf(fact_2678__C5_OIH_C_I2_J,axiom,
( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ summary @ x ) )
=> ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ summary @ x ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).
% "5.IH"(2)
thf(fact_2679_Leaf__0__not,axiom,
! [A2: $o,B2: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) ) ).
% Leaf_0_not
thf(fact_2680_deg1Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
= ( ? [A5: $o,B4: $o] :
( T2
= ( vEBT_Leaf @ A5 @ B4 ) ) ) ) ).
% deg1Leaf
thf(fact_2681_deg__1__Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
=> ? [A6: $o,B5: $o] :
( T2
= ( vEBT_Leaf @ A6 @ B5 ) ) ) ).
% deg_1_Leaf
thf(fact_2682_deg__1__Leafy,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( N
= ( one_one @ nat ) )
=> ? [A6: $o,B5: $o] :
( T2
= ( vEBT_Leaf @ A6 @ B5 ) ) ) ) ).
% deg_1_Leafy
thf(fact_2683_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_2684_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_2685_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_2686_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_2687_semiring__norm_I84_J,axiom,
! [N: num] :
( one2
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_2688_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one2 ) ).
% semiring_norm(86)
thf(fact_2689_tdeletemimi,axiom,
! [Deg: nat,Mi: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: 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 @ TreeList2 @ Summary ) @ X3 ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).
% tdeletemimi
thf(fact_2690_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_2691_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_2692_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_2693_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_2694_list__update__beyond,axiom,
! [A: $tType,Xs2: list @ A,I: nat,X3: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ I )
=> ( ( list_update @ A @ Xs2 @ I @ X3 )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_2695_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_2696_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_2697_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_2698_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_2699_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_2700_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit1 @ M ) @ one2 ) ).
% semiring_norm(70)
thf(fact_2701_or__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X3 ) ) ) ) ).
% or_numerals(8)
thf(fact_2702_or__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) ) ) ).
% or_numerals(2)
thf(fact_2703_zdiv__numeral__Bit1,axiom,
! [V2: num,W: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit1 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W ) ) ) ).
% zdiv_numeral_Bit1
thf(fact_2704_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_2705_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% semiring_norm(4)
thf(fact_2706_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ one2 )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_2707_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ one2 )
= ( bit0 @ ( plus_plus @ num @ M @ one2 ) ) ) ).
% semiring_norm(8)
thf(fact_2708_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_2709_nth__list__update__eq,axiom,
! [A: $tType,I: nat,Xs2: list @ A,X3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs2 @ I @ X3 ) @ I )
= X3 ) ) ).
% nth_list_update_eq
thf(fact_2710_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_2711_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_2712_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_2713_or__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X3 ) ) ) ) ).
% or_numerals(5)
thf(fact_2714_or__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) ) ) ).
% or_numerals(1)
thf(fact_2715_xor__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) ) ) ).
% xor_numerals(1)
thf(fact_2716_xor__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) ) ) ).
% xor_numerals(2)
thf(fact_2717_xor__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X3 ) ) ) ) ).
% xor_numerals(5)
thf(fact_2718_xor__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ X3 ) ) ) ) ).
% xor_numerals(8)
thf(fact_2719_set__swap,axiom,
! [A: $tType,I: nat,Xs2: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs2 @ I @ ( nth @ A @ Xs2 @ J ) ) @ J @ ( nth @ A @ Xs2 @ I ) ) )
= ( set2 @ A @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_2720_or__nat__numerals_I4_J,axiom,
! [X3: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) ) ).
% or_nat_numerals(4)
thf(fact_2721_or__nat__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) ) ).
% or_nat_numerals(2)
thf(fact_2722_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_2723_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_2724_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_2725_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_2726_xor__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% xor_numerals(7)
thf(fact_2727_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_2728_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_2729_or__nat__numerals_I3_J,axiom,
! [X3: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) ) ).
% or_nat_numerals(3)
thf(fact_2730_or__nat__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) ) ).
% or_nat_numerals(1)
thf(fact_2731_xor__nat__numerals_I4_J,axiom,
! [X3: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) ) ).
% xor_nat_numerals(4)
thf(fact_2732_xor__nat__numerals_I3_J,axiom,
! [X3: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) ) ).
% xor_nat_numerals(3)
thf(fact_2733_xor__nat__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) ) ).
% xor_nat_numerals(2)
thf(fact_2734_xor__nat__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) ) ).
% xor_nat_numerals(1)
thf(fact_2735_or__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ) ).
% or_numerals(4)
thf(fact_2736_or__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ) ).
% or_numerals(6)
thf(fact_2737_or__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ) ).
% or_numerals(7)
thf(fact_2738_xor__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_2739_xor__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_2740_zmod__numeral__Bit1,axiom,
! [V2: num,W: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit1 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W ) ) ) @ ( one_one @ int ) ) ) ).
% zmod_numeral_Bit1
thf(fact_2741_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_2742_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_2743_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_2744_verit__eq__simplify_I14_J,axiom,
! [X2: num,X32: num] :
( ( bit0 @ X2 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_2745_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one2
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_2746_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_I3_J,axiom,
! [A2: $o,B2: $o,N: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ N ) ) )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
thf(fact_2747_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_2748_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_2749_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_2750_xor__num_Ocases,axiom,
! [X3: product_prod @ num @ num] :
( ( X3
!= ( product_Pair @ num @ num @ one2 @ one2 ) )
=> ( ! [N2: num] :
( X3
!= ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) )
=> ( ! [N2: num] :
( X3
!= ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) )
=> ( ! [M4: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ one2 ) )
=> ( ! [M4: num,N2: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) )
=> ( ! [M4: num,N2: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) )
=> ( ! [M4: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ one2 ) )
=> ( ! [M4: num,N2: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) )
=> ~ ! [M4: num,N2: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_2751_num_Oexhaust,axiom,
! [Y3: num] :
( ( Y3 != one2 )
=> ( ! [X22: num] :
( Y3
!= ( bit0 @ X22 ) )
=> ~ ! [X33: num] :
( Y3
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_2752_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_2753_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ ( zero_zero @ nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_2754_VEBT__internal_Onaive__member_Ocases,axiom,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [A6: $o,B5: $o,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ X4 ) )
=> ( ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,Ux2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Ux2 ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList3: list @ vEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList3 @ S3 ) @ X4 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_2755_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_2756_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_2757_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_2758_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_2759_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_2760_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_2761_power__minus__Bit1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: A,K: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_2762_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_I4_J,axiom,
! [Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList2 @ Summary ) @ Uu2 )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
thf(fact_2763_set__update__memI,axiom,
! [A: $tType,N: nat,Xs2: list @ A,X3: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( member @ A @ X3 @ ( set2 @ A @ ( list_update @ A @ Xs2 @ N @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_2764_nth__list__update,axiom,
! [A: $tType,I: nat,Xs2: list @ A,J: nat,X3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( I = J )
=> ( ( nth @ A @ ( list_update @ A @ Xs2 @ I @ X3 ) @ J )
= X3 ) )
& ( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs2 @ I @ X3 ) @ J )
= ( nth @ A @ Xs2 @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_2765_list__update__same__conv,axiom,
! [A: $tType,I: nat,Xs2: list @ A,X3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( list_update @ A @ Xs2 @ I @ X3 )
= Xs2 )
= ( ( nth @ A @ Xs2 @ I )
= X3 ) ) ) ).
% list_update_same_conv
thf(fact_2766_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_2767_vebt__pred_Osimps_I1_J,axiom,
! [Uu2: $o,Uv2: $o] :
( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) )
= ( none @ nat ) ) ).
% vebt_pred.simps(1)
thf(fact_2768_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_2769_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_2770_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_2771_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_2772_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_2773_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_2774_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_2775_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu: $o,Uv: $o] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) ) )
=> ( ! [A6: $o,Uw: $o] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [A6: $o,B5: $o,Va: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ ( suc @ Va ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT,Vb: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) @ Vb ) )
=> ( ! [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Vf2 ) )
=> ( ! [V4: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Vj2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
thf(fact_2776_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu: $o,B5: $o] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ B5 ) @ ( zero_zero @ nat ) ) )
=> ( ! [Uv: $o,Uw: $o,N2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) )
=> ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Va3 ) )
=> ( ! [V4: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) @ Ve2 ) )
=> ( ! [V4: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Vi2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
thf(fact_2777_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,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [A6: $o,B5: $o,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ X4 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S3 ) @ X4 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S3 ) @ X4 ) )
=> ( ! [V4: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList3 @ Summary2 ) @ X4 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
thf(fact_2778_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,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [A6: $o,B5: $o,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ X4 ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X4 ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ X4 ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc2 ) @ X4 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
thf(fact_2779_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_Ocases,axiom,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [A6: $o,B5: $o] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( zero_zero @ nat ) ) )
=> ( ! [A6: $o,B5: $o] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [A6: $o,B5: $o,N2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ ( suc @ N2 ) ) ) )
=> ( ! [Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Uu: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList3 @ Summary2 ) @ Uu ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList3 @ Summary2 ) @ X4 ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList3 @ Summary2 ) @ X4 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.cases
thf(fact_2780_VEBT__internal_Omembermima_Ocases,axiom,
! [X3: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu: $o,Uv: $o,Uw: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Uz2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb: vEBT_VEBT,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb ) @ X4 ) )
=> ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList3 @ Vc2 ) @ X4 ) )
=> ~ ! [V4: nat,TreeList3: list @ vEBT_VEBT,Vd2: vEBT_VEBT,X4: nat] :
( X3
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList3 @ Vd2 ) @ X4 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_2781_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_2782_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_2783_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_2784_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_2785_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_2786_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_2787_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_2788_vebt__succ_Osimps_I1_J,axiom,
! [B2: $o,Uu2: $o] :
( ( B2
=> ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu2 @ B2 ) @ ( zero_zero @ nat ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu2 @ B2 ) @ ( zero_zero @ nat ) )
= ( none @ nat ) ) ) ) ).
% vebt_succ.simps(1)
thf(fact_2789_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_2790_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_2791_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,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: 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 ) @ TreeList2 @ Summary ) @ X3 )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
thf(fact_2792_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_2793_vebt__mint_Oelims,axiom,
! [X3: vEBT_VEBT,Y3: option @ nat] :
( ( ( vEBT_vebt_mint @ X3 )
= Y3 )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ~ ( ( A6
=> ( Y3
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( ( B5
=> ( Y3
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B5
=> ( Y3
= ( none @ nat ) ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y3
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat] :
( ? [Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( some @ nat @ Mi2 ) ) ) ) ) ) ).
% vebt_mint.elims
thf(fact_2794_vebt__maxt_Oelims,axiom,
! [X3: vEBT_VEBT,Y3: option @ nat] :
( ( ( vEBT_vebt_maxt @ X3 )
= Y3 )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ~ ( ( B5
=> ( Y3
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B5
=> ( ( A6
=> ( Y3
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( Y3
= ( none @ nat ) ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y3
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat] :
( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( some @ nat @ Ma2 ) ) ) ) ) ) ).
% vebt_maxt.elims
thf(fact_2795_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,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: 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 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
thf(fact_2796_odd__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% odd_mod_4_div_2
thf(fact_2797_vebt__delete_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,TrLst: list @ vEBT_VEBT,Smry: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) ) ).
% vebt_delete.simps(5)
thf(fact_2798_invar__vebt_Ocases,axiom,
! [A1: vEBT_VEBT,A22: nat] :
( ( vEBT_invar_vebt @ A1 @ A22 )
=> ( ( ? [A6: $o,B5: $o] :
( A1
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( A22
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [TreeList3: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X6 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4 = N2 )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M4 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList3: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X6 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4
= ( suc @ N2 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M4 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList3: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X6 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4 = N2 )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M4 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X6: nat] :
( ( ( ( vEBT_VEBT_high @ X6 @ N2 )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X6 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X6 )
& ( ord_less_eq @ nat @ X6 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList3: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X6 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4
= ( suc @ N2 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M4 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X6: nat] :
( ( ( ( vEBT_VEBT_high @ X6 @ N2 )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X6 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X6 )
& ( ord_less_eq @ nat @ X6 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_2799_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A12: vEBT_VEBT,A23: nat] :
( ( ? [A5: $o,B4: $o] :
( A12
= ( vEBT_Leaf @ A5 @ B4 ) )
& ( A23
= ( suc @ ( zero_zero @ nat ) ) ) )
| ? [TreeList: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A23 @ TreeList @ Summary3 ) )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X @ N3 ) )
& ( vEBT_invar_vebt @ Summary3 @ N3 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
& ( A23
= ( plus_plus @ nat @ N3 @ N3 ) )
& ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X9 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X @ X9 ) ) )
| ? [TreeList: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A23 @ TreeList @ Summary3 ) )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X @ N3 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) )
& ( A23
= ( plus_plus @ nat @ N3 @ ( suc @ N3 ) ) )
& ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X9 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X @ X9 ) ) )
| ? [TreeList: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList @ Summary3 ) )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X @ N3 ) )
& ( vEBT_invar_vebt @ Summary3 @ N3 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
& ( A23
= ( plus_plus @ nat @ N3 @ N3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X @ X9 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A23 ) )
& ( ( Mi3 != Ma3 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
& ! [X: nat] :
( ( ( ( vEBT_VEBT_high @ X @ N3 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) )
| ? [TreeList: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList @ Summary3 ) )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X @ N3 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) )
& ( A23
= ( plus_plus @ nat @ N3 @ ( suc @ N3 ) ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X @ X9 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A23 ) )
& ( ( Mi3 != Ma3 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
& ! [X: nat] :
( ( ( ( vEBT_VEBT_high @ X @ N3 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_2800_insert__simp__norm,axiom,
! [X3: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( ord_less @ nat @ Mi @ X3 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X3 != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( ord_max @ nat @ X3 @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_norm
thf(fact_2801_insert__simp__excp,axiom,
! [Mi: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,X3: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( ord_less @ nat @ X3 @ Mi )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X3 != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X3 @ ( ord_max @ nat @ Mi @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_excp
thf(fact_2802_member__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U
= ( 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 @ X3 ) ) @ ( 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 ) ) @ U ) ) ) ) ) ) ) ).
% member_bound_size_univ
thf(fact_2803_vebt__member_Osimps_I4_J,axiom,
! [V2: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) @ X3 ) ).
% vebt_member.simps(4)
thf(fact_2804_succ__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U
= ( 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 @ X3 ) ) @ ( 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 ) ) @ U ) ) ) ) ) ) ) ).
% succ_bound_size_univ
thf(fact_2805_pred__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U
= ( 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 @ X3 ) ) @ ( 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 ) ) @ U ) ) ) ) ) ) ) ).
% pred_bound_size_univ
thf(fact_2806_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_2807_max__0R,axiom,
! [N: nat] :
( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
= N ) ).
% max_0R
thf(fact_2808_max__0L,axiom,
! [N: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% max_0L
thf(fact_2809_max__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ A2 @ ( zero_zero @ nat ) )
= A2 ) ).
% max_nat.right_neutral
thf(fact_2810_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_2811_max__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A2 )
= A2 ) ).
% max_nat.left_neutral
thf(fact_2812_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_2813_max__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(1)
thf(fact_2814_max__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_max @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X3 ) )
= ( numeral_numeral @ A @ X3 ) ) ) ).
% max_0_1(3)
thf(fact_2815_max__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X3 ) @ ( zero_zero @ A ) )
= ( numeral_numeral @ A @ X3 ) ) ) ).
% max_0_1(4)
thf(fact_2816_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_2817_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_2818_max__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_max @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X3 ) )
= ( numeral_numeral @ A @ X3 ) ) ) ).
% max_0_1(5)
thf(fact_2819_max__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X3 ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ X3 ) ) ) ).
% max_0_1(6)
thf(fact_2820_max__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(2)
thf(fact_2821_max__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(3)
thf(fact_2822_max__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(4)
thf(fact_2823_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B4 ) @ B4 @ A5 ) ) ) ) ).
% max_def
thf(fact_2824_max__absorb1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_max @ A @ X3 @ Y3 )
= X3 ) ) ) ).
% max_absorb1
thf(fact_2825_max__absorb2,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_max @ A @ X3 @ Y3 )
= Y3 ) ) ) ).
% max_absorb2
thf(fact_2826_of__int__max,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: int,Y3: int] :
( ( ring_1_of_int @ A @ ( ord_max @ int @ X3 @ Y3 ) )
= ( ord_max @ A @ ( ring_1_of_int @ A @ X3 ) @ ( ring_1_of_int @ A @ Y3 ) ) ) ) ).
% of_int_max
thf(fact_2827_of__nat__max,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: nat,Y3: nat] :
( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X3 @ Y3 ) )
= ( ord_max @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( semiring_1_of_nat @ A @ Y3 ) ) ) ) ).
% of_nat_max
thf(fact_2828_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X3 @ Y3 ) @ Z )
= ( ord_max @ A @ ( plus_plus @ A @ X3 @ Z ) @ ( plus_plus @ A @ Y3 @ Z ) ) ) ) ).
% max_add_distrib_left
thf(fact_2829_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( plus_plus @ A @ X3 @ ( ord_max @ A @ Y3 @ Z ) )
= ( ord_max @ A @ ( plus_plus @ A @ X3 @ Y3 ) @ ( plus_plus @ A @ X3 @ Z ) ) ) ) ).
% max_add_distrib_right
thf(fact_2830_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X3 @ Y3 ) @ Z )
= ( ord_max @ A @ ( minus_minus @ A @ X3 @ Z ) @ ( minus_minus @ A @ Y3 @ Z ) ) ) ) ).
% max_diff_distrib_left
thf(fact_2831_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_2832_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_2833_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_2834_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_2835_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_2836_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
! [Uu2: $o,Uv2: $o] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
thf(fact_2837_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_2838_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
! [Uu2: $o,B2: $o] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uu2 @ 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_2839_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_2840_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_2841_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_2842_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_2843_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,
! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X3 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
thf(fact_2844_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_2845_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,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X3: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X3 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
thf(fact_2846_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,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) @ X3 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
thf(fact_2847_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,X3: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Leaf @ A2 @ B2 ) @ X3 )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( X3
= ( 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_2848_vebt__member_Osimps_I3_J,axiom,
! [V2: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X3: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X3 ) ).
% vebt_member.simps(3)
thf(fact_2849_vebt__member_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X3: nat] :
( ( vEBT_vebt_member @ ( vEBT_Leaf @ A2 @ B2 ) @ X3 )
= ( ( ( X3
= ( zero_zero @ nat ) )
=> A2 )
& ( ( X3
!= ( zero_zero @ nat ) )
=> ( ( ( X3
= ( one_one @ nat ) )
=> B2 )
& ( X3
= ( one_one @ nat ) ) ) ) ) ) ).
% vebt_member.simps(1)
thf(fact_2850_pred__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_p_r_e_d @ T2 @ X3 ) @ ( 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_2851_succ__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_s_u_c_c @ T2 @ X3 ) @ ( 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_2852_member__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_m_e_m_b_e_r @ T2 @ X3 ) @ ( 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_2853_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_2854_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_2855_max__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( ord_less @ A @ ( ord_max @ A @ X3 @ Y3 ) @ Z )
= ( ( ord_less @ A @ X3 @ Z )
& ( ord_less @ A @ Y3 @ Z ) ) ) ) ).
% max_less_iff_conj
thf(fact_2856_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_2857_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_2858_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_2859_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_2860_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_2861_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_2862_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_2863_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_max @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% max.absorb_iff2
thf(fact_2864_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_max @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% max.absorb_iff1
thf(fact_2865_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X3: A,Y3: A] :
( ( ord_less_eq @ A @ Z @ ( ord_max @ A @ X3 @ Y3 ) )
= ( ( ord_less_eq @ A @ Z @ X3 )
| ( ord_less_eq @ A @ Z @ Y3 ) ) ) ) ).
% le_max_iff_disj
thf(fact_2866_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_2867_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_2868_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( A5
= ( ord_max @ A @ A5 @ B4 ) ) ) ) ) ).
% max.order_iff
thf(fact_2869_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_2870_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_2871_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_2872_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_2873_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A2: A,D3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ A @ D3 @ B2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C2 @ D3 ) @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ) ).
% max.mono
thf(fact_2874_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_2875_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_2876_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( A5
= ( ord_max @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_2877_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_2878_less__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X3: A,Y3: A] :
( ( ord_less @ A @ Z @ ( ord_max @ A @ X3 @ Y3 ) )
= ( ( ord_less @ A @ Z @ X3 )
| ( ord_less @ A @ Z @ Y3 ) ) ) ) ).
% less_max_iff_disj
thf(fact_2879_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_2880_insert__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U
= ( 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 @ X3 ) ) @ ( 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 ) ) @ U ) ) ) ) ) ) ) ).
% insert_bound_size_univ
thf(fact_2881_pred__less__length__list,axiom,
! [Deg: nat,X3: nat,Ma: nat,TreeList2: list @ vEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X3 @ Ma )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X3 ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% pred_less_length_list
thf(fact_2882_pred__lesseq__max,axiom,
! [Deg: nat,X3: nat,Ma: nat,Mi: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X3 @ Ma )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X3 ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% pred_lesseq_max
thf(fact_2883_succ__greatereq__min,axiom,
! [Deg: nat,Mi: nat,X3: nat,Ma: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X3 )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% succ_greatereq_min
thf(fact_2884_succ__less__length__list,axiom,
! [Deg: nat,Mi: nat,X3: nat,TreeList2: list @ vEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X3 )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% succ_less_length_list
thf(fact_2885_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% set_vebt'_def
thf(fact_2886_del__x__not__mia,axiom,
! [Mi: nat,X3: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X3 = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) )
= ( none @ nat ) )
@ Mi
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) )
@ ( vEBT_vebt_delete @ Summary @ H ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X3 = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) ) @ H ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).
% del_x_not_mia
thf(fact_2887_del__x__not__mi__new__node__nil,axiom,
! [Mi: nat,X3: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list @ vEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) )
=> ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Sn
= ( vEBT_vebt_delete @ Summary @ H ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList2 @ H @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X3 = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ Sn )
= ( none @ nat ) )
@ Mi
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_not_mi_new_node_nil
thf(fact_2888_del__x__not__mi,axiom,
! [Mi: nat,X3: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list @ vEBT_VEBT,Newlist: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList2 @ H @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X3 = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) )
= ( none @ nat ) )
@ Mi
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ ( vEBT_vebt_delete @ Summary @ H ) ) ) )
& ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X3 = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_not_mi
thf(fact_2889_del__x__mia,axiom,
! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X3 = Mi )
& ( ord_less @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
@ ( if @ nat
@ ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
@ ( if @ nat
@ ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ) ) ).
% del_x_mia
thf(fact_2890_del__x__mi__lets__in__minNull,axiom,
! [X3: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list @ vEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT,Sn: vEBT_VEBT] :
( ( ( X3 = Mi )
& ( ord_less @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList2 @ H @ Newnode ) )
=> ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Sn
= ( vEBT_vebt_delete @ Summary @ H ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Xn
@ ( if @ nat @ ( Xn = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ Sn )
= ( none @ nat ) )
@ Xn
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi_lets_in_minNull
thf(fact_2891_del__x__mi__lets__in,axiom,
! [X3: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list @ vEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( X3 = Mi )
& ( ord_less @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList2 @ H @ Newnode ) )
=> ( ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Xn
@ ( if @ nat @ ( Xn = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) )
= ( none @ nat ) )
@ Xn
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ ( vEBT_vebt_delete @ Summary @ H ) ) ) )
& ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi_lets_in
thf(fact_2892_del__x__mi,axiom,
! [X3: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list @ vEBT_VEBT,L: nat] :
( ( ( X3 = Mi )
& ( ord_less @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Xn
@ ( if @ nat @ ( Xn = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) )
= ( none @ nat ) )
@ Xn
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) )
@ ( vEBT_vebt_delete @ Summary @ H ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) ) @ H ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi
thf(fact_2893_del__in__range,axiom,
! [Mi: nat,X3: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less_eq @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X3 = Mi ) @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ Mi )
@ ( if @ nat
@ ( ( ( X3 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X3 != Mi )
=> ( X3 = Ma ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( X3 = Mi ) @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ Mi )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X3 = Mi ) @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ Mi )
@ ( if @ nat
@ ( ( ( X3 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X3 != Mi )
=> ( X3 = Ma ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ) ) ).
% del_in_range
thf(fact_2894_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B4 ) @ B4 @ A5 ) ) ) ) ).
% max_def_raw
thf(fact_2895_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H2: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_2896_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_2897_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B4: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_2898_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_2899_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_2900_power__numeral__even,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z: A,W: num] :
( ( power_power @ A @ Z @ ( numeral_numeral @ nat @ ( bit0 @ W ) ) )
= ( times_times @ A @ ( power_power @ A @ Z @ ( numeral_numeral @ nat @ W ) ) @ ( power_power @ A @ Z @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).
% power_numeral_even
thf(fact_2901_power__numeral__odd,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z: A,W: num] :
( ( power_power @ A @ Z @ ( numeral_numeral @ nat @ ( bit1 @ W ) ) )
= ( times_times @ A @ ( times_times @ A @ Z @ ( power_power @ A @ Z @ ( numeral_numeral @ nat @ W ) ) ) @ ( power_power @ A @ Z @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).
% power_numeral_odd
thf(fact_2902_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_2903_nat__times__as__int,axiom,
( ( times_times @ nat )
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_2904_nat__minus__as__int,axiom,
( ( minus_minus @ nat )
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_2905_nat__div__as__int,axiom,
( ( divide_divide @ nat )
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_2906_nat__mod__as__int,axiom,
( ( modulo_modulo @ nat )
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ).
% nat_mod_as_int
thf(fact_2907_diff__nat__eq__if,axiom,
! [Z5: int,Z: int] :
( ( ( ord_less @ int @ Z5 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z5 ) )
= ( nat2 @ Z ) ) )
& ( ~ ( ord_less @ int @ Z5 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z5 ) )
= ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z @ Z5 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z @ Z5 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_2908_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 ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts2 @ S ) @ X3 )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
thf(fact_2909_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 ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) @ X3 )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
thf(fact_2910_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,X3: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Leaf @ A2 @ B2 ) @ X3 )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( X3
= ( 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_2911_vebt__member_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( ( X3 != Mi )
=> ( ( X3 != Ma )
=> ( ~ ( ord_less @ nat @ X3 @ Mi )
& ( ~ ( ord_less @ nat @ X3 @ Mi )
=> ( ~ ( ord_less @ nat @ Ma @ X3 )
& ( ~ ( ord_less @ nat @ Ma @ X3 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_2912_vebt__member_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa: nat] :
( ( vEBT_vebt_member @ X3 @ Xa )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B5 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_2913_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,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: 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 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( X3 = Mi ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( X3 = Ma ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma @ X3 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( 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_2914_vebt__member_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: $o] :
( ( ( vEBT_vebt_member @ X3 @ Xa )
= Y3 )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( Y3
= ( ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B5 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> Y3 )
=> ( ( ? [V4: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> Y3 )
=> ( ( ? [V4: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc2 ) )
=> Y3 )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( Y3
= ( ~ ( ( 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 ) @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_2915_vebt__member_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_vebt_member @ X3 @ Xa )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B5 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_2916_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,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
thf(fact_2917_insersimp,axiom,
! [T2: vEBT_VEBT,N: nat,Y3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_1 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ Y3 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ).
% insersimp
thf(fact_2918_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_2919_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,
! [X3: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_m_e_m_b_e_r @ X3 @ Xa )
= Y3 )
=> ( ( ? [A6: $o,B5: $o] :
( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( Y3
!= ( 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 ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y3
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc2 ) )
=> ( Y3
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( Y3
!= ( 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 ) @ TreeList3 ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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_2920_vebt__succ_Osimps_I6_J,axiom,
! [X3: nat,Mi: nat,Ma: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X3 @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( some @ nat @ Mi ) ) )
& ( ~ ( ord_less @ nat @ X3 @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( 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 @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( 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 @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% vebt_succ.simps(6)
thf(fact_2921_vebt__pred_Osimps_I7_J,axiom,
! [Ma: nat,X3: nat,Mi: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Ma @ X3 )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( some @ nat @ Ma ) ) )
& ( ~ ( ord_less @ nat @ Ma @ X3 )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( 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 @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X3 ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% vebt_pred.simps(7)
thf(fact_2922_vebt__delete_Osimps_I7_J,axiom,
! [X3: nat,Mi: nat,Ma: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ( ord_less @ nat @ X3 @ Mi )
| ( ord_less @ nat @ Ma @ X3 ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) ) )
& ( ~ ( ( ord_less @ nat @ X3 @ Mi )
| ( ord_less @ nat @ Ma @ X3 ) )
=> ( ( ( ( X3 = Mi )
& ( X3 = Ma ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) ) )
& ( ~ ( ( X3 = Mi )
& ( X3 = Ma ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X3 = Mi ) @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ Mi )
@ ( if @ nat
@ ( ( ( X3 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X3 != Mi )
=> ( X3 = Ma ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( X3 = Mi ) @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ Mi )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( 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 @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ ( suc @ ( suc @ Va2 ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X3 = Mi ) @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ Mi )
@ ( if @ nat
@ ( ( ( X3 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X3 != Mi )
=> ( X3 = Ma ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ ( suc @ ( suc @ Va2 ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( 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 ) ) @ TreeList2 @ Summary ) ) ) ) ) ) ) ).
% vebt_delete.simps(7)
thf(fact_2923_vebt__delete_Oelims,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
( ( ( vEBT_vebt_delete @ X3 @ Xa )
= Y3 )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
!= ( vEBT_Leaf @ $false @ B5 ) ) ) )
=> ( ! [A6: $o] :
( ? [B5: $o] :
( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y3
!= ( vEBT_Leaf @ A6 @ $false ) ) ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( Y3
!= ( vEBT_Leaf @ A6 @ B5 ) ) ) )
=> ( ! [Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( Y3
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList3 @ Summary2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TrLst2: list @ vEBT_VEBT,Smry2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ( Y3
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Tr2: list @ vEBT_VEBT,Sm2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ( Y3
!= ( 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,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ~ ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( Y3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) ) )
& ( ~ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y3
= ( 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 @ TreeList3 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_delete.elims
thf(fact_2924_vebt__succ_Oelims,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: option @ nat] :
( ( ( vEBT_vebt_succ @ X3 @ Xa )
= Y3 )
=> ( ! [Uu: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ B5 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ~ ( ( B5
=> ( Y3
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B5
=> ( Y3
= ( none @ nat ) ) ) ) ) )
=> ( ( ? [Uv: $o,Uw: $o] :
( X3
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ N2 ) )
=> ( Y3
!= ( none @ nat ) ) ) )
=> ( ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( none @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( Y3
!= ( none @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( Y3
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y3
= ( some @ nat @ Mi2 ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y3
= ( 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 ) @ TreeList3 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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_2925_vebt__pred_Oelims,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: option @ nat] :
( ( ( vEBT_vebt_pred @ X3 @ Xa )
= Y3 )
=> ( ( ? [Uu: $o,Uv: $o] :
( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
!= ( none @ nat ) ) ) )
=> ( ! [A6: $o] :
( ? [Uw: $o] :
( X3
= ( vEBT_Leaf @ A6 @ Uw ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( ( A6
=> ( Y3
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( Y3
= ( none @ nat ) ) ) ) ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ? [Va: nat] :
( Xa
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( B5
=> ( Y3
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B5
=> ( ( A6
=> ( Y3
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( Y3
= ( none @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) )
=> ( Y3
!= ( none @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y3
!= ( none @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( Y3
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y3
= ( some @ nat @ Ma2 ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y3
= ( 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 ) @ TreeList3 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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_2926_insert__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ X3 ) @ ( 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_2927_vebt__insert_Oelims,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X3 @ Xa )
= Y3 )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
= ( vEBT_Leaf @ $true @ B5 ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> ( Y3
= ( vEBT_Leaf @ A6 @ $true ) ) )
& ( ( Xa
!= ( one_one @ nat ) )
=> ( Y3
= ( vEBT_Leaf @ A6 @ B5 ) ) ) ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S3 ) )
=> ( Y3
!= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S3 ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S3 ) )
=> ( Y3
!= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S3 ) ) )
=> ( ! [V4: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList3 @ Summary2 ) )
=> ( Y3
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList3 @ Summary2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( Y3
!= ( 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 ) @ TreeList3 ) )
& ~ ( ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% vebt_insert.elims
thf(fact_2928_vebt__insert_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( X3 = Mi )
| ( X3 = Ma ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ X3 @ Mi ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) ) ) ).
% vebt_insert.simps(5)
thf(fact_2929_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_2930_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_s_u_c_c2 @ X3 @ Xa )
= Y3 )
=> ( ( ? [Uu: $o,B5: $o] :
( X3
= ( vEBT_Leaf @ Uu @ B5 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Uv: $o,Uw: $o] :
( X3
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ N2 ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y3
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y3
= ( 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 ) @ TreeList3 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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_2931_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_p_r_e_d2 @ X3 @ Xa )
= Y3 )
=> ( ( ? [Uu: $o,Uv: $o] :
( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,Uw: $o] :
( X3
= ( vEBT_Leaf @ A6 @ Uw ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,B5: $o] :
( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ? [Va: nat] :
( Xa
= ( suc @ ( suc @ Va ) ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y3
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y3
= ( 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 ) @ TreeList3 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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_2932_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
! [Uu2: $o,Uv2: $o] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
thf(fact_2933_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
! [Uu2: $o,B2: $o] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uu2 @ B2 ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
thf(fact_2934_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_2935_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_2936_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_2937_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_2938_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_2939_pred__bound__height_H,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_p_r_e_d2 @ T2 @ X3 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% pred_bound_height'
thf(fact_2940_succ_H__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_s_u_c_c2 @ T2 @ X3 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% succ'_bound_height
thf(fact_2941_vebt__insert_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts2 @ S ) @ X3 )
= ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts2 @ S ) ) ).
% vebt_insert.simps(2)
thf(fact_2942_pred__bound__size__univ_H,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U
= ( 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 @ X3 ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ).
% pred_bound_size_univ'
thf(fact_2943_succ__bound__size__univ_H,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U
= ( 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 @ X3 ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ).
% succ_bound_size_univ'
thf(fact_2944_vebt__insert_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) @ X3 )
= ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) ) ).
% vebt_insert.simps(3)
thf(fact_2945_vebt__insert_Osimps_I1_J,axiom,
! [X3: nat,A2: $o,B2: $o] :
( ( ( X3
= ( zero_zero @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X3 )
= ( vEBT_Leaf @ $true @ B2 ) ) )
& ( ( X3
!= ( zero_zero @ nat ) )
=> ( ( ( X3
= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X3 )
= ( vEBT_Leaf @ A2 @ $true ) ) )
& ( ( X3
!= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X3 )
= ( vEBT_Leaf @ A2 @ B2 ) ) ) ) ) ) ).
% vebt_insert.simps(1)
thf(fact_2946_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
! [Ma: nat,X3: nat,Mi: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Ma @ X3 )
=> ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma @ X3 )
=> ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary @ ( vEBT_VEBT_high @ X3 @ ( 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_2947_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
! [X3: nat,Mi: nat,Ma: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X3 @ Mi )
=> ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ X3 @ Mi )
=> ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary @ ( vEBT_VEBT_high @ X3 @ ( 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_2948_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,
! [X3: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_i_n_s_e_r_t @ X3 @ Xa )
= Y3 )
=> ( ( ? [A6: $o,B5: $o] :
( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( Y3
!= ( 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 ),Ts: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S3 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S3 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList3 @ Summary2 ) )
=> ( Y3
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( Y3
!= ( 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 ) @ TreeList3 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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_2949_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,
! [X3: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_m_e_m_b_e_r2 @ X3 @ Xa )
= Y3 )
=> ( ( ? [A6: $o,B5: $o] :
( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( Y3
!= ( 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 ) @ TreeList3 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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_2950_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,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: 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 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( X3 = Mi )
| ( X3 = Ma ) ) )
@ ( 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 @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( 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 @ X3 @ Mi ) @ Mi @ X3 ) @ ( 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_2951_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,
! [X3: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_i_n_s_e_r_t2 @ X3 @ Xa )
= Y3 )
=> ( ( ? [A6: $o,B5: $o] :
( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S3 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S3 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList3 @ Summary2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( Y3
!= ( 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 ) @ TreeList3 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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_2952_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,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: 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 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( X3 = Mi ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( X3 = Ma ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma @ X3 ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less @ nat @ X3 @ Ma ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( 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_2953_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_2954_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 ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts2 @ S ) @ X3 )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
thf(fact_2955_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 ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) @ X3 )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
thf(fact_2956_insersimp_H,axiom,
! [T2: vEBT_VEBT,N: nat,Y3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_1 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ Y3 ) @ ( one_one @ nat ) ) ) ) ).
% insersimp'
thf(fact_2957_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_2958_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,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X3: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X3 )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
thf(fact_2959_insert_H__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ X3 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% insert'_bound_height
thf(fact_2960_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,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) @ X3 )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
thf(fact_2961_member__bound__height_H,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_m_e_m_b_e_r2 @ T2 @ X3 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% member_bound_height'
thf(fact_2962_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,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: 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 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( X3 = Mi )
| ( X3 = Ma ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( 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 @ X3 @ Mi ) @ Mi @ X3 ) @ ( 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_2963_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,
! [X3: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_d_e_l_e_t_e @ X3 @ Xa )
= Y3 )
=> ( ( ? [A6: $o,B5: $o] :
( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,B5: $o] :
( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,B5: $o] :
( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList3 @ Summary2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList3 @ Summary2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( Y3
!= ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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_2964_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: $o] :
( ( ( vEBT_VEBT_membermima @ X3 @ Xa )
= Y3 )
=> ( ( ? [Uu: $o,Uv: $o] :
( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> Y3 )
=> ( ( ? [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> Y3 )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb ) )
=> ( Y3
= ( ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList3 @ Vc2 ) )
=> ( Y3
= ( ~ ( ( 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 ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) )
=> ~ ! [V4: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList3 @ Vd2 ) )
=> ( Y3
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_2965_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_membermima @ X3 @ Xa )
=> ( ! [Uu: $o,Uv: $o] :
( X3
!= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb ) )
=> ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) )
=> ~ ! [V4: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList3 @ Vd2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_2966_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: $o] :
( ( ( vEBT_V5719532721284313246member @ X3 @ Xa )
= Y3 )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( Y3
= ( ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B5 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) )
=> Y3 )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList3: list @ vEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList3 @ S3 ) )
=> ( Y3
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_2967_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa: nat] :
( ( vEBT_V5719532721284313246member @ X3 @ Xa )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B5 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList3: list @ vEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList3 @ S3 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_2968_buildup__nothing__in__min__max,axiom,
! [N: nat,X3: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X3 ) ).
% buildup_nothing_in_min_max
thf(fact_2969_buildup__nothing__in__leaf,axiom,
! [N: nat,X3: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X3 ) ).
% buildup_nothing_in_leaf
thf(fact_2970_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T3: vEBT_VEBT,X: nat] :
( ( vEBT_V5719532721284313246member @ T3 @ X )
| ( vEBT_VEBT_membermima @ T3 @ X ) ) ) ) ).
% both_member_options_def
thf(fact_2971_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ Tree @ N )
=> ( ( vEBT_vebt_member @ Tree @ X3 )
=> ( ( vEBT_V5719532721284313246member @ Tree @ X3 )
| ( vEBT_VEBT_membermima @ Tree @ X3 ) ) ) ) ).
% member_valid_both_member_options
thf(fact_2972_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,Ux: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Ux ) ).
% VEBT_internal.naive_member.simps(2)
thf(fact_2973_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) @ Uz ) ).
% VEBT_internal.membermima.simps(2)
thf(fact_2974_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X3: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A2 @ B2 ) @ X3 )
= ( ( ( X3
= ( zero_zero @ nat ) )
=> A2 )
& ( ( X3
!= ( zero_zero @ nat ) )
=> ( ( ( X3
= ( one_one @ nat ) )
=> B2 )
& ( X3
= ( one_one @ nat ) ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
thf(fact_2975_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) @ X3 )
= ( ( X3 = Mi )
| ( X3 = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_2976_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_2977_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_2978_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_2979_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
! [X3: vEBT_VEBT,Y3: nat] :
( ( ( vEBT_T_m_i_n_t @ X3 )
= Y3 )
=> ( ! [A6: $o] :
( ? [B5: $o] :
( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( Y3
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ A6 @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
thf(fact_2980_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V2: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT,X3: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList2 @ Vd ) @ X3 )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_2981_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy: option @ ( product_prod @ nat @ nat ),V2: nat,TreeList2: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList2 @ S ) @ X3 )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_2982_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi: nat,Ma: nat,V2: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc ) @ X3 )
= ( ( X3 = Mi )
| ( X3 = Ma )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_2983_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,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma @ X3 ) @ ( 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 @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( 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 @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( 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_2984_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,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ X3 @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( 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 @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( 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_2985_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_p_r_e_d @ X3 @ Xa )
= Y3 )
=> ( ( ? [Uu: $o,Uv: $o] :
( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,Uw: $o] :
( X3
= ( vEBT_Leaf @ A6 @ Uw ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y3
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ? [Va: nat] :
( Xa
= ( suc @ ( suc @ Va ) ) )
=> ( Y3
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B5 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( Y3
!= ( 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 ) @ TreeList3 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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_2986_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_s_u_c_c @ X3 @ Xa )
= Y3 )
=> ( ( ? [Uu: $o,B5: $o] :
( X3
= ( vEBT_Leaf @ Uu @ B5 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ( ( ? [Uv: $o,Uw: $o] :
( X3
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ N2 ) )
=> ( Y3
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( Y3
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( Y3
!= ( 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 ) @ TreeList3 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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_2987_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_membermima @ X3 @ Xa )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb ) )
=> ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) )
=> ~ ! [V4: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList3 @ Vd2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_2988_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,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: 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 ) ) @ TreeList2 @ Summary ) @ X3 )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ X3 @ Mi )
| ( ord_less @ nat @ Ma @ X3 ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( X3 = Mi )
& ( X3 = Ma ) )
@ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_m_i_n_t @ Summary ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( 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 @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( 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 @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( X3 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X3 != Mi )
=> ( X3 = 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 @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( 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 @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( 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 @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( X3 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X3 != Mi )
=> ( X3 = Ma ) ) )
@ ( 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 @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X3 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( 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_2989_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B5 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList3: list @ vEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList3 @ S3 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_2990_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,
! [X3: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_d_e_l_e_t_e @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( Y3
= ( 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 @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList3 @ Summary2 ) )
=> ( ( Y3
= ( 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 ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y3
= ( 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 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y3
= ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
thf(fact_2991_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_s_u_c_c @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [Uu: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ B5 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( 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 @ Uu @ B5 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv: $o,Uw: $o] :
( ( X3
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ N2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) ) ) ) )
=> ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( 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 ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( ( Y3
= ( 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] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( ( Y3
= ( 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,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y3
= ( 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 ) @ TreeList3 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
thf(fact_2992_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_p_r_e_d @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A6: $o,Uw: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ Uw ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y3
= ( 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,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ! [Va: nat] :
( ( Xa
= ( suc @ ( suc @ Va ) ) )
=> ( ( Y3
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B5 @ ( 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 @ B5 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) )
=> ( ( Y3
= ( 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 ) ) @ Uy2 @ Uz2 @ Va3 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y3
= ( 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] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( ( Y3
= ( 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,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y3
= ( 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 ) @ TreeList3 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
thf(fact_2993_pred__empty,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X3 )
= ( none @ nat ) )
= ( ( collect @ nat
@ ^ [Y: nat] :
( ( vEBT_vebt_member @ T2 @ Y )
& ( ord_less @ nat @ Y @ X3 ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% pred_empty
thf(fact_2994_succ__empty,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X3 )
= ( none @ nat ) )
= ( ( collect @ nat
@ ^ [Y: nat] :
( ( vEBT_vebt_member @ T2 @ Y )
& ( ord_less @ nat @ X3 @ Y ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% succ_empty
thf(fact_2995_buildup__gives__empty,axiom,
! [N: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% buildup_gives_empty
thf(fact_2996_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_2997_mint__corr__help__empty,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) )
=> ( ( vEBT_VEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mint_corr_help_empty
thf(fact_2998_maxt__corr__help__empty,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( none @ nat ) )
=> ( ( vEBT_VEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% maxt_corr_help_empty
thf(fact_2999_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X3 )
= X3 ) ) ).
% max_bot
thf(fact_3000_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ord_max @ A @ X3 @ ( bot_bot @ A ) )
= X3 ) ) ).
% max_bot2
thf(fact_3001_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_3002_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_3003_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_3004_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_3005_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_3006_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_3007_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_3008_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_3009_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_3010_diff__shunt__var,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y3: A] :
( ( ( minus_minus @ A @ X3 @ Y3 )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% diff_shunt_var
thf(fact_3011_vebt__succ_Opelims,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: option @ nat] :
( ( ( vEBT_vebt_succ @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [Uu: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ B5 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( ( B5
=> ( Y3
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B5
=> ( Y3
= ( none @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ B5 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv: $o,Uw: $o] :
( ( X3
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ N2 ) )
=> ( ( Y3
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) ) ) ) )
=> ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( ( Y3
= ( 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] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( ( Y3
= ( 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,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y3
= ( some @ nat @ Mi2 ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y3
= ( 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 ) @ TreeList3 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.pelims
thf(fact_3012_vebt__pred_Opelims,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: option @ nat] :
( ( ( vEBT_vebt_pred @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A6: $o,Uw: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ Uw ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( ( A6
=> ( Y3
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( Y3
= ( 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,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ! [Va: nat] :
( ( Xa
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( B5
=> ( Y3
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B5
=> ( ( A6
=> ( Y3
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( Y3
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) )
=> ( ( Y3
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y3
= ( 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] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( ( Y3
= ( 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,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y3
= ( some @ nat @ Ma2 ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y3
= ( 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 ) @ TreeList3 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.pelims
thf(fact_3013_vebt__delete_Opelims,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
( ( ( vEBT_vebt_delete @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( vEBT_Leaf @ $false @ B5 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y3
= ( vEBT_Leaf @ A6 @ $false ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( ( Y3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( Y3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TrLst2: list @ vEBT_VEBT,Smry2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ( ( Y3
= ( 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] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ( ( Y3
= ( 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,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( Y3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) ) )
& ( ~ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y3
= ( 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 @ TreeList3 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) ) @ TreeList3 @ 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 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_delete.pelims
thf(fact_3014_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_s_u_c_c2 @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [Uu: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ B5 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ B5 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv: $o,Uw: $o] :
( ( X3
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ N2 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) ) ) ) )
=> ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( 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 ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( ( Y3
= ( 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] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( ( Y3
= ( 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,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y3
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y3
= ( 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 ) @ TreeList3 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
thf(fact_3015_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,
! [X3: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_i_n_s_e_r_t @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Y3
= ( 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 @ B5 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S3 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S3 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S3 ) )
=> ( ( Y3
= ( 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 ) ) @ Ts @ S3 ) @ Xa ) ) ) )
=> ( ! [V4: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y3
= ( 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 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y3
= ( 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 ) @ TreeList3 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
thf(fact_3016_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_3017_bot__enat__def,axiom,
( ( bot_bot @ extended_enat )
= ( zero_zero @ extended_enat ) ) ).
% bot_enat_def
thf(fact_3018_vebt__insert_Opelims,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
= ( vEBT_Leaf @ $true @ B5 ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> ( Y3
= ( vEBT_Leaf @ A6 @ $true ) ) )
& ( ( Xa
!= ( one_one @ nat ) )
=> ( Y3
= ( vEBT_Leaf @ A6 @ B5 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S3 ) )
=> ( ( Y3
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S3 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S3 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S3 ) )
=> ( ( Y3
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S3 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S3 ) @ Xa ) ) ) )
=> ( ! [V4: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList3 @ 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 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y3
= ( 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 ) @ TreeList3 ) )
& ~ ( ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) ) @ TreeList3 @ 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 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% vebt_insert.pelims
thf(fact_3019_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_p_r_e_d2 @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A6: $o,Uw: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ Uw ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y3
= ( 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,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ! [Va: nat] :
( ( Xa
= ( suc @ ( suc @ Va ) ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) )
=> ( ( Y3
= ( 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 ) ) @ Uy2 @ Uz2 @ Va3 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y3
= ( 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] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( ( Y3
= ( 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,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y3
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y3
= ( 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 ) @ TreeList3 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
thf(fact_3020_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,
! [X3: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_i_n_s_e_r_t2 @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S3 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S3 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S3 ) )
=> ( ( Y3
= ( 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 ) ) @ Ts @ S3 ) @ Xa ) ) ) )
=> ( ! [V4: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y3
= ( 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 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y3
= ( 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 ) @ TreeList3 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
thf(fact_3021_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,
! [X3: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_m_e_m_b_e_r @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Y3
= ( 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 @ B5 ) @ Xa ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y3
= ( 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 ) ) @ Uu @ Uv @ Uw ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( 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 ) @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc2 ) )
=> ( ( Y3
= ( 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 ) ) @ Vb @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y3
= ( 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 ) @ TreeList3 ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
thf(fact_3022_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,
! [X3: vEBT_VEBT,Xa: nat,Y3: nat] :
( ( ( vEBT_T_m_e_m_b_e_r2 @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y3
= ( 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 ) ) @ Uu @ Uv @ Uw ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( 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 ) @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc2 ) )
=> ( ( Y3
= ( 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 ) ) @ Vb @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y3
= ( 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 ) @ TreeList3 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
thf(fact_3023_vebt__member_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: $o] :
( ( ( vEBT_vebt_member @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Y3
= ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B5 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ~ Y3
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ~ Y3
=> ~ ( 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 ) @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc2 ) )
=> ( ~ Y3
=> ~ ( 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 ) ) @ Vb @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y3
= ( ( 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 ) @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_3024_vebt__member_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_vebt_member @ X3 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B5 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ Xa ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ 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 ) ) @ Vb @ Vc2 ) @ Xa ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_3025_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: $o] :
( ( ( vEBT_V5719532721284313246member @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Y3
= ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B5 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa ) ) ) )
=> ( ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) )
=> ( ~ Y3
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Xa ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList3: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList3 @ S3 ) )
=> ( ( Y3
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList3 @ S3 ) @ Xa ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_3026_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa: nat] :
( ( vEBT_V5719532721284313246member @ X3 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B5 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList3: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList3 @ S3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList3 @ S3 ) @ Xa ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_3027_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B5 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Xa ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList3: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList3 @ S3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V4 ) @ TreeList3 @ S3 ) @ Xa ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_3028_vebt__member_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa: nat] :
( ( vEBT_vebt_member @ X3 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B5 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_3029_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: $o] :
( ( ( vEBT_VEBT_membermima @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ~ Y3
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa ) ) ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ( ~ Y3
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb ) )
=> ( ( Y3
= ( ( 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 @ Vb ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList3 @ Vc2 ) )
=> ( ( Y3
= ( ( 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 ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList3 @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [V4: nat,TreeList3: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList3 @ Vd2 ) )
=> ( ( Y3
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList3 @ Vd2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_3030_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_membermima @ X3 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa ) ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Xa ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb ) @ Xa ) )
=> ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList3 @ Vc2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) )
=> ~ ! [V4: nat,TreeList3: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList3 @ Vd2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList3 @ Vd2 ) @ Xa ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_3031_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_membermima @ X3 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb ) @ Xa ) )
=> ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList3 @ Vc2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList3 @ 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 ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) )
=> ~ ! [V4: nat,TreeList3: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList3 @ Vd2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList3 @ Vd2 ) @ Xa ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( 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 ) @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_3032_monoseq__arctan__series,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( topological_monoseq @ real
@ ^ [N3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_3033_delete__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X3 ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert @ nat @ X3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% delete_correct
thf(fact_3034_delete__correct_H,axiom,
! [T2: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X3 ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( insert @ nat @ X3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% delete_correct'
thf(fact_3035_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_3036_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_3037_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_3038_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_3039_monoseq__realpow,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( topological_monoseq @ real @ ( power_power @ real @ X3 ) ) ) ) ).
% monoseq_realpow
thf(fact_3040_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z: A,N: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_3041_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_3042_ln__series,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ( ln_ln @ real @ X3 )
= ( suminf @ real
@ ^ [N3: nat] : ( times_times @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) @ ( power_power @ real @ ( minus_minus @ real @ X3 @ ( one_one @ real ) ) @ ( suc @ N3 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_3043_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_3044_arctan__series,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( ( arctan @ X3 )
= ( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_3045_of__nat__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F2: B > nat,A4: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( semiring_1_of_nat @ A @ ( F2 @ X ) )
@ A4 ) ) ) ).
% of_nat_prod
thf(fact_3046_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_3047_of__int__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_ring_1 @ A )
=> ! [F2: B > int,A4: set @ B] :
( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F2 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( ring_1_of_int @ A @ ( F2 @ X ) )
@ A4 ) ) ) ).
% of_int_prod
thf(fact_3048_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one2 )
= ( zero_zero @ nat ) ) ).
% pred_numeral_simps(1)
thf(fact_3049_eq__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral @ nat @ K )
= ( suc @ N ) )
= ( ( pred_numeral @ K )
= N ) ) ).
% eq_numeral_Suc
thf(fact_3050_Suc__eq__numeral,axiom,
! [N: nat,K: num] :
( ( ( suc @ N )
= ( numeral_numeral @ nat @ K ) )
= ( N
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_3051_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_3052_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_3053_pred__numeral__simps_I3_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit1 @ K ) )
= ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ).
% pred_numeral_simps(3)
thf(fact_3054_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_3055_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_3056_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_3057_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_3058_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_3059_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_3060_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_3061_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_3062_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_3063_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_3064_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_3065_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_3066_norm__prod__le,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [F2: B > A,A4: set @ B] :
( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) )
@ ( groups7121269368397514597t_prod @ B @ real
@ ^ [A5: B] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ A5 ) )
@ A4 ) ) ) ).
% norm_prod_le
thf(fact_3067_numeral__eq__Suc,axiom,
( ( numeral_numeral @ nat )
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_3068_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K3: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K3 ) @ ( one_one @ nat ) ) ) ) ).
% pred_numeral_def
thf(fact_3069_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_3070_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_3071_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_3072_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_3073_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A,P6: 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 @ P6 ) ) )
= ( 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 @ P6 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_3074_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X: A] : ( minus_minus @ A @ ( plus_plus @ A @ X @ X ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_3075_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_3076_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_3077_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_3078_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_3079_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_3080_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_3081_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_3082_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_3083_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_3084_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_3085_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_3086_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X4 ) )
& ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_3087_pred__numeral__inc,axiom,
! [K: num] :
( ( pred_numeral @ ( inc @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% pred_numeral_inc
thf(fact_3088_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_3089_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_3090_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_3091_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_3092_int__prod,axiom,
! [B: $tType,F2: B > nat,A4: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ int
@ ^ [X: B] : ( semiring_1_of_nat @ int @ ( F2 @ X ) )
@ A4 ) ) ).
% int_prod
thf(fact_3093_pi__neq__zero,axiom,
( pi
!= ( zero_zero @ real ) ) ).
% pi_neq_zero
thf(fact_3094_num__induct,axiom,
! [P: num > $o,X3: num] :
( ( P @ one2 )
=> ( ! [X4: num] :
( ( P @ X4 )
=> ( P @ ( inc @ X4 ) ) )
=> ( P @ X3 ) ) ) ).
% num_induct
thf(fact_3095_add__inc,axiom,
! [X3: num,Y3: num] :
( ( plus_plus @ num @ X3 @ ( inc @ Y3 ) )
= ( inc @ ( plus_plus @ num @ X3 @ Y3 ) ) ) ).
% add_inc
thf(fact_3096_pi__gt__zero,axiom,
ord_less @ real @ ( zero_zero @ real ) @ pi ).
% pi_gt_zero
thf(fact_3097_pi__not__less__zero,axiom,
~ ( ord_less @ real @ pi @ ( zero_zero @ real ) ) ).
% pi_not_less_zero
thf(fact_3098_pi__ge__zero,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ pi ).
% pi_ge_zero
thf(fact_3099_inc_Osimps_I1_J,axiom,
( ( inc @ one2 )
= ( bit0 @ one2 ) ) ).
% inc.simps(1)
thf(fact_3100_inc_Osimps_I3_J,axiom,
! [X3: num] :
( ( inc @ ( bit1 @ X3 ) )
= ( bit0 @ ( inc @ X3 ) ) ) ).
% inc.simps(3)
thf(fact_3101_inc_Osimps_I2_J,axiom,
! [X3: num] :
( ( inc @ ( bit0 @ X3 ) )
= ( bit1 @ X3 ) ) ).
% inc.simps(2)
thf(fact_3102_add__One,axiom,
! [X3: num] :
( ( plus_plus @ num @ X3 @ one2 )
= ( inc @ X3 ) ) ).
% add_One
thf(fact_3103_mult__inc,axiom,
! [X3: num,Y3: num] :
( ( times_times @ num @ X3 @ ( inc @ Y3 ) )
= ( plus_plus @ num @ ( times_times @ num @ X3 @ Y3 ) @ X3 ) ) ).
% mult_inc
thf(fact_3104_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
@ ^ [X: int] : X
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ J ) ) ) ) ).
% prod_int_eq
thf(fact_3105_numeral__inc,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X3: num] :
( ( numeral_numeral @ A @ ( inc @ X3 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X3 ) @ ( one_one @ A ) ) ) ) ).
% numeral_inc
thf(fact_3106_pi__less__4,axiom,
ord_less @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ).
% pi_less_4
thf(fact_3107_pi__ge__two,axiom,
ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ).
% pi_ge_two
thf(fact_3108_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_3109_prod__int__plus__eq,axiom,
! [I: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J ) ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X: int] : X
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_3110_pi__half__neq__zero,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% pi_half_neq_zero
thf(fact_3111_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_3112_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_3113_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_3114_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_3115_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_3116_arctan__ubound,axiom,
! [Y3: real] : ( ord_less @ real @ ( arctan @ Y3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arctan_ubound
thf(fact_3117_arctan__one,axiom,
( ( arctan @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_one
thf(fact_3118_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_3119_arctan__bounded,axiom,
! [Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y3 ) )
& ( ord_less @ real @ ( arctan @ Y3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_bounded
thf(fact_3120_arctan__lbound,axiom,
! [Y3: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y3 ) ) ).
% arctan_lbound
thf(fact_3121_prod__power__distrib,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ B )
=> ! [F2: A > B,A4: set @ A,N: nat] :
( ( power_power @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A4 ) @ N )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N )
@ A4 ) ) ) ).
% prod_power_distrib
thf(fact_3122_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_3123_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_3124_prod__mono,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F2: B > A,G: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) )
& ( ord_less_eq @ A @ ( F2 @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) ) ) ) ).
% prod_mono
thf(fact_3125_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) ) ) ) ).
% prod_nonneg
thf(fact_3126_prod__pos,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ X4 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) ) ) ) ).
% prod_pos
thf(fact_3127_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F2 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) ) ) ) ).
% prod_ge_1
thf(fact_3128_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_3129_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_3130_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_3131_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_3132_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q3: A,R2: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q3 @ R2 ) )
= ( R2
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_3133_summable__arctan__series,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_3134_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_3135_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_3136_summable__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( summable @ A
@ ^ [N3: nat] : ( zero_zero @ A ) ) ) ).
% summable_zero
thf(fact_3137_sin__pi,axiom,
( ( sin @ real @ pi )
= ( zero_zero @ real ) ) ).
% sin_pi
thf(fact_3138_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_3139_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_3140_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_3141_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_3142_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_3143_sin__npi,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_npi
thf(fact_3144_sin__npi2,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) )
= ( zero_zero @ real ) ) ).
% sin_npi2
thf(fact_3145_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_3146_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_3147_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_3148_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_3149_sin__two__pi,axiom,
( ( sin @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_two_pi
thf(fact_3150_sin__pi__half,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) ) ).
% sin_pi_half
thf(fact_3151_sin__periodic,axiom,
! [X3: real] :
( ( sin @ real @ ( plus_plus @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( sin @ real @ X3 ) ) ).
% sin_periodic
thf(fact_3152_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_3153_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_3154_sin__2pi__minus,axiom,
! [X3: real] :
( ( sin @ real @ ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ X3 ) )
= ( uminus_uminus @ real @ ( sin @ real @ X3 ) ) ) ).
% sin_2pi_minus
thf(fact_3155_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_3156_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_3157_summable__comparison__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,G: nat > real] :
( ? [N7: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N7 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) @ ( G @ N2 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_comparison_test
thf(fact_3158_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_3159_summable__const__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C2: A] :
( ( summable @ A
@ ^ [Uu3: nat] : C2 )
= ( C2
= ( zero_zero @ A ) ) ) ) ).
% summable_const_iff
thf(fact_3160_powser__insidea,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,X3: A,Z: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) )
=> ( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) ) ) ) ) ) ).
% powser_insidea
thf(fact_3161_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_3162_sin__x__le__x,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( sin @ real @ X3 ) @ X3 ) ) ).
% sin_x_le_x
thf(fact_3163_sin__le__one,axiom,
! [X3: real] : ( ord_less_eq @ real @ ( sin @ real @ X3 ) @ ( one_one @ real ) ) ).
% sin_le_one
thf(fact_3164_abs__sin__x__le__abs__x,axiom,
! [X3: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X3 ) ) @ ( abs_abs @ real @ X3 ) ) ).
% abs_sin_x_le_abs_x
thf(fact_3165_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_3166_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_3167_sin__int__times__real,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [M: int,X3: real] :
( ( sin @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ M ) @ ( real_Vector_of_real @ A @ X3 ) ) )
= ( real_Vector_of_real @ A @ ( sin @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ M ) @ X3 ) ) ) ) ) ).
% sin_int_times_real
thf(fact_3168_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_3169_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_3170_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_3171_sin__gt__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ pi )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X3 ) ) ) ) ).
% sin_gt_zero
thf(fact_3172_sin__x__ge__neg__x,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( uminus_uminus @ real @ X3 ) @ ( sin @ real @ X3 ) ) ) ).
% sin_x_ge_neg_x
thf(fact_3173_sin__ge__zero,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ pi )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X3 ) ) ) ) ).
% sin_ge_zero
thf(fact_3174_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_3175_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_3176_sin__ge__minus__one,axiom,
! [X3: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( sin @ real @ X3 ) ) ).
% sin_ge_minus_one
thf(fact_3177_summable__powser__split__head,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,Z: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ ( suc @ N3 ) ) @ ( power_power @ A @ Z @ N3 ) ) )
= ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) ) ) ) ).
% summable_powser_split_head
thf(fact_3178_powser__split__head_I3_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ ( suc @ N3 ) ) @ ( power_power @ A @ Z @ N3 ) ) ) ) ) ).
% powser_split_head(3)
thf(fact_3179_abs__sin__le__one,axiom,
! [X3: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X3 ) ) @ ( one_one @ real ) ) ).
% abs_sin_le_one
thf(fact_3180_summable__powser__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,M: nat,Z: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ ( plus_plus @ nat @ N3 @ M ) ) @ ( power_power @ A @ Z @ N3 ) ) )
= ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_3181_summable__norm__comparison__test,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > real] :
( ? [N7: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N7 @ 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_3182_summable__rabs__comparison__test,axiom,
! [F2: nat > real,G: nat > real] :
( ? [N7: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N7 @ 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_3183_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_3184_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_3185_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_3186_powser__inside,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,X3: A,Z: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) ) ) ) ) ).
% powser_inside
thf(fact_3187_sin__eq__0__pi,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X3 )
=> ( ( ord_less @ real @ X3 @ pi )
=> ( ( ( sin @ real @ X3 )
= ( zero_zero @ real ) )
=> ( X3
= ( zero_zero @ real ) ) ) ) ) ).
% sin_eq_0_pi
thf(fact_3188_complete__algebra__summable__geometric,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( one_one @ real ) )
=> ( summable @ A @ ( power_power @ A @ X3 ) ) ) ) ).
% complete_algebra_summable_geometric
thf(fact_3189_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_3190_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_3191_sin__zero__pi__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ pi )
=> ( ( ( sin @ real @ X3 )
= ( zero_zero @ real ) )
= ( X3
= ( zero_zero @ real ) ) ) ) ).
% sin_zero_pi_iff
thf(fact_3192_sin__zero__iff__int2,axiom,
! [X3: real] :
( ( ( sin @ real @ X3 )
= ( zero_zero @ real ) )
= ( ? [I4: int] :
( X3
= ( times_times @ real @ ( ring_1_of_int @ real @ I4 ) @ pi ) ) ) ) ).
% sin_zero_iff_int2
thf(fact_3193_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_3194_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_3195_sin__gt__zero__02,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X3 ) ) ) ) ).
% sin_gt_zero_02
thf(fact_3196_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_3197_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_3198_powser__split__head_I1_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) )
= ( plus_plus @ A @ ( F2 @ ( zero_zero @ nat ) )
@ ( times_times @ A
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ ( suc @ N3 ) ) @ ( power_power @ A @ Z @ N3 ) ) )
@ Z ) ) ) ) ) ).
% powser_split_head(1)
thf(fact_3199_powser__split__head_I2_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) )
=> ( ( times_times @ A
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ ( suc @ N3 ) ) @ ( power_power @ A @ Z @ N3 ) ) )
@ Z )
= ( minus_minus @ A
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) )
@ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% powser_split_head(2)
thf(fact_3200_suminf__exist__split,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real,F2: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( 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 ) ) ) )
@ R2 ) ) ) ) ) ).
% suminf_exist_split
thf(fact_3201_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_3202_summable__power__series,axiom,
! [F2: nat > real,Z: real] :
( ! [I3: nat] : ( ord_less_eq @ real @ ( F2 @ I3 ) @ ( one_one @ real ) )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ I3 ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Z )
=> ( ( ord_less @ real @ Z @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [I4: nat] : ( times_times @ real @ ( F2 @ I4 ) @ ( power_power @ real @ Z @ I4 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_3203_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_3204_Abel__lemma,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real,R0: real,A2: nat > A,M7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( ord_less @ real @ R2 @ R0 )
=> ( ! [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 @ R2 @ N3 ) ) ) ) ) ) ) ).
% Abel_lemma
thf(fact_3205_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_3206_sin__gt__zero2,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X3 ) ) ) ) ).
% sin_gt_zero2
thf(fact_3207_sin__lt__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ pi @ X3 )
=> ( ( ord_less @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less @ real @ ( sin @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ).
% sin_lt_zero
thf(fact_3208_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_3209_sin__inj__pi,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( sin @ real @ X3 )
= ( sin @ real @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% sin_inj_pi
thf(fact_3210_sin__mono__le__eq,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( sin @ real @ X3 ) @ ( sin @ real @ Y3 ) )
= ( ord_less_eq @ real @ X3 @ Y3 ) ) ) ) ) ) ).
% sin_mono_le_eq
thf(fact_3211_sin__monotone__2pi__le,axiom,
! [Y3: real,X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sin @ real @ Y3 ) @ ( sin @ real @ X3 ) ) ) ) ) ).
% sin_monotone_2pi_le
thf(fact_3212_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_3213_sin__le__zero,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ pi @ X3 )
=> ( ( ord_less @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less_eq @ real @ ( sin @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ).
% sin_le_zero
thf(fact_3214_sin__less__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( sin @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ).
% sin_less_zero
thf(fact_3215_sin__mono__less__eq,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( sin @ real @ X3 ) @ ( sin @ real @ Y3 ) )
= ( ord_less @ real @ X3 @ Y3 ) ) ) ) ) ) ).
% sin_mono_less_eq
thf(fact_3216_sin__monotone__2pi,axiom,
! [Y3: real,X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
=> ( ( ord_less @ real @ Y3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sin @ real @ Y3 ) @ ( sin @ real @ X3 ) ) ) ) ) ).
% sin_monotone_2pi
thf(fact_3217_sin__total,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ? [X4: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X4 )
& ( ord_less_eq @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X4 )
= Y3 )
& ! [Y5: real] :
( ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y5 )
& ( ord_less_eq @ real @ Y5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ Y5 )
= Y3 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% sin_total
thf(fact_3218_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_3219_sin__arctan,axiom,
! [X3: real] :
( ( sin @ real @ ( arctan @ X3 ) )
= ( divide_divide @ real @ X3 @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_arctan
thf(fact_3220_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_3221_sin__zero__iff__int,axiom,
! [X3: real] :
( ( ( sin @ real @ X3 )
= ( zero_zero @ real ) )
= ( ? [I4: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I4 )
& ( X3
= ( times_times @ real @ ( ring_1_of_int @ real @ I4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_iff_int
thf(fact_3222_sin__zero__lemma,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ( sin @ real @ X3 )
= ( zero_zero @ real ) )
=> ? [N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X3
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_lemma
thf(fact_3223_sin__zero__iff,axiom,
! [X3: real] :
( ( ( sin @ real @ X3 )
= ( zero_zero @ real ) )
= ( ? [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X3
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X3
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% sin_zero_iff
thf(fact_3224_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_3225_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_3226_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_3227_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A2: int,Q3: int,R2: int] :
( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A2 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q3 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_3228_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs2: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs2 @ Ys ) @ N )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ ( divide_divide @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth @ B @ Ys @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3229_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_3230_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_3231_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_3232_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_3233_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X3: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y3: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X3 @ Xa @ Xb @ Xc )
= Y3 )
=> ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y3 = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y3
= ( set_fo6178422350223883121st_nat @ A @ X3 @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X3 @ Xa @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_3234_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F3: nat > A > A,A5: nat,B4: nat,Acc: A] : ( if @ A @ ( ord_less @ nat @ B4 @ A5 ) @ Acc @ ( set_fo6178422350223883121st_nat @ A @ F3 @ ( plus_plus @ nat @ A5 @ ( one_one @ nat ) ) @ B4 @ ( F3 @ A5 @ Acc ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_3235_zminus1__lemma,axiom,
! [A2: int,B2: int,Q3: int,R2: int] :
( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( ( B2
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A2 ) @ B2
@ ( product_Pair @ int @ int
@ ( if @ int
@ ( R2
= ( zero_zero @ int ) )
@ ( uminus_uminus @ int @ Q3 )
@ ( minus_minus @ int @ ( uminus_uminus @ int @ Q3 ) @ ( one_one @ int ) ) )
@ ( if @ int
@ ( R2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ B2 @ R2 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_3236_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q3: int,R2: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
= ( ( K
= ( plus_plus @ int @ ( times_times @ int @ L @ Q3 ) @ R2 ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
& ( ord_less @ int @ R2 @ L ) ) )
& ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ R2 )
& ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) ) ) )
& ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( Q3
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_3237_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A2: int,Q3: int,R2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q3 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_3238_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_3239_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_3240_sincos__total__2pi,axiom,
! [X3: real,Y3: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( 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 ) )
=> ( ( X3
= ( cos @ real @ T4 ) )
=> ( Y3
!= ( sin @ real @ T4 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_3241_sin__tan,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( sin @ real @ X3 )
= ( divide_divide @ real @ ( tan @ real @ X3 ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_tan
thf(fact_3242_vebt__buildup_Oelims,axiom,
! [X3: nat,Y3: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X3 )
= Y3 )
=> ( ( ( X3
= ( zero_zero @ nat ) )
=> ( Y3
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X3
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y3
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va: nat] :
( ( X3
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y3
= ( 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 ) ) )
=> ( Y3
= ( 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_3243_intind,axiom,
! [A: $tType,I: nat,N: nat,P: A > $o,X3: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( P @ X3 )
=> ( P @ ( nth @ A @ ( replicate @ A @ N @ X3 ) @ I ) ) ) ) ).
% intind
thf(fact_3244_bit_Ocompl__eq__compl__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A,Y3: A] :
( ( ( bit_ri4277139882892585799ns_not @ A @ X3 )
= ( bit_ri4277139882892585799ns_not @ A @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% bit.compl_eq_compl_iff
thf(fact_3245_bit_Odouble__compl,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( bit_ri4277139882892585799ns_not @ A @ X3 ) )
= X3 ) ) ).
% bit.double_compl
thf(fact_3246_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_3247_replicate__eq__replicate,axiom,
! [A: $tType,M: nat,X3: A,N: nat,Y3: A] :
( ( ( replicate @ A @ M @ X3 )
= ( replicate @ A @ N @ Y3 ) )
= ( ( M = N )
& ( ( M
!= ( zero_zero @ nat ) )
=> ( X3 = Y3 ) ) ) ) ).
% replicate_eq_replicate
thf(fact_3248_bit_Oxor__compl__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A,Y3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X3 @ ( bit_ri4277139882892585799ns_not @ A @ Y3 ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344971392196577ns_xor @ A @ X3 @ Y3 ) ) ) ) ).
% bit.xor_compl_right
thf(fact_3249_bit_Oxor__compl__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A,Y3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X3 ) @ Y3 )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344971392196577ns_xor @ A @ X3 @ Y3 ) ) ) ) ).
% bit.xor_compl_left
thf(fact_3250_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_3251_in__set__replicate,axiom,
! [A: $tType,X3: A,N: nat,Y3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( replicate @ A @ N @ Y3 ) ) )
= ( ( X3 = Y3 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% in_set_replicate
thf(fact_3252_Bex__set__replicate,axiom,
! [A: $tType,N: nat,A2: A,P: A > $o] :
( ( ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N @ A2 ) ) )
& ( P @ X ) ) )
= ( ( P @ A2 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_3253_Ball__set__replicate,axiom,
! [A: $tType,N: nat,A2: A,P: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N @ A2 ) ) )
=> ( P @ X ) ) )
= ( ( P @ A2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_3254_nth__replicate,axiom,
! [A: $tType,I: nat,N: nat,X3: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( replicate @ A @ N @ X3 ) @ I )
= X3 ) ) ).
% nth_replicate
thf(fact_3255_tan__pi,axiom,
( ( tan @ real @ pi )
= ( zero_zero @ real ) ) ).
% tan_pi
thf(fact_3256_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_3257_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_3258_bit_Odisj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se1065995026697491101ons_or @ A @ X3 @ ( bit_ri4277139882892585799ns_not @ A @ X3 ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_right
thf(fact_3259_bit_Odisj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X3 ) @ X3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_left
thf(fact_3260_bit_Oxor__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X3 @ ( bit_ri4277139882892585799ns_not @ A @ X3 ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_right
thf(fact_3261_bit_Oxor__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X3 ) @ X3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_left
thf(fact_3262_bit_Oxor__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ X3 ) ) ) ).
% bit.xor_one_right
thf(fact_3263_bit_Oxor__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X3 )
= ( bit_ri4277139882892585799ns_not @ A @ X3 ) ) ) ).
% bit.xor_one_left
thf(fact_3264_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_3265_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_3266_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_3267_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_3268_set__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X3 ) )
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_3269_tan__npi,axiom,
! [N: nat] :
( ( tan @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% tan_npi
thf(fact_3270_tan__periodic__n,axiom,
! [X3: real,N: num] :
( ( tan @ real @ ( plus_plus @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ N ) @ pi ) ) )
= ( tan @ real @ X3 ) ) ).
% tan_periodic_n
thf(fact_3271_tan__periodic__nat,axiom,
! [X3: real,N: nat] :
( ( tan @ real @ ( plus_plus @ real @ X3 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) ) )
= ( tan @ real @ X3 ) ) ).
% tan_periodic_nat
thf(fact_3272_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_3273_cos__pi__half,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ).
% cos_pi_half
thf(fact_3274_cos__two__pi,axiom,
( ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ real ) ) ).
% cos_two_pi
thf(fact_3275_cos__periodic,axiom,
! [X3: real] :
( ( cos @ real @ ( plus_plus @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( cos @ real @ X3 ) ) ).
% cos_periodic
thf(fact_3276_cos__2pi__minus,axiom,
! [X3: real] :
( ( cos @ real @ ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ X3 ) )
= ( cos @ real @ X3 ) ) ).
% cos_2pi_minus
thf(fact_3277_tan__periodic,axiom,
! [X3: real] :
( ( tan @ real @ ( plus_plus @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( tan @ real @ X3 ) ) ).
% tan_periodic
thf(fact_3278_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_3279_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_3280_sin__cos__squared__add2,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( plus_plus @ A @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add2
thf(fact_3281_sin__cos__squared__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( plus_plus @ A @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add
thf(fact_3282_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_3283_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_3284_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_3285_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_3286_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_3287_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_3288_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_3289_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_3290_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_3291_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_3292_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_3293_cos__le__one,axiom,
! [X3: real] : ( ord_less_eq @ real @ ( cos @ real @ X3 ) @ ( one_one @ real ) ) ).
% cos_le_one
thf(fact_3294_cos__arctan__not__zero,axiom,
! [X3: real] :
( ( cos @ real @ ( arctan @ X3 ) )
!= ( zero_zero @ real ) ) ).
% cos_arctan_not_zero
thf(fact_3295_cos__int__times__real,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [M: int,X3: real] :
( ( cos @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ M ) @ ( real_Vector_of_real @ A @ X3 ) ) )
= ( real_Vector_of_real @ A @ ( cos @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ M ) @ X3 ) ) ) ) ) ).
% cos_int_times_real
thf(fact_3296_add__tan__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y3: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y3 )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y3 ) )
= ( divide_divide @ A @ ( sin @ A @ ( plus_plus @ A @ X3 @ Y3 ) ) @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y3 ) ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_3297_cos__one__sin__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( cos @ A @ X3 )
= ( one_one @ A ) )
=> ( ( sin @ A @ X3 )
= ( zero_zero @ A ) ) ) ) ).
% cos_one_sin_zero
thf(fact_3298_tan__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y3: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( plus_plus @ A @ X3 @ Y3 ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( plus_plus @ A @ X3 @ Y3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y3 ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y3 ) ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_3299_tan__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y3: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( minus_minus @ A @ X3 @ Y3 ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( minus_minus @ A @ X3 @ Y3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y3 ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y3 ) ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_3300_lemma__tan__add1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y3: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y3 )
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y3 ) ) )
= ( divide_divide @ A @ ( cos @ A @ ( plus_plus @ A @ X3 @ Y3 ) ) @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y3 ) ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_3301_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_3302_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_3303_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_3304_cos__monotone__0__pi__le,axiom,
! [Y3: real,X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ pi )
=> ( ord_less_eq @ real @ ( cos @ real @ X3 ) @ ( cos @ real @ Y3 ) ) ) ) ) ).
% cos_monotone_0_pi_le
thf(fact_3305_cos__mono__le__eq,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ pi )
=> ( ( ord_less_eq @ real @ ( cos @ real @ X3 ) @ ( cos @ real @ Y3 ) )
= ( ord_less_eq @ real @ Y3 @ X3 ) ) ) ) ) ) ).
% cos_mono_le_eq
thf(fact_3306_cos__inj__pi,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ pi )
=> ( ( ( cos @ real @ X3 )
= ( cos @ real @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% cos_inj_pi
thf(fact_3307_cos__ge__minus__one,axiom,
! [X3: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( cos @ real @ X3 ) ) ).
% cos_ge_minus_one
thf(fact_3308_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_3309_abs__cos__le__one,axiom,
! [X3: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( cos @ real @ X3 ) ) @ ( one_one @ real ) ) ).
% abs_cos_le_one
thf(fact_3310_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_3311_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_3312_sin__zero__norm__cos__one,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( sin @ A @ X3 )
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( cos @ A @ X3 ) )
= ( one_one @ real ) ) ) ) ).
% sin_zero_norm_cos_one
thf(fact_3313_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_3314_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_3315_cos__two__neq__zero,axiom,
( ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% cos_two_neq_zero
thf(fact_3316_cos__mono__less__eq,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ pi )
=> ( ( ord_less @ real @ ( cos @ real @ X3 ) @ ( cos @ real @ Y3 ) )
= ( ord_less @ real @ Y3 @ X3 ) ) ) ) ) ) ).
% cos_mono_less_eq
thf(fact_3317_cos__monotone__0__pi,axiom,
! [Y3: real,X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less @ real @ Y3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ pi )
=> ( ord_less @ real @ ( cos @ real @ X3 ) @ ( cos @ real @ Y3 ) ) ) ) ) ).
% cos_monotone_0_pi
thf(fact_3318_tan__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A )
= ( ^ [X: A] : ( divide_divide @ A @ ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) @ ( plus_plus @ A @ ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% tan_half
thf(fact_3319_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_3320_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_3321_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_3322_cos__monotone__minus__pi__0_H,axiom,
! [Y3: real,X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( cos @ real @ Y3 ) @ ( cos @ real @ X3 ) ) ) ) ) ).
% cos_monotone_minus_pi_0'
thf(fact_3323_set__replicate__conv__if,axiom,
! [A: $tType,N: nat,X3: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X3 ) )
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_3324_sin__zero__abs__cos__one,axiom,
! [X3: real] :
( ( ( sin @ real @ X3 )
= ( zero_zero @ real ) )
=> ( ( abs_abs @ real @ ( cos @ real @ X3 ) )
= ( one_one @ real ) ) ) ).
% sin_zero_abs_cos_one
thf(fact_3325_sin__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ X3 ) ) @ ( cos @ A @ X3 ) ) ) ) ).
% sin_double
thf(fact_3326_cos__two__less__zero,axiom,
ord_less @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).
% cos_two_less_zero
thf(fact_3327_cos__is__zero,axiom,
? [X4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less_eq @ real @ X4 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X4 )
= ( zero_zero @ real ) )
& ! [Y5: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y5 )
& ( ord_less_eq @ real @ Y5 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ Y5 )
= ( zero_zero @ real ) ) )
=> ( Y5 = X4 ) ) ) ).
% cos_is_zero
thf(fact_3328_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_3329_tan__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( tan @ A @ X3 ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% tan_double
thf(fact_3330_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_3331_cos__monotone__minus__pi__0,axiom,
! [Y3: real,X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y3 )
=> ( ( ord_less @ real @ Y3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cos @ real @ Y3 ) @ ( cos @ real @ X3 ) ) ) ) ) ).
% cos_monotone_minus_pi_0
thf(fact_3332_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_3333_cos__total,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ? [X4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less_eq @ real @ X4 @ pi )
& ( ( cos @ real @ X4 )
= Y3 )
& ! [Y5: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y5 )
& ( ord_less_eq @ real @ Y5 @ pi )
& ( ( cos @ real @ Y5 )
= Y3 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% cos_total
thf(fact_3334_sincos__principal__value,axiom,
! [X3: real] :
? [Y4: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ Y4 )
& ( ord_less_eq @ real @ Y4 @ pi )
& ( ( sin @ real @ Y4 )
= ( sin @ real @ X3 ) )
& ( ( cos @ real @ Y4 )
= ( cos @ real @ X3 ) ) ) ).
% sincos_principal_value
thf(fact_3335_cos__tan,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( cos @ real @ X3 )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_tan
thf(fact_3336_tan__45,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
= ( one_one @ real ) ) ).
% tan_45
thf(fact_3337_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_3338_tan__60,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
= ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) ).
% tan_60
thf(fact_3339_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_3340_sin__cos__le1,axiom,
! [X3: real,Y3: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( plus_plus @ real @ ( times_times @ real @ ( sin @ real @ X3 ) @ ( sin @ real @ Y3 ) ) @ ( times_times @ real @ ( cos @ real @ X3 ) @ ( cos @ real @ Y3 ) ) ) ) @ ( one_one @ real ) ) ).
% sin_cos_le1
thf(fact_3341_cos__plus__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z: A] :
( ( plus_plus @ A @ ( cos @ A @ W ) @ ( cos @ A @ Z ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_3342_cos__times__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z: A] :
( ( times_times @ A @ ( cos @ A @ W ) @ ( cos @ A @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( cos @ A @ ( minus_minus @ A @ W @ Z ) ) @ ( cos @ A @ ( plus_plus @ A @ W @ Z ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% cos_times_cos
thf(fact_3343_sin__squared__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_squared_eq
thf(fact_3344_cos__squared__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cos_squared_eq
thf(fact_3345_tan__gt__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( tan @ real @ X3 ) ) ) ) ).
% tan_gt_zero
thf(fact_3346_lemma__tan__total,axiom,
! [Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ? [X4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ord_less @ real @ Y3 @ ( tan @ real @ X4 ) ) ) ) ).
% lemma_tan_total
thf(fact_3347_tan__total,axiom,
! [Y3: real] :
? [X4: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X4 )
& ( ord_less @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X4 )
= Y3 )
& ! [Y5: real] :
( ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y5 )
& ( ord_less @ real @ Y5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ Y5 )
= Y3 ) )
=> ( Y5 = X4 ) ) ) ).
% tan_total
thf(fact_3348_tan__monotone,axiom,
! [Y3: real,X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
=> ( ( ord_less @ real @ Y3 @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( tan @ real @ Y3 ) @ ( tan @ real @ X3 ) ) ) ) ) ).
% tan_monotone
thf(fact_3349_tan__monotone_H,axiom,
! [Y3: real,X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
=> ( ( ord_less @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y3 @ X3 )
= ( ord_less @ real @ ( tan @ real @ Y3 ) @ ( tan @ real @ X3 ) ) ) ) ) ) ) ).
% tan_monotone'
thf(fact_3350_tan__mono__lt__eq,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
=> ( ( ord_less @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( tan @ real @ X3 ) @ ( tan @ real @ Y3 ) )
= ( ord_less @ real @ X3 @ Y3 ) ) ) ) ) ) ).
% tan_mono_lt_eq
thf(fact_3351_lemma__tan__total1,axiom,
! [Y3: real] :
? [X4: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X4 )
& ( ord_less @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X4 )
= Y3 ) ) ).
% lemma_tan_total1
thf(fact_3352_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_3353_cos__double__less__one,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 ) ) @ ( one_one @ real ) ) ) ) ).
% cos_double_less_one
thf(fact_3354_tan__inverse,axiom,
! [Y3: real] :
( ( divide_divide @ real @ ( one_one @ real ) @ ( tan @ real @ Y3 ) )
= ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y3 ) ) ) ).
% tan_inverse
thf(fact_3355_cos__gt__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X3 ) ) ) ) ).
% cos_gt_zero
thf(fact_3356_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_3357_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_3358_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_3359_cos__one__2pi__int,axiom,
! [X3: real] :
( ( ( cos @ real @ X3 )
= ( one_one @ real ) )
= ( ? [X: int] :
( X3
= ( times_times @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ X ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ).
% cos_one_2pi_int
thf(fact_3360_cos__double__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ W ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ ( cos @ A @ W ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) ) ) ) ).
% cos_double_cos
thf(fact_3361_cos__treble__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) @ X3 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) @ ( cos @ A @ X3 ) ) ) ) ) ).
% cos_treble_cos
thf(fact_3362_cos__diff__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z: A] :
( ( minus_minus @ A @ ( cos @ A @ W ) @ ( cos @ A @ Z ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ Z @ W ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_3363_sin__diff__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z: A] :
( ( minus_minus @ A @ ( sin @ A @ W ) @ ( sin @ A @ Z ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_3364_sin__plus__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z: A] :
( ( plus_plus @ A @ ( sin @ A @ W ) @ ( sin @ A @ Z ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_3365_cos__times__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z: A] :
( ( times_times @ A @ ( cos @ A @ W ) @ ( sin @ A @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( sin @ A @ ( plus_plus @ A @ W @ Z ) ) @ ( sin @ A @ ( minus_minus @ A @ W @ Z ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% cos_times_sin
thf(fact_3366_sin__times__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z: A] :
( ( times_times @ A @ ( sin @ A @ W ) @ ( cos @ A @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( sin @ A @ ( plus_plus @ A @ W @ Z ) ) @ ( sin @ A @ ( minus_minus @ A @ W @ Z ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% sin_times_cos
thf(fact_3367_sin__times__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z: A] :
( ( times_times @ A @ ( sin @ A @ W ) @ ( sin @ A @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( cos @ A @ ( minus_minus @ A @ W @ Z ) ) @ ( cos @ A @ ( plus_plus @ A @ W @ Z ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% sin_times_sin
thf(fact_3368_cos__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) )
= ( minus_minus @ A @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cos_double
thf(fact_3369_cos__sin__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cos @ A )
= ( ^ [X: A] : ( sin @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X ) ) ) ) ) ).
% cos_sin_eq
thf(fact_3370_sin__cos__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( sin @ A )
= ( ^ [X: A] : ( cos @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X ) ) ) ) ) ).
% sin_cos_eq
thf(fact_3371_tan__total__pos,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ? [X4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X4 )
= Y3 ) ) ) ).
% tan_total_pos
thf(fact_3372_tan__pos__pi2__le,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tan @ real @ X3 ) ) ) ) ).
% tan_pos_pi2_le
thf(fact_3373_tan__less__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( tan @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ).
% tan_less_zero
thf(fact_3374_tan__mono__le,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ Y3 )
=> ( ( ord_less @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( tan @ real @ X3 ) @ ( tan @ real @ Y3 ) ) ) ) ) ).
% tan_mono_le
thf(fact_3375_tan__mono__le__eq,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
=> ( ( ord_less @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( tan @ real @ X3 ) @ ( tan @ real @ Y3 ) )
= ( ord_less_eq @ real @ X3 @ Y3 ) ) ) ) ) ) ).
% tan_mono_le_eq
thf(fact_3376_tan__bound__pi2,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( abs_abs @ real @ ( tan @ real @ X3 ) ) @ ( one_one @ real ) ) ) ).
% tan_bound_pi2
thf(fact_3377_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_3378_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_3379_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_3380_cos__gt__zero__pi,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X3 ) ) ) ) ).
% cos_gt_zero_pi
thf(fact_3381_cos__ge__zero,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cos @ real @ X3 ) ) ) ) ).
% cos_ge_zero
thf(fact_3382_arctan,axiom,
! [Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y3 ) )
& ( ord_less @ real @ ( arctan @ Y3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ ( arctan @ Y3 ) )
= Y3 ) ) ).
% arctan
thf(fact_3383_arctan__tan,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arctan @ ( tan @ real @ X3 ) )
= X3 ) ) ) ).
% arctan_tan
thf(fact_3384_arctan__unique,axiom,
! [X3: real,Y3: 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 )
= Y3 )
=> ( ( arctan @ Y3 )
= X3 ) ) ) ) ).
% arctan_unique
thf(fact_3385_cos__one__2pi,axiom,
! [X3: real] :
( ( ( cos @ real @ X3 )
= ( one_one @ real ) )
= ( ? [X: nat] :
( X3
= ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
| ? [X: nat] :
( X3
= ( uminus_uminus @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ) ).
% cos_one_2pi
thf(fact_3386_cos__double__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ W ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ ( sin @ A @ W ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_double_sin
thf(fact_3387_minus__sin__cos__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( uminus_uminus @ A @ ( sin @ A @ X3 ) )
= ( cos @ A @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% minus_sin_cos_eq
thf(fact_3388_tan__total__pi4,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ? [Z3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) @ Z3 )
& ( ord_less @ real @ Z3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
& ( ( tan @ real @ Z3 )
= X3 ) ) ) ).
% tan_total_pi4
thf(fact_3389_cos__arctan,axiom,
! [X3: real] :
( ( cos @ real @ ( arctan @ X3 ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_arctan
thf(fact_3390_sincos__total__pi,axiom,
! [Y3: real,X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ pi )
& ( X3
= ( cos @ real @ T4 ) )
& ( Y3
= ( sin @ real @ T4 ) ) ) ) ) ).
% sincos_total_pi
thf(fact_3391_sin__cos__sqrt,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X3 ) )
=> ( ( sin @ real @ X3 )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( cos @ real @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_cos_sqrt
thf(fact_3392_sin__expansion__lemma,axiom,
! [X3: real,M: nat] :
( ( sin @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( cos @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_expansion_lemma
thf(fact_3393_cos__zero__iff__int,axiom,
! [X3: real] :
( ( ( cos @ real @ X3 )
= ( zero_zero @ real ) )
= ( ? [I4: int] :
( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I4 )
& ( X3
= ( times_times @ real @ ( ring_1_of_int @ real @ I4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_iff_int
thf(fact_3394_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_3395_cos__zero__lemma,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ( cos @ real @ X3 )
= ( zero_zero @ real ) )
=> ? [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X3
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_lemma
thf(fact_3396_cos__zero__iff,axiom,
! [X3: real] :
( ( ( cos @ real @ X3 )
= ( zero_zero @ real ) )
= ( ? [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X3
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X3
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% cos_zero_iff
thf(fact_3397_cos__expansion__lemma,axiom,
! [X3: real,M: nat] :
( ( cos @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( uminus_uminus @ real @ ( sin @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_expansion_lemma
thf(fact_3398_sincos__total__pi__half,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( 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 ) ) ) )
& ( X3
= ( cos @ real @ T4 ) )
& ( Y3
= ( sin @ real @ T4 ) ) ) ) ) ) ).
% sincos_total_pi_half
thf(fact_3399_sincos__total__2pi__le,axiom,
! [X3: real,Y3: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( 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 ) )
& ( X3
= ( cos @ real @ T4 ) )
& ( Y3
= ( sin @ real @ T4 ) ) ) ) ).
% sincos_total_2pi_le
thf(fact_3400_complex__unimodular__polar,axiom,
! [Z: complex] :
( ( ( real_V7770717601297561774m_norm @ complex @ Z )
= ( 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 ) )
=> ( Z
!= ( complex2 @ ( cos @ real @ T4 ) @ ( sin @ real @ T4 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_3401_obtain__set__succ,axiom,
! [X3: nat,Z: nat,A4: set @ nat,B6: set @ nat] :
( ( ord_less @ nat @ X3 @ Z )
=> ( ( vEBT_VEBT_max_in_set @ A4 @ Z )
=> ( ( finite_finite @ nat @ B6 )
=> ( ( A4 = B6 )
=> ? [X_1: nat] : ( vEBT_is_succ_in_set @ A4 @ X3 @ X_1 ) ) ) ) ) ).
% obtain_set_succ
thf(fact_3402_obtain__set__pred,axiom,
! [Z: nat,X3: nat,A4: set @ nat] :
( ( ord_less @ nat @ Z @ X3 )
=> ( ( vEBT_VEBT_min_in_set @ A4 @ Z )
=> ( ( finite_finite @ nat @ A4 )
=> ? [X_1: nat] : ( vEBT_is_pred_in_set @ A4 @ X3 @ X_1 ) ) ) ) ).
% obtain_set_pred
thf(fact_3403_cos__arcsin,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X3 ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_arcsin
thf(fact_3404_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N: nat,M: nat,X3: A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( ( X3
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) )
& ( ( X3
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X3 @ M ) @ ( power_power @ A @ X3 @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_3405_set__vebt__finite,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( finite_finite @ nat @ ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).
% set_vebt_finite
thf(fact_3406_pred__none__empty,axiom,
! [Xs2: set @ nat,A2: nat] :
( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A2 @ X_1 )
=> ( ( finite_finite @ nat @ Xs2 )
=> ~ ? [X6: nat] :
( ( member @ nat @ X6 @ Xs2 )
& ( ord_less @ nat @ X6 @ A2 ) ) ) ) ).
% pred_none_empty
thf(fact_3407_succ__none__empty,axiom,
! [Xs2: set @ nat,A2: nat] :
( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A2 @ X_1 )
=> ( ( finite_finite @ nat @ Xs2 )
=> ~ ? [X6: nat] :
( ( member @ nat @ X6 @ Xs2 )
& ( ord_less @ nat @ A2 @ X6 ) ) ) ) ).
% succ_none_empty
thf(fact_3408_arcsin__0,axiom,
( ( arcsin @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% arcsin_0
thf(fact_3409_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [Uu3: B] : ( zero_zero @ A )
@ A4 )
= ( zero_zero @ A ) ) ) ).
% sum.neutral_const
thf(fact_3410_of__nat__sum,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [F2: B > nat,A4: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( semiring_1_of_nat @ A @ ( F2 @ X ) )
@ A4 ) ) ) ).
% of_nat_sum
thf(fact_3411_of__int__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ A )
=> ! [F2: B > int,A4: set @ B] :
( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F2 @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( ring_1_of_int @ A @ ( F2 @ X ) )
@ A4 ) ) ) ).
% of_int_sum
thf(fact_3412_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_3413_sum_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A] :
( ~ ( finite_finite @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( zero_zero @ A ) ) ) ) ).
% sum.infinite
thf(fact_3414_sum__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [F4: set @ B,F2: B > A] :
( ( finite_finite @ B @ F4 )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ F4 )
= ( zero_zero @ A ) )
= ( ! [X: B] :
( ( member @ B @ X @ F4 )
=> ( ( F2 @ X )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_3415_prod__zero__iff,axiom,
! [A: $tType,B: $tType] :
( ( semidom @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 )
= ( zero_zero @ A ) )
= ( ? [X: B] :
( ( member @ B @ X @ A4 )
& ( ( F2 @ X )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% prod_zero_iff
thf(fact_3416_infinite__Icc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Icc_iff
thf(fact_3417_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,A2: B,B2: B > A] :
( ( finite_finite @ 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_3418_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,A2: B,B2: B > A] :
( ( finite_finite @ 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_3419_sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A4: set @ A] :
( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A4 ) )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( abs_abs @ B @ ( F2 @ I4 ) )
@ A4 ) ) ) ).
% sum_abs
thf(fact_3420_summable__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A4: set @ nat,F2: nat > A] :
( ( finite_finite @ nat @ A4 )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A4 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite_set
thf(fact_3421_summable__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P: nat > $o,F2: nat > A] :
( ( finite_finite @ nat @ ( collect @ nat @ P ) )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( P @ R5 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite
thf(fact_3422_prod__pos__nat__iff,axiom,
! [A: $tType,A4: set @ A,F2: A > nat] :
( ( finite_finite @ A @ A4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F2 @ A4 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_3423_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A4: set @ A] :
( ord_less_eq @ B @ ( zero_zero @ B )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( abs_abs @ B @ ( F2 @ I4 ) )
@ A4 ) ) ) ).
% sum_abs_ge_zero
thf(fact_3424_sin__arcsin,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arcsin @ Y3 ) )
= Y3 ) ) ) ).
% sin_arcsin
thf(fact_3425_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_3426_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: set @ nat,C2: nat > A] :
( ( ( ( finite_finite @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A4 )
= ( C2 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_3427_arcsin__1,axiom,
( ( arcsin @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arcsin_1
thf(fact_3428_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: set @ nat,C2: nat > A,D3: nat > A] :
( ( ( ( finite_finite @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D3 @ I4 ) )
@ A4 )
= ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D3 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D3 @ I4 ) )
@ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_3429_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_3430_suminf__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [N4: set @ nat,F2: nat > A] :
( ( finite_finite @ 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_3431_sum__norm__le,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [S2: set @ B,F2: B > A,G: B > real] :
( ! [X4: B] :
( ( member @ B @ X4 @ S2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X4 ) ) @ ( G @ X4 ) ) )
=> ( 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_3432_norm__sum,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,A4: set @ B] :
( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) )
@ ( groups7311177749621191930dd_sum @ B @ real
@ ^ [I4: B] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ I4 ) )
@ A4 ) ) ) ).
% norm_sum
thf(fact_3433_complex__eq__cancel__iff2,axiom,
! [X3: real,Y3: real,Xa: real] :
( ( ( complex2 @ X3 @ Y3 )
= ( real_Vector_of_real @ complex @ Xa ) )
= ( ( X3 = Xa )
& ( Y3
= ( zero_zero @ real ) ) ) ) ).
% complex_eq_cancel_iff2
thf(fact_3434_complex__of__real__code,axiom,
( ( real_Vector_of_real @ complex )
= ( ^ [X: real] : ( complex2 @ X @ ( zero_zero @ real ) ) ) ) ).
% complex_of_real_code
thf(fact_3435_complex__of__real__def,axiom,
( ( real_Vector_of_real @ complex )
= ( ^ [R5: real] : ( complex2 @ R5 @ ( zero_zero @ real ) ) ) ) ).
% complex_of_real_def
thf(fact_3436_zero__complex_Ocode,axiom,
( ( zero_zero @ complex )
= ( complex2 @ ( zero_zero @ real ) @ ( zero_zero @ real ) ) ) ).
% zero_complex.code
thf(fact_3437_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_3438_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S4: set @ B,T5: set @ C,S2: set @ B,I: C > B,J: B > C,T6: set @ C,G: B > A,H: C > A] :
( ( finite_finite @ B @ S4 )
=> ( ( finite_finite @ C @ T5 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S2 @ S4 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S2 @ S4 ) )
=> ( member @ C @ ( J @ A6 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( ( J @ ( I @ B5 ) )
= B5 ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S2 @ S4 ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S4 )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T5 )
=> ( ( H @ B5 )
= ( zero_zero @ A ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S2 )
=> ( ( H @ ( J @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 )
= ( groups7311177749621191930dd_sum @ C @ A @ H @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_3439_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [X: B] :
( ( G @ X )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_3440_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A4: set @ B,F2: B > A,G: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less @ A @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_3441_sum_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A,P: B > $o] :
( ( finite_finite @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A4 )
& ( P @ X ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( G @ X ) @ ( zero_zero @ A ) )
@ A4 ) ) ) ) ).
% sum.inter_filter
thf(fact_3442_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F2: B > A,I: B] :
( ( finite_finite @ B @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I @ S )
=> ( ( F2 @ I )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_3443_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F2: B > A,B6: A,I: B] :
( ( finite_finite @ B @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S )
= B6 )
=> ( ( member @ B @ I @ S )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ B6 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_3444_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_finite @ B @ S )
=> ( ( finite_finite @ C @ T2 )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ T2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G @ X4 ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S )
=> ? [Xa2: C] :
( ( member @ C @ Xa2 @ T2 )
& ( ( I @ Xa2 )
= X4 )
& ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( G @ Xa2 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S ) @ ( groups7311177749621191930dd_sum @ C @ A @ G @ T2 ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_3445_sum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X4 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 )
= ( zero_zero @ A ) )
= ( ! [X: B] :
( ( member @ B @ X @ A4 )
=> ( ( F2 @ X )
= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_3446_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 ) )
=> ( ! [I3: I5] :
( ( member @ I5 @ I3 @ I6 )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member @ I5 @ I @ I6 )
=> ( ( finite_finite @ I5 @ I6 )
=> ( ( F2 @ I )
= ( G @ I ) ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_3447_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R3: A > A > $o,S2: set @ B,H: B > A,G: B > A] :
( ( R3 @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X1: A,Y1: A,X22: A,Y22: A] :
( ( ( R3 @ X1 @ X22 )
& ( R3 @ Y1 @ Y22 ) )
=> ( R3 @ ( plus_plus @ A @ X1 @ Y1 ) @ ( plus_plus @ A @ X22 @ Y22 ) ) )
=> ( ( finite_finite @ B @ S2 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S2 )
=> ( R3 @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R3 @ ( groups7311177749621191930dd_sum @ B @ A @ H @ S2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 ) ) ) ) ) ) ) ).
% sum.related
thf(fact_3448_sum_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,A4: set @ B] :
( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
!= ( zero_zero @ A ) )
=> ~ ! [A6: B] :
( ( member @ B @ A6 @ A4 )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_3449_sum_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( zero_zero @ A ) ) ) ) ).
% sum.neutral
thf(fact_3450_sum__strict__mono__ex1,axiom,
! [A: $tType,I5: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A4: set @ I5,F2: I5 > A,G: I5 > A] :
( ( finite_finite @ I5 @ A4 )
=> ( ! [X4: I5] :
( ( member @ I5 @ X4 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X6: I5] :
( ( member @ I5 @ X6 @ A4 )
& ( ord_less @ A @ ( F2 @ X6 ) @ ( G @ X6 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I5 @ A @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ I5 @ A @ G @ A4 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_3451_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite @ nat )
= ( ^ [N6: set @ nat] :
? [M3: nat] :
! [X: nat] :
( ( member @ nat @ X @ N6 )
=> ( ord_less_eq @ nat @ X @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_3452_bounded__nat__set__is__finite,axiom,
! [N4: set @ nat,N: nat] :
( ! [X4: nat] :
( ( member @ nat @ X4 @ N4 )
=> ( ord_less @ nat @ X4 @ N ) )
=> ( finite_finite @ nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_3453_finite__nat__set__iff__bounded,axiom,
( ( finite_finite @ nat )
= ( ^ [N6: set @ nat] :
? [M3: nat] :
! [X: nat] :
( ( member @ nat @ X @ N6 )
=> ( ord_less @ nat @ X @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_3454_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,I: B,F2: B > A] :
( ( finite_finite @ B @ I6 )
=> ( ( member @ B @ I @ I6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I6 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_3455_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,F2: B > A] :
( ( finite_finite @ B @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I6 ) ) ) ) ) ) ).
% sum_pos
thf(fact_3456_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K5: set @ B,F2: B > A,G: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ K5 )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ K5 ) ) ) ) ).
% sum_mono
thf(fact_3457_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S2: set @ B,G: B > A,H: B > A] :
( ( finite_finite @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ S2 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_3458_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S2: set @ B,H: B > A,G: B > A] :
( ( finite_finite @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( H @ X4 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ T6 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_3459_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S2: set @ B,G: B > A] :
( ( finite_finite @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_3460_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S2: set @ B,G: B > A] :
( ( finite_finite @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_3461_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ B,A4: set @ B,B6: set @ B,G: B > A,H: B > A] :
( ( finite_finite @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C5 @ A4 ) )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
=> ( ( H @ B5 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ C5 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ B6 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_3462_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ B,A4: set @ B,B6: set @ B,G: B > A,H: B > A] :
( ( finite_finite @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C5 @ A4 ) )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
=> ( ( H @ B5 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ B6 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ C5 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_3463_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_3464_finite__less__ub,axiom,
! [F2: nat > nat,U: nat] :
( ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( F2 @ N2 ) )
=> ( finite_finite @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ ( F2 @ N3 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_3465_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B6: set @ B,A4: set @ B,F2: B > A] :
( ( finite_finite @ B @ B6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B6 )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ B6 @ A4 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ B5 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B6 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_3466_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) ) ) ) ).
% sum_nonneg
thf(fact_3467_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_3468_sum__le__suminf,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,I6: set @ nat] :
( ( summable @ A @ F2 )
=> ( ( finite_finite @ 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_3469_sum__cong__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ nat,F2: nat > A,G: nat > A] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ! [X4: nat] :
( ( member @ nat @ ( suc @ X4 ) @ A4 )
=> ( ( F2 @ ( suc @ X4 ) )
= ( G @ ( suc @ X4 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ A4 )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ A4 ) ) ) ) ) ).
% sum_cong_Suc
thf(fact_3470_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_3471_one__complex_Ocode,axiom,
( ( one_one @ complex )
= ( complex2 @ ( one_one @ real ) @ ( zero_zero @ real ) ) ) ).
% one_complex.code
thf(fact_3472_Complex__eq__numeral,axiom,
! [A2: real,B2: real,W: num] :
( ( ( complex2 @ A2 @ B2 )
= ( numeral_numeral @ complex @ W ) )
= ( ( A2
= ( numeral_numeral @ real @ W ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_numeral
thf(fact_3473_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X8: set @ A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ X8 )
& ( ord_less @ A @ X4 @ Xa2 ) ) )
=> ~ ( finite_finite @ A @ X8 ) ) ) ) ).
% infinite_growing
thf(fact_3474_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S2: set @ A] :
( ( finite_finite @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ S2 )
& ~ ? [Xa2: A] :
( ( member @ A @ Xa2 @ S2 )
& ( ord_less @ A @ Xa2 @ X4 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_3475_prod__zero,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ? [X6: B] :
( ( member @ B @ X6 @ A4 )
& ( ( F2 @ X6 )
= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% prod_zero
thf(fact_3476_infinite__Icc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Icc
thf(fact_3477_finite__lists__length__le,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( finite_finite @ A @ A4 )
=> ( finite_finite @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A4 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3478_summable__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N4: set @ nat,F2: nat > A] :
( ( finite_finite @ nat @ N4 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N4 )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_finite
thf(fact_3479_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B6: set @ A,A4: set @ A,B2: A,F2: A > B] :
( ( finite_finite @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B6 @ A4 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F2 @ B2 ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ B6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X4 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ B6 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_3480_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I: C,A4: set @ C,F2: C > B] :
( ( member @ C @ I @ A4 )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ ( minus_minus @ ( set @ C ) @ A4 @ ( insert @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X4 ) ) )
=> ( ( finite_finite @ C @ A4 )
=> ( ord_less_eq @ B @ ( F2 @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F2 @ A4 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_3481_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,X3: B > A,Y3: B > A] :
( ( finite_finite @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( X3 @ I4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( Y3 @ I4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( plus_plus @ A @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3482_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_3483_Complex__eq__neg__numeral,axiom,
! [A2: real,B2: real,W: num] :
( ( ( complex2 @ A2 @ B2 )
= ( uminus_uminus @ complex @ ( numeral_numeral @ complex @ W ) ) )
= ( ( A2
= ( uminus_uminus @ real @ ( numeral_numeral @ real @ W ) ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_neg_numeral
thf(fact_3484_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S2: set @ B,P: ( set @ B ) > $o,F2: B > A] :
( ( finite_finite @ B @ S2 )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X4: B,S5: set @ B] :
( ( finite_finite @ B @ S5 )
=> ( ! [Y5: B] :
( ( member @ B @ Y5 @ S5 )
=> ( ord_less_eq @ A @ ( F2 @ Y5 ) @ ( F2 @ X4 ) ) )
=> ( ( P @ S5 )
=> ( P @ ( insert @ B @ X4 @ S5 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_3485_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite @ A @ A4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B5: A,A7: set @ A] :
( ( finite_finite @ A @ A7 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A7 )
=> ( ord_less @ A @ B5 @ X6 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert @ A @ B5 @ A7 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_3486_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite @ A @ A4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B5: A,A7: set @ A] :
( ( finite_finite @ A @ A7 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A7 )
=> ( ord_less @ A @ X6 @ B5 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert @ A @ B5 @ A7 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_3487_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,M: nat,I6: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( power_power @ A @ X3 @ ( plus_plus @ nat @ M @ I4 ) )
@ I6 )
= ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ I6 ) ) ) ) ).
% sum_power_add
thf(fact_3488_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( finite_finite @ nat
@ ( collect @ nat
@ ^ [D4: nat] : ( dvd_dvd @ nat @ D4 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_3489_arcsin__minus,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( ( arcsin @ ( uminus_uminus @ real @ X3 ) )
= ( uminus_uminus @ real @ ( arcsin @ X3 ) ) ) ) ) ).
% arcsin_minus
thf(fact_3490_arcsin__le__arcsin,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% arcsin_le_arcsin
thf(fact_3491_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_finite @ nat @ N4 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_3492_arcsin__eq__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y3 ) @ ( one_one @ real ) )
=> ( ( ( arcsin @ X3 )
= ( arcsin @ Y3 ) )
= ( X3 = Y3 ) ) ) ) ).
% arcsin_eq_iff
thf(fact_3493_arcsin__le__mono,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y3 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) )
= ( ord_less_eq @ real @ X3 @ Y3 ) ) ) ) ).
% arcsin_le_mono
thf(fact_3494_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_3495_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_3496_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_3497_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_3498_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,I: A,F2: A > B] :
( ( finite_finite @ A @ I6 )
=> ( ( member @ A @ I @ I6 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I6 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_3499_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,F2: A > B] :
( ( finite_finite @ A @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I6 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_3500_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_3501_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_3502_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_3503_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_3504_finite__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( finite_finite @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( power_power @ A @ Z6 @ N )
= ( one_one @ A ) ) ) ) ) ) ).
% finite_roots_unity
thf(fact_3505_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_3506_arcsin__less__arcsin,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% arcsin_less_arcsin
thf(fact_3507_arcsin__less__mono,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y3 ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) )
= ( ord_less @ real @ X3 @ Y3 ) ) ) ) ).
% arcsin_less_mono
thf(fact_3508_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A,P6: 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 @ P6 ) ) )
= ( 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 @ P6 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_3509_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F2: B > A,G: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) )
& ( ord_less @ A @ ( F2 @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_3510_even__prod__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) )
= ( ? [X: B] :
( ( member @ B @ X @ A4 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_3511_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,X3: A > B,A2: A > B,B2: B,Delta: B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X3 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X3 @ I6 )
= ( one_one @ B ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A2 @ I3 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( times_times @ B @ ( A2 @ I4 ) @ ( X3 @ I4 ) )
@ I6 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_3512_cos__arcsin__nonzero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X3 ) )
!= ( zero_zero @ real ) ) ) ) ).
% cos_arcsin_nonzero
thf(fact_3513_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B6: set @ A,A4: set @ A,F2: A > B] :
( ( finite_finite @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B6 @ A4 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ B5 ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ A6 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ B6 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_3514_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_3515_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_3516_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A4: set @ B,F2: B > A,A2: B] :
( ( finite_finite @ B @ A4 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A2 @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( F2 @ A2 ) ) ) )
& ( ~ ( member @ B @ A2 @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_3517_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_3518_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
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_3519_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X3: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X3 @ M ) @ ( power_power @ A @ X3 @ ( suc @ N ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_3520_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_3521_complex__norm,axiom,
! [X3: real,Y3: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( complex2 @ X3 @ Y3 ) )
= ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_norm
thf(fact_3522_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_3523_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ ( 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_3524_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,D3: 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 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) ) ) ).
% double_arith_series
thf(fact_3525_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_3526_arith__series__nat,axiom,
! [A2: nat,D3: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( plus_plus @ nat @ A2 @ ( times_times @ nat @ I4 @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ nat @ N @ D3 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_3527_Sum__Icc__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ ( 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_3528_arcsin__lt__bounded,axiom,
! [Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less @ real @ Y3 @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y3 ) )
& ( ord_less @ real @ ( arcsin @ Y3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% arcsin_lt_bounded
thf(fact_3529_arcsin__bounded,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y3 ) )
& ( ord_less_eq @ real @ ( arcsin @ Y3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% arcsin_bounded
thf(fact_3530_arcsin__ubound,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arcsin @ Y3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arcsin_ubound
thf(fact_3531_arcsin__lbound,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y3 ) ) ) ) ).
% arcsin_lbound
thf(fact_3532_arcsin__sin,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arcsin @ ( sin @ real @ X3 ) )
= X3 ) ) ) ).
% arcsin_sin
thf(fact_3533_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_3534_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_3535_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,D3: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I4 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_3536_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,M: nat,N: nat] :
( ( ( X3
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) )
& ( ( X3
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_3537_arcsin,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y3 ) )
& ( ord_less_eq @ real @ ( arcsin @ Y3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ ( arcsin @ Y3 ) )
= Y3 ) ) ) ) ).
% arcsin
thf(fact_3538_arcsin__pi,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y3 ) )
& ( ord_less_eq @ real @ ( arcsin @ Y3 ) @ pi )
& ( ( sin @ real @ ( arcsin @ Y3 ) )
= Y3 ) ) ) ) ).
% arcsin_pi
thf(fact_3539_arcsin__le__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( arcsin @ X3 ) @ Y3 )
= ( ord_less_eq @ real @ X3 @ ( sin @ real @ Y3 ) ) ) ) ) ) ) ).
% arcsin_le_iff
thf(fact_3540_le__arcsin__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ Y3 @ ( arcsin @ X3 ) )
= ( ord_less_eq @ real @ ( sin @ real @ Y3 ) @ X3 ) ) ) ) ) ) ).
% le_arcsin_iff
thf(fact_3541_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_3542_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_3543_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less @ nat @ N3 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_3544_set__encode__insert,axiom,
! [A4: set @ nat,N: nat] :
( ( finite_finite @ nat @ A4 )
=> ( ~ ( member @ nat @ N @ A4 )
=> ( ( nat_set_encode @ ( insert @ nat @ N @ A4 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A4 ) ) ) ) ) ).
% set_encode_insert
thf(fact_3545_lemma__termdiff2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [H: A,Z: A,N: nat] :
( ( H
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H ) @ N ) @ ( power_power @ A @ Z @ N ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
= ( times_times @ A @ H
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [Q5: nat] : ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H ) @ Q5 ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q5 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ P5 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_3546_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_3547_finite__interval__int1,axiom,
! [A2: int,B2: int] :
( finite_finite @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less_eq @ int @ A2 @ I4 )
& ( ord_less_eq @ int @ I4 @ B2 ) ) ) ) ).
% finite_interval_int1
thf(fact_3548_finite__interval__int4,axiom,
! [A2: int,B2: int] :
( finite_finite @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less @ int @ A2 @ I4 )
& ( ord_less @ int @ I4 @ B2 ) ) ) ) ).
% finite_interval_int4
thf(fact_3549_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_3550_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_3551_finite__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite @ complex
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= C2 ) ) ) ) ).
% finite_nth_roots
thf(fact_3552_finite__interval__int2,axiom,
! [A2: int,B2: int] :
( finite_finite @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less_eq @ int @ A2 @ I4 )
& ( ord_less @ int @ I4 @ B2 ) ) ) ) ).
% finite_interval_int2
thf(fact_3553_finite__interval__int3,axiom,
! [A2: int,B2: int] :
( finite_finite @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less @ int @ A2 @ I4 )
& ( ord_less_eq @ int @ I4 @ B2 ) ) ) ) ).
% finite_interval_int3
thf(fact_3554_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X3 ) @ ( set_ord_lessThan @ A @ Y3 ) )
= ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% lessThan_subset_iff
thf(fact_3555_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X3 ) @ ( set_ord_atMost @ A @ Y3 ) )
= ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% atMost_subset_iff
thf(fact_3556_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_3557_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_3558_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H: A,H3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H ) @ ( set_ord_atMost @ A @ H3 ) )
= ( ~ ( ord_less_eq @ A @ L @ H )
| ( ord_less_eq @ A @ H @ H3 ) ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_3559_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_3560_int__sum,axiom,
! [B: $tType,F2: B > nat,A4: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ int
@ ^ [X: B] : ( semiring_1_of_nat @ int @ ( F2 @ X ) )
@ A4 ) ) ).
% int_sum
thf(fact_3561_Complex__sum_H,axiom,
! [A: $tType,F2: A > real,S: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ complex
@ ^ [X: A] : ( complex2 @ ( F2 @ X ) @ ( zero_zero @ real ) )
@ S )
= ( complex2 @ ( groups7311177749621191930dd_sum @ A @ real @ F2 @ S ) @ ( zero_zero @ real ) ) ) ).
% Complex_sum'
thf(fact_3562_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q: A > nat,P: A > nat,N: A] :
( ! [X4: A] : ( ord_less_eq @ nat @ ( Q @ X4 ) @ ( P @ X4 ) )
=> ( ( 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
@ ^ [X: A] : ( minus_minus @ nat @ ( P @ X ) @ ( Q @ X ) )
@ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_3563_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_3564_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X: A] : ( ord_less @ A @ X @ U2 ) ) ) ) ) ).
% lessThan_def
thf(fact_3565_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X: A] : ( ord_less_eq @ A @ X @ U2 ) ) ) ) ) ).
% atMost_def
thf(fact_3566_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_3567_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_3568_atMost__atLeast0,axiom,
( ( set_ord_atMost @ nat )
= ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% atMost_atLeast0
thf(fact_3569_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_3570_finite__maxlen,axiom,
! [A: $tType,M7: set @ ( list @ A )] :
( ( finite_finite @ ( 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_3571_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan @ nat @ N )
= ( bot_bot @ ( set @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_3572_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X: complex] : X
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= ( one_one @ complex ) ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_roots_unity
thf(fact_3573_sum__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X: complex] : X
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= C2 ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_nth_roots
thf(fact_3574_power__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C2: A,F2: B > nat,A4: set @ B] :
( ( power_power @ A @ C2 @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [A5: B] : ( power_power @ A @ C2 @ ( F2 @ A5 ) )
@ A4 ) ) ) ).
% power_sum
thf(fact_3575_sum__subtractf__nat,axiom,
! [A: $tType,A4: set @ A,G: A > nat,F2: A > nat] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ nat @ ( G @ X4 ) @ ( F2 @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X: A] : ( minus_minus @ nat @ ( F2 @ X ) @ ( G @ X ) )
@ A4 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_3576_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 ) )
=> ~ ! [B5: 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 @ ( B5 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_3577_polyfun__linear__factor,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,N: nat,A2: A] :
? [B5: 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 @ ( B5 @ 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_3578_finite__divisors__int,axiom,
! [I: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( finite_finite @ int
@ ( collect @ int
@ ^ [D4: int] : ( dvd_dvd @ int @ D4 @ I ) ) ) ) ).
% finite_divisors_int
thf(fact_3579_sum__eq__Suc0__iff,axiom,
! [A: $tType,A4: set @ A,F2: A > nat] :
( ( finite_finite @ A @ A4 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X: A] :
( ( member @ A @ X @ A4 )
& ( ( F2 @ X )
= ( suc @ ( zero_zero @ nat ) ) )
& ! [Y: A] :
( ( member @ A @ Y @ A4 )
=> ( ( X != Y )
=> ( ( F2 @ Y )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_3580_sum__SucD,axiom,
! [A: $tType,F2: A > nat,A4: set @ A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 )
= ( suc @ N ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X4 ) ) ) ) ).
% sum_SucD
thf(fact_3581_sum__eq__1__iff,axiom,
! [A: $tType,A4: set @ A,F2: A > nat] :
( ( finite_finite @ A @ A4 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 )
= ( one_one @ nat ) )
= ( ? [X: A] :
( ( member @ A @ X @ A4 )
& ( ( F2 @ X )
= ( one_one @ nat ) )
& ! [Y: A] :
( ( member @ A @ Y @ A4 )
=> ( ( X != Y )
=> ( ( F2 @ Y )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_3582_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_3583_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_3584_set__encode__inf,axiom,
! [A4: set @ nat] :
( ~ ( finite_finite @ nat @ A4 )
=> ( ( nat_set_encode @ A4 )
= ( zero_zero @ nat ) ) ) ).
% set_encode_inf
thf(fact_3585_polyfun__diff__alt,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A2: nat > A,X3: A,Y3: 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 @ X3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( power_power @ A @ Y3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X3 @ Y3 )
@ ( 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 @ Y3 @ K3 ) ) @ ( power_power @ A @ X3 @ J3 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ J3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_3586_suminf__le__const,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,X3: A] :
( ( summable @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) @ X3 )
=> ( ord_less_eq @ A @ ( suminf @ A @ F2 ) @ X3 ) ) ) ) ).
% suminf_le_const
thf(fact_3587_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_3588_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_3589_polyfun__eq__coeffs,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat,D3: nat > A] :
( ( ! [X: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( D3 @ I4 ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
= ( ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
=> ( ( C2 @ I4 )
= ( D3 @ I4 ) ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_3590_bounded__imp__summable,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linord2810124833399127020strict @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A2: nat > A,B6: 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 ) ) @ B6 )
=> ( summable @ A @ A2 ) ) ) ) ).
% bounded_imp_summable
thf(fact_3591_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_3592_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_3593_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_3594_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_3595_sumr__diff__mult__const2,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [F2: nat > A,N: nat,R2: A] :
( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ R2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F2 @ I4 ) @ R2 )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sumr_diff_mult_const2
thf(fact_3596_summableI__nonneg__bounded,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,X3: 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 ) ) @ X3 )
=> ( summable @ A @ F2 ) ) ) ) ).
% summableI_nonneg_bounded
thf(fact_3597_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_3598_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_3599_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_3600_polyfun__diff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A2: nat > A,X3: A,Y3: 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 @ X3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( power_power @ A @ Y3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X3 @ Y3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ ( minus_minus @ nat @ I4 @ J3 ) @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( power_power @ A @ X3 @ J3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff
thf(fact_3601_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_3602_norm__prod__diff,axiom,
! [A: $tType,I5: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [I6: set @ I5,Z: I5 > A,W: I5 > A] :
( ! [I3: I5] :
( ( member @ I5 @ I3 @ I6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( Z @ I3 ) ) @ ( one_one @ real ) ) )
=> ( ! [I3: I5] :
( ( member @ I5 @ I3 @ I6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( W @ I3 ) ) @ ( one_one @ real ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( groups7121269368397514597t_prod @ I5 @ A @ Z @ I6 ) @ ( groups7121269368397514597t_prod @ I5 @ A @ W @ I6 ) ) )
@ ( groups7311177749621191930dd_sum @ I5 @ real
@ ^ [I4: I5] : ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( Z @ I4 ) @ ( W @ I4 ) ) )
@ I6 ) ) ) ) ) ).
% norm_prod_diff
thf(fact_3603_zero__polynom__imp__zero__coeffs,axiom,
! [A: $tType] :
( ( ( ab_semigroup_mult @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [C2: nat > A,N: nat,K: nat] :
( ! [W2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ W2 @ 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_3604_polyfun__eq__0,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat] :
( ( ! [X: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ X @ 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_3605_one__diff__power__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq
thf(fact_3606_power__diff__1__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ X3 @ N ) @ ( one_one @ A ) )
= ( times_times @ A @ ( minus_minus @ A @ X3 @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_1_eq
thf(fact_3607_geometric__sum,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,N: nat] :
( ( X3
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X3 @ N ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X3 @ ( one_one @ A ) ) ) ) ) ) ).
% geometric_sum
thf(fact_3608_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_3609_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_3610_ln__prod,axiom,
! [A: $tType,I6: set @ A,F2: A > real] :
( ( finite_finite @ A @ I6 )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ I3 ) ) )
=> ( ( ln_ln @ real @ ( groups7121269368397514597t_prod @ A @ real @ F2 @ I6 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X: A] : ( ln_ln @ real @ ( F2 @ X ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_3611_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_3612_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ N ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ ( suc @ N ) ) ) ) ) ).
% sum_gp_basic
thf(fact_3613_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_finite @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z6 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_3614_polyfun__finite__roots,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat] :
( ( finite_finite @ A
@ ( collect @ A
@ ^ [X: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ X @ 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_3615_sum__gp__strict,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat] :
( ( ( X3
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_lessThan @ nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) )
& ( ( X3
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ N ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ).
% sum_gp_strict
thf(fact_3616_lemma__termdiff1,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [Z: A,H: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] : ( minus_minus @ A @ ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H ) @ ( minus_minus @ nat @ M @ P5 ) ) @ ( power_power @ A @ Z @ P5 ) ) @ ( power_power @ A @ Z @ M ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] : ( times_times @ A @ ( power_power @ A @ Z @ P5 ) @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H ) @ ( minus_minus @ nat @ M @ P5 ) ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ M @ P5 ) ) ) )
@ ( set_ord_lessThan @ nat @ M ) ) ) ) ).
% lemma_termdiff1
thf(fact_3617_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat,Y3: A] :
( ( minus_minus @ A @ ( power_power @ A @ X3 @ N ) @ ( power_power @ A @ Y3 @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ X3 @ Y3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) ) @ ( power_power @ A @ X3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_3618_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat,Y3: A] :
( ( minus_minus @ A @ ( power_power @ A @ X3 @ ( suc @ N ) ) @ ( power_power @ A @ Y3 @ ( suc @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X3 @ Y3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] : ( times_times @ A @ ( power_power @ A @ X3 @ P5 ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ N @ P5 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_3619_polynomial__product__nat,axiom,
! [M: nat,A2: nat > nat,N: nat,B2: nat > nat,X3: nat] :
( ! [I3: nat] :
( ( ord_less @ nat @ M @ I3 )
=> ( ( A2 @ I3 )
= ( zero_zero @ nat ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B2 @ J2 )
= ( zero_zero @ nat ) ) )
=> ( ( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ ( A2 @ I4 ) @ ( power_power @ nat @ X3 @ I4 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( times_times @ nat @ ( B2 @ J3 ) @ ( power_power @ nat @ X3 @ J3 ) )
@ ( set_ord_atMost @ nat @ 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 @ X3 @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_3620_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X3: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_3621_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_3622_set__encode__def,axiom,
( nat_set_encode
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% set_encode_def
thf(fact_3623_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_3624_real__sum__nat__ivl__bounded2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,F2: nat > A,K5: A,K: nat] :
( ! [P7: nat] :
( ( ord_less @ nat @ P7 @ N )
=> ( ord_less_eq @ A @ ( F2 @ P7 ) @ K5 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ K5 )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ K ) ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ K5 ) ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_3625_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: set @ A,A2: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ A2 @ A4 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( ord_less_eq @ A @ X4 @ A2 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A4 )
=> ( ( ord_less_eq @ A @ Xa2 @ X4 )
=> ( X4 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_3626_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: set @ A,A2: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ A2 @ A4 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( ord_less_eq @ A @ A2 @ X4 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A4 )
=> ( ( ord_less_eq @ A @ X4 @ Xa2 )
=> ( X4 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_3627_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_3628_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_3629_polynomial__product,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [M: nat,A2: nat > A,N: nat,B2: nat > A,X3: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ M @ I3 )
=> ( ( A2 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B2 @ J2 )
= ( zero_zero @ A ) ) )
=> ( ( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( power_power @ A @ X3 @ I4 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( times_times @ A @ ( B2 @ J3 ) @ ( power_power @ A @ X3 @ J3 ) )
@ ( set_ord_atMost @ nat @ 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 @ X3 @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ) ).
% polynomial_product
thf(fact_3630_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_3631_polyfun__eq__const,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat,K: A] :
( ( ! [X: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= K ) )
= ( ( ( C2 @ ( zero_zero @ nat ) )
= K )
& ! [X: nat] :
( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) )
=> ( ( C2 @ X )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_const
thf(fact_3632_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( power_power @ A @ X3 @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_3633_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_3634_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_3635_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P6: nat,K: nat,G: nat > A,H: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P6 )
=> ( ( ord_less_eq @ nat @ K @ P6 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( zero_zero @ A ) @ ( H @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P6 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P6 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_3636_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P6: nat,K: nat,G: nat > A,H: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P6 )
=> ( ( ord_less_eq @ nat @ K @ P6 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P6 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P6 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_3637_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_3638_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_3639_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_3640_root__polyfun,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,Z: A,A2: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( ( power_power @ A @ Z @ 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 @ Z @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% root_polyfun
thf(fact_3641_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat] :
( ( ( X3
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ N ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) )
& ( ( X3
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ).
% sum_gp0
thf(fact_3642_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_3643_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_3644_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_3645_even__set__encode__iff,axiom,
! [A4: set @ nat] :
( ( finite_finite @ nat @ A4 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A4 ) )
= ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A4 ) ) ) ) ).
% even_set_encode_iff
thf(fact_3646_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_3647_Sum__Icc__int,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X: int] : X
@ ( 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_3648_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A4 )
=> ( ( ord_less_eq @ A @ Xa2 @ X4 )
=> ( X4 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_3649_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A4 )
=> ( ( ord_less_eq @ A @ X4 @ Xa2 )
=> ( X4 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_3650_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_3651_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_3652_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_3653_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_3654_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_3655_sin__arccos__abs,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Y3 ) @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ Y3 ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_arccos_abs
thf(fact_3656_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_3657_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_3658_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_3659_arccos__1,axiom,
( ( arccos @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% arccos_1
thf(fact_3660_cos__coeff__0,axiom,
( ( cos_coeff @ ( zero_zero @ nat ) )
= ( one_one @ real ) ) ).
% cos_coeff_0
thf(fact_3661_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_3662_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_3663_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_3664_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_3665_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_3666_cos__arccos,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arccos @ Y3 ) )
= Y3 ) ) ) ).
% cos_arccos
thf(fact_3667_arccos__0,axiom,
( ( arccos @ ( zero_zero @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arccos_0
thf(fact_3668_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_3669_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_3670_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_3671_arccos__le__arccos,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y3 ) @ ( arccos @ X3 ) ) ) ) ) ).
% arccos_le_arccos
thf(fact_3672_arccos__eq__iff,axiom,
! [X3: real,Y3: real] :
( ( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
& ( ord_less_eq @ real @ ( abs_abs @ real @ Y3 ) @ ( one_one @ real ) ) )
=> ( ( ( arccos @ X3 )
= ( arccos @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% arccos_eq_iff
thf(fact_3673_arccos__le__mono,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y3 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( arccos @ X3 ) @ ( arccos @ Y3 ) )
= ( ord_less_eq @ real @ Y3 @ X3 ) ) ) ) ).
% arccos_le_mono
thf(fact_3674_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_3675_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_3676_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_3677_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_3678_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X: A] : ( plus_plus @ A @ ( plus_plus @ A @ X @ X ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_3679_arccos__lbound,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y3 ) ) ) ) ).
% arccos_lbound
thf(fact_3680_arccos__less__arccos,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arccos @ Y3 ) @ ( arccos @ X3 ) ) ) ) ) ).
% arccos_less_arccos
thf(fact_3681_arccos__less__mono,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y3 ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arccos @ X3 ) @ ( arccos @ Y3 ) )
= ( ord_less @ real @ Y3 @ X3 ) ) ) ) ).
% arccos_less_mono
thf(fact_3682_arccos__ubound,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y3 ) @ pi ) ) ) ).
% arccos_ubound
thf(fact_3683_arccos__cos,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ pi )
=> ( ( arccos @ ( cos @ real @ X3 ) )
= X3 ) ) ) ).
% arccos_cos
thf(fact_3684_cos__arccos__abs,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Y3 ) @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arccos @ Y3 ) )
= Y3 ) ) ).
% cos_arccos_abs
thf(fact_3685_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_3686_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_3687_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_3688_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_3689_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_3690_arccos__lt__bounded,axiom,
! [Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less @ real @ Y3 @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( arccos @ Y3 ) )
& ( ord_less @ real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% arccos_lt_bounded
thf(fact_3691_arccos__bounded,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y3 ) )
& ( ord_less_eq @ real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% arccos_bounded
thf(fact_3692_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_3693_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_3694_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_3695_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_3696_sin__arccos__nonzero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X3 ) )
!= ( zero_zero @ real ) ) ) ) ).
% sin_arccos_nonzero
thf(fact_3697_arccos__cos2,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ X3 )
=> ( ( arccos @ ( cos @ real @ X3 ) )
= ( uminus_uminus @ real @ X3 ) ) ) ) ).
% arccos_cos2
thf(fact_3698_arccos__minus,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( ( arccos @ ( uminus_uminus @ real @ X3 ) )
= ( minus_minus @ real @ pi @ ( arccos @ X3 ) ) ) ) ) ).
% arccos_minus
thf(fact_3699_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_3700_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_3701_arccos,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y3 ) )
& ( ord_less_eq @ real @ ( arccos @ Y3 ) @ pi )
& ( ( cos @ real @ ( arccos @ Y3 ) )
= Y3 ) ) ) ) ).
% arccos
thf(fact_3702_arccos__minus__abs,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( ( arccos @ ( uminus_uminus @ real @ X3 ) )
= ( minus_minus @ real @ pi @ ( arccos @ X3 ) ) ) ) ).
% arccos_minus_abs
thf(fact_3703_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_3704_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_3705_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A2: A,X3: A,Y3: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K3 ) @ ( power_power @ A @ X3 @ K3 ) ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X3 ) @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X3 @ Y3 ) @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_3706_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_3707_arccos__le__pi2,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arccos_le_pi2
thf(fact_3708_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_3709_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A2: A,X3: A,Y3: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K3 ) @ ( power_power @ A @ X3 @ K3 ) ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A2 ) @ ( one_one @ A ) ) @ K3 ) @ ( power_power @ A @ X3 @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X3 @ Y3 ) @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_3710_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_3711_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ R2 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R2 @ ( suc @ M ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_3712_sin__arccos,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X3 ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_arccos
thf(fact_3713_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ? [T4: real] :
( ( ord_less @ real @ X3 @ T4 )
& ( ord_less @ real @ T4 @ ( zero_zero @ real ) )
& ( ( cos @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( cos_coeff @ M3 ) @ ( power_power @ real @ X3 @ 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 @ X3 @ N ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_3714_Maclaurin__cos__expansion2,axiom,
! [X3: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less @ real @ T4 @ X3 )
& ( ( cos @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( cos_coeff @ M3 ) @ ( power_power @ real @ X3 @ 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 @ X3 @ N ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_3715_Maclaurin__cos__expansion,axiom,
! [X3: real,N: nat] :
? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X3 ) )
& ( ( cos @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( cos_coeff @ M3 ) @ ( power_power @ real @ X3 @ 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 @ X3 @ N ) ) ) ) ) ).
% Maclaurin_cos_expansion
thf(fact_3716_infinite__int__iff__unbounded,axiom,
! [S2: set @ int] :
( ( ~ ( finite_finite @ int @ S2 ) )
= ( ! [M3: int] :
? [N3: int] :
( ( ord_less @ int @ M3 @ ( abs_abs @ int @ N3 ) )
& ( member @ int @ N3 @ S2 ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_3717_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_3718_of__int__fact,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( ring_1 @ A ) )
=> ! [N: nat] :
( ( ring_1_of_int @ A @ ( semiring_char_0_fact @ int @ N ) )
= ( semiring_char_0_fact @ A @ N ) ) ) ).
% of_int_fact
thf(fact_3719_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_3720_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_3721_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_3722_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_3723_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_3724_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_3725_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_3726_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_3727_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_3728_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_3729_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_3730_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_3731_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_3732_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_3733_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
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N3 ) ) ) ) ) ) ).
% fact_prod
thf(fact_3734_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_3735_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_3736_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_3737_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_3738_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_3739_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_3740_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_3741_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_3742_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_3743_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_3744_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_3745_infinite__nat__iff__unbounded,axiom,
! [S2: set @ nat] :
( ( ~ ( finite_finite @ nat @ S2 ) )
= ( ! [M3: nat] :
? [N3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ( member @ nat @ N3 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_3746_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_finite @ nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_3747_infinite__nat__iff__unbounded__le,axiom,
! [S2: set @ nat] :
( ( ~ ( finite_finite @ nat @ S2 ) )
= ( ! [M3: nat] :
? [N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
& ( member @ nat @ N3 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_3748_Maclaurin__zero,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X3: real,N: nat,Diff: nat > A > real] :
( ( X3
= ( zero_zero @ real ) )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ X3 @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% Maclaurin_zero
thf(fact_3749_Maclaurin__lemma,axiom,
! [H: real,F2: real > real,J: nat > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H )
=> ? [B7: real] :
( ( F2 @ H )
= ( 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 @ H @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ B7 @ ( divide_divide @ real @ ( power_power @ real @ H @ N ) @ ( semiring_char_0_fact @ real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_3750_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_3751_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_3752_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_3753_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_3754_infinite__int__iff__unbounded__le,axiom,
! [S2: set @ int] :
( ( ~ ( finite_finite @ 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_3755_Maclaurin__sin__expansion3,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less @ real @ T4 @ X3 )
& ( ( sin @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( sin_coeff @ M3 ) @ ( power_power @ real @ X3 @ 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 @ X3 @ N ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_3756_Maclaurin__sin__expansion4,axiom,
! [X3: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ X3 )
& ( ( sin @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( sin_coeff @ M3 ) @ ( power_power @ real @ X3 @ 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 @ X3 @ N ) ) ) ) ) ) ).
% Maclaurin_sin_expansion4
thf(fact_3757_Maclaurin__sin__expansion2,axiom,
! [X3: real,N: nat] :
? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X3 ) )
& ( ( sin @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( sin_coeff @ M3 ) @ ( power_power @ real @ X3 @ 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 @ X3 @ N ) ) ) ) ) ).
% Maclaurin_sin_expansion2
thf(fact_3758_Maclaurin__sin__expansion,axiom,
! [X3: real,N: nat] :
? [T4: real] :
( ( sin @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( sin_coeff @ M3 ) @ ( power_power @ real @ X3 @ 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 @ X3 @ N ) ) ) ) ).
% Maclaurin_sin_expansion
thf(fact_3759_sin__coeff__0,axiom,
( ( sin_coeff @ ( zero_zero @ nat ) )
= ( zero_zero @ real ) ) ).
% sin_coeff_0
thf(fact_3760_fact__ge__self,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( semiring_char_0_fact @ nat @ N ) ) ).
% fact_ge_self
thf(fact_3761_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_3762_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_3763_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_3764_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_3765_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_3766_fact__div__fact__le__pow,axiom,
! [R2: nat,N: nat] :
( ( ord_less_eq @ nat @ R2 @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ R2 ) ) ) @ ( power_power @ nat @ N @ R2 ) ) ) ).
% fact_div_fact_le_pow
thf(fact_3767_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_3768_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
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_3769_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_3770_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
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_3771_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_3772_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_3773_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_3774_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_3775_Maclaurin__exp__lt,axiom,
! [X3: real,N: nat] :
( ( X3
!= ( 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 @ X3 ) )
& ( ( exp @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( divide_divide @ real @ ( power_power @ real @ X3 @ 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 @ X3 @ N ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_3776_sin__paired,axiom,
! [X3: real] :
( sums @ real
@ ^ [N3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) )
@ ( sin @ real @ X3 ) ) ).
% sin_paired
thf(fact_3777_geometric__deriv__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Z: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( one_one @ real ) )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) @ ( power_power @ A @ Z @ N3 ) )
@ ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( minus_minus @ A @ ( one_one @ A ) @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_3778_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_3779_exp__less__cancel__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( exp @ real @ X3 ) @ ( exp @ real @ Y3 ) )
= ( ord_less @ real @ X3 @ Y3 ) ) ).
% exp_less_cancel_iff
thf(fact_3780_exp__less__mono,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ X3 @ Y3 )
=> ( ord_less @ real @ ( exp @ real @ X3 ) @ ( exp @ real @ Y3 ) ) ) ).
% exp_less_mono
thf(fact_3781_exp__le__cancel__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( exp @ real @ X3 ) @ ( exp @ real @ Y3 ) )
= ( ord_less_eq @ real @ X3 @ Y3 ) ) ).
% exp_le_cancel_iff
thf(fact_3782_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_3783_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_3784_exp__eq__one__iff,axiom,
! [X3: real] :
( ( ( exp @ real @ X3 )
= ( one_one @ real ) )
= ( X3
= ( zero_zero @ real ) ) ) ).
% exp_eq_one_iff
thf(fact_3785_exp__less__one__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( exp @ real @ X3 ) @ ( one_one @ real ) )
= ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ).
% exp_less_one_iff
thf(fact_3786_one__less__exp__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X3 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ).
% one_less_exp_iff
thf(fact_3787_exp__le__one__iff,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( exp @ real @ X3 ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) ) ) ).
% exp_le_one_iff
thf(fact_3788_one__le__exp__iff,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( exp @ real @ X3 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) ) ).
% one_le_exp_iff
thf(fact_3789_exp__ln,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( exp @ real @ ( ln_ln @ real @ X3 ) )
= X3 ) ) ).
% exp_ln
thf(fact_3790_exp__ln__iff,axiom,
! [X3: real] :
( ( ( exp @ real @ ( ln_ln @ real @ X3 ) )
= X3 )
= ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ).
% exp_ln_iff
thf(fact_3791_powser__sums__zero__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: nat > A,X3: A] :
( ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( A2 @ N3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N3 ) )
@ X3 )
= ( ( A2 @ ( zero_zero @ nat ) )
= X3 ) ) ) ).
% powser_sums_zero_iff
thf(fact_3792_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_3793_exp__less__cancel,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( exp @ real @ X3 ) @ ( exp @ real @ Y3 ) )
=> ( ord_less @ real @ X3 @ Y3 ) ) ).
% exp_less_cancel
thf(fact_3794_norm__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ X3 ) ) @ ( exp @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) ) ) ) ).
% norm_exp
thf(fact_3795_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_3796_exp__not__eq__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( ( exp @ A @ X3 )
!= ( zero_zero @ A ) ) ) ).
% exp_not_eq_zero
thf(fact_3797_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_3798_not__exp__less__zero,axiom,
! [X3: real] :
~ ( ord_less @ real @ ( exp @ real @ X3 ) @ ( zero_zero @ real ) ) ).
% not_exp_less_zero
thf(fact_3799_exp__gt__zero,axiom,
! [X3: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( exp @ real @ X3 ) ) ).
% exp_gt_zero
thf(fact_3800_exp__total,axiom,
! [Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
=> ? [X4: real] :
( ( exp @ real @ X4 )
= Y3 ) ) ).
% exp_total
thf(fact_3801_not__exp__le__zero,axiom,
! [X3: real] :
~ ( ord_less_eq @ real @ ( exp @ real @ X3 ) @ ( zero_zero @ real ) ) ).
% not_exp_le_zero
thf(fact_3802_exp__ge__zero,axiom,
! [X3: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( exp @ real @ X3 ) ) ).
% exp_ge_zero
thf(fact_3803_sums__mult2__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C2: A,F2: nat > A,D3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ C2 )
@ ( times_times @ A @ D3 @ C2 ) )
= ( sums @ A @ F2 @ D3 ) ) ) ) ).
% sums_mult2_iff
thf(fact_3804_sums__mult__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C2: A,F2: nat > A,D3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ C2 @ ( F2 @ N3 ) )
@ ( times_times @ A @ C2 @ D3 ) )
= ( sums @ A @ F2 @ D3 ) ) ) ) ).
% sums_mult_iff
thf(fact_3805_exp__gt__one,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X3 ) ) ) ).
% exp_gt_one
thf(fact_3806_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_3807_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_3808_exp__ge__add__one__self,axiom,
! [X3: real] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) @ ( exp @ real @ X3 ) ) ).
% exp_ge_add_one_self
thf(fact_3809_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_3810_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_3811_sums__zero__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [N: nat,F2: nat > A,S: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ( F2 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [I4: nat] : ( F2 @ ( plus_plus @ nat @ I4 @ N ) )
@ S )
= ( sums @ A @ F2 @ S ) ) ) ) ).
% sums_zero_iff_shift
thf(fact_3812_exp__of__nat__mult,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N: nat,X3: A] :
( ( exp @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ X3 ) )
= ( power_power @ A @ ( exp @ A @ X3 ) @ N ) ) ) ).
% exp_of_nat_mult
thf(fact_3813_exp__of__nat2__mult,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,N: nat] :
( ( exp @ A @ ( times_times @ A @ X3 @ ( semiring_1_of_nat @ A @ N ) ) )
= ( power_power @ A @ ( exp @ A @ X3 ) @ N ) ) ) ).
% exp_of_nat2_mult
thf(fact_3814_sums__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N4: set @ nat,F2: nat > A] :
( ( finite_finite @ 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_3815_sums__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P: nat > $o,F2: nat > A] :
( ( finite_finite @ 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_3816_sums__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A4: set @ nat,F2: nat > A] :
( ( finite_finite @ nat @ A4 )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A4 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ A4 ) ) ) ) ).
% sums_If_finite_set
thf(fact_3817_powser__sums__if,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [M: nat,Z: A] :
( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( if @ A @ ( N3 = M ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ ( power_power @ A @ Z @ N3 ) )
@ ( power_power @ A @ Z @ M ) ) ) ).
% powser_sums_if
thf(fact_3818_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_3819_exp__ge__add__one__self__aux,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) @ ( exp @ real @ X3 ) ) ) ).
% exp_ge_add_one_self_aux
thf(fact_3820_lemma__exp__total,axiom,
! [Y3: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ Y3 )
=> ? [X4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less_eq @ real @ X4 @ ( minus_minus @ real @ Y3 @ ( one_one @ real ) ) )
& ( ( exp @ real @ X4 )
= Y3 ) ) ) ).
% lemma_exp_total
thf(fact_3821_ln__ge__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ Y3 @ ( ln_ln @ real @ X3 ) )
= ( ord_less_eq @ real @ ( exp @ real @ Y3 ) @ X3 ) ) ) ).
% ln_ge_iff
thf(fact_3822_ln__x__over__x__mono,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ Y3 )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( ln_ln @ real @ Y3 ) @ Y3 ) @ ( divide_divide @ real @ ( ln_ln @ real @ X3 ) @ X3 ) ) ) ) ).
% ln_x_over_x_mono
thf(fact_3823_powr__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( powr @ A )
= ( ^ [X: A,A5: A] :
( if @ A
@ ( X
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( exp @ A @ ( times_times @ A @ A5 @ ( ln_ln @ A @ X ) ) ) ) ) ) ) ).
% powr_def
thf(fact_3824_exp__le,axiom,
ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ).
% exp_le
thf(fact_3825_exp__divide__power__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N: nat,X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( power_power @ A @ ( exp @ A @ ( divide_divide @ A @ X3 @ ( semiring_1_of_nat @ A @ N ) ) ) @ N )
= ( exp @ A @ X3 ) ) ) ) ).
% exp_divide_power_eq
thf(fact_3826_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_3827_exp__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z: A] :
( ( exp @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z ) )
= ( power_power @ A @ ( exp @ A @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% exp_double
thf(fact_3828_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_3829_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_3830_exp__bound__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ Z ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% exp_bound_half
thf(fact_3831_sums__if_H,axiom,
! [G: nat > real,X3: real] :
( ( sums @ real @ G @ X3 )
=> ( 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 ) ) ) ) )
@ X3 ) ) ).
% sums_if'
thf(fact_3832_sums__if,axiom,
! [G: nat > real,X3: real,F2: nat > real,Y3: real] :
( ( sums @ real @ G @ X3 )
=> ( ( sums @ real @ F2 @ Y3 )
=> ( 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 @ X3 @ Y3 ) ) ) ) ).
% sums_if
thf(fact_3833_exp__bound,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X3 ) @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% exp_bound
thf(fact_3834_real__exp__bound__lemma,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X3 ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 ) ) ) ) ) ).
% real_exp_bound_lemma
thf(fact_3835_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N: nat,X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N ) ) @ X3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X3 @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ X3 ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_3836_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X3: real,N: nat] :
( ( ord_less_eq @ real @ X3 @ ( semiring_1_of_nat @ real @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X3 @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ ( uminus_uminus @ real @ X3 ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_3837_exp__bound__lemma,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ Z ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( real_V7770717601297561774m_norm @ A @ Z ) ) ) ) ) ) ).
% exp_bound_lemma
thf(fact_3838_Maclaurin__exp__le,axiom,
! [X3: real,N: nat] :
? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X3 ) )
& ( ( exp @ real @ X3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( divide_divide @ real @ ( power_power @ real @ X3 @ 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 @ X3 @ N ) ) ) ) ) ).
% Maclaurin_exp_le
thf(fact_3839_exp__lower__Taylor__quadratic,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) @ ( divide_divide @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ X3 ) ) ) ).
% exp_lower_Taylor_quadratic
thf(fact_3840_log__base__10__eq2,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X3 )
= ( times_times @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ X3 ) ) ) ) ).
% log_base_10_eq2
thf(fact_3841_tanh__real__altdef,axiom,
( ( tanh @ real )
= ( ^ [X: real] : ( divide_divide @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X ) ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X ) ) ) ) ) ) ).
% tanh_real_altdef
thf(fact_3842_cos__paired,axiom,
! [X3: real] :
( sums @ real
@ ^ [N3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) @ ( semiring_char_0_fact @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) @ ( power_power @ real @ X3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( cos @ real @ X3 ) ) ).
% cos_paired
thf(fact_3843_log__base__10__eq1,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X3 )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( ln_ln @ real @ X3 ) ) ) ) ).
% log_base_10_eq1
thf(fact_3844_sin__x__sin__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y3: A] :
( sums @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
@ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P5 @ N3 ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) ) @ ( power_power @ A @ X3 @ N3 ) ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ P5 @ N3 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P5 ) )
@ ( times_times @ A @ ( sin @ A @ X3 ) @ ( sin @ A @ Y3 ) ) ) ) ).
% sin_x_sin_y
thf(fact_3845_sums__cos__x__plus__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y3: A] :
( sums @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 ) @ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P5 @ N3 ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ A @ X3 @ N3 ) ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ P5 @ N3 ) ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P5 ) )
@ ( cos @ A @ ( plus_plus @ A @ X3 @ Y3 ) ) ) ) ).
% sums_cos_x_plus_y
thf(fact_3846_cos__x__cos__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y3: A] :
( sums @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 )
& ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
@ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P5 @ N3 ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ A @ X3 @ N3 ) ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ P5 @ N3 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P5 ) )
@ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y3 ) ) ) ) ).
% cos_x_cos_y
thf(fact_3847_diffs__equiv,axiom,
! [A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( ring_1 @ A ) )
=> ! [C2: nat > A,X3: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( C2 @ N3 ) ) @ ( power_power @ A @ X3 @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) ) ) ) ) ).
% diffs_equiv
thf(fact_3848_scaleR__cancel__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X3: A,B2: real] :
( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X3 )
= ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) )
= ( ( A2 = B2 )
| ( X3
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_cancel_right
thf(fact_3849_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_3850_scaleR__cancel__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X3: A,Y3: A] :
( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X3 )
= ( real_V8093663219630862766scaleR @ A @ A2 @ Y3 ) )
= ( ( X3 = Y3 )
| ( A2
= ( zero_zero @ real ) ) ) ) ) ).
% scaleR_cancel_left
thf(fact_3851_scaleR__zero__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( zero_zero @ real ) @ X3 )
= ( zero_zero @ A ) ) ) ).
% scaleR_zero_left
thf(fact_3852_scaleR__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X3: A] :
( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X3 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ real ) )
| ( X3
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_eq_0_iff
thf(fact_3853_scaleR__power,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X3: real,Y3: A,N: nat] :
( ( power_power @ A @ ( real_V8093663219630862766scaleR @ A @ X3 @ Y3 ) @ N )
= ( real_V8093663219630862766scaleR @ A @ ( power_power @ real @ X3 @ N ) @ ( power_power @ A @ Y3 @ N ) ) ) ) ).
% scaleR_power
thf(fact_3854_scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [U: num,W: num,A2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ U ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A2 ) )
= ( real_V8093663219630862766scaleR @ A @ ( times_times @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ W ) ) @ A2 ) ) ) ).
% scaleR_times
thf(fact_3855_inverse__scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [V2: num,W: num,A2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ V2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A2 ) )
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ W ) @ ( numeral_numeral @ real @ V2 ) ) @ A2 ) ) ) ).
% inverse_scaleR_times
thf(fact_3856_fraction__scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [U: num,V2: num,W: num,A2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ V2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A2 ) )
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ W ) ) @ ( numeral_numeral @ real @ V2 ) ) @ A2 ) ) ) ).
% fraction_scaleR_times
thf(fact_3857_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_3858_scaleR__left__imp__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X3: A,Y3: A] :
( ( A2
!= ( zero_zero @ real ) )
=> ( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X3 )
= ( real_V8093663219630862766scaleR @ A @ A2 @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ).
% scaleR_left_imp_eq
thf(fact_3859_scaleR__right__imp__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A,A2: real,B2: real] :
( ( X3
!= ( zero_zero @ A ) )
=> ( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X3 )
= ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) )
=> ( A2 = B2 ) ) ) ) ).
% scaleR_right_imp_eq
thf(fact_3860_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_3861_scaleR__right__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,X3: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) ) ) ) ) ).
% scaleR_right_mono
thf(fact_3862_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_3863_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_3864_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_3865_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_3866_scaleR__left__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X3: A,Y3: A,A2: real] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y3 ) ) ) ) ) ).
% scaleR_left_mono
thf(fact_3867_eq__vector__fraction__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A,U: real,V2: real,A2: A] :
( ( X3
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V2 ) @ A2 ) )
= ( ( ( V2
= ( zero_zero @ real ) )
=> ( X3
= ( zero_zero @ A ) ) )
& ( ( V2
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ V2 @ X3 )
= ( real_V8093663219630862766scaleR @ A @ U @ A2 ) ) ) ) ) ) ).
% eq_vector_fraction_iff
thf(fact_3868_vector__fraction__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U: real,V2: real,A2: A,X3: A] :
( ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V2 ) @ A2 )
= X3 )
= ( ( ( V2
= ( zero_zero @ real ) )
=> ( X3
= ( zero_zero @ A ) ) )
& ( ( V2
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ U @ A2 )
= ( real_V8093663219630862766scaleR @ A @ V2 @ X3 ) ) ) ) ) ) ).
% vector_fraction_eq_iff
thf(fact_3869_Real__Vector__Spaces_Ole__add__iff1,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,E2: A,C2: A,B2: real,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E2 ) @ D3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ E2 ) @ C2 ) @ D3 ) ) ) ).
% Real_Vector_Spaces.le_add_iff1
thf(fact_3870_Real__Vector__Spaces_Ole__add__iff2,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,E2: A,C2: A,B2: real,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E2 ) @ D3 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ B2 @ A2 ) @ E2 ) @ D3 ) ) ) ) ).
% Real_Vector_Spaces.le_add_iff2
thf(fact_3871_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_3872_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_3873_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_3874_scaleR__nonpos__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X3: A] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_3875_scaleR__nonneg__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X3: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_3876_scaleR__nonneg__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X3: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_3877_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_3878_split__scaleR__neg__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X3: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( zero_zero @ A ) ) ) ) ).
% split_scaleR_neg_le
thf(fact_3879_scaleR__mono_H,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,C2: A,D3: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( 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 @ D3 ) ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_3880_scaleR__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,X3: A,Y3: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ Y3 ) ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_3881_scaleR__left__le__one__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X3: A,A2: real] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ real @ A2 @ ( one_one @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ X3 ) ) ) ) ).
% scaleR_left_le_one_le
thf(fact_3882_scaleR__2,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 )
= ( plus_plus @ A @ X3 @ X3 ) ) ) ).
% scaleR_2
thf(fact_3883_diffs__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( diffs @ A )
= ( ^ [C4: nat > A,N3: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) @ ( C4 @ ( suc @ N3 ) ) ) ) ) ) ).
% diffs_def
thf(fact_3884_termdiff__converges__all,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X3: A] :
( ! [X4: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X4 @ N3 ) ) )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) ) ) ) ).
% termdiff_converges_all
thf(fact_3885_sin__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ X3 @ N3 ) )
@ ( sin @ A @ X3 ) ) ) ).
% sin_converges
thf(fact_3886_sin__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A )
= ( ^ [X: A] :
( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ) ).
% sin_def
thf(fact_3887_cos__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ X3 @ N3 ) )
@ ( cos @ A @ X3 ) ) ) ).
% cos_converges
thf(fact_3888_cos__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cos @ A )
= ( ^ [X: A] :
( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ) ).
% cos_def
thf(fact_3889_summable__norm__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) ) ) ) ).
% summable_norm_sin
thf(fact_3890_summable__norm__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) ) ) ) ).
% summable_norm_cos
thf(fact_3891_sin__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( sums @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X3 ) @ N3 ) ) )
@ ( sin @ A @ X3 ) ) ) ).
% sin_minus_converges
thf(fact_3892_cos__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X3 ) @ N3 ) )
@ ( cos @ A @ X3 ) ) ) ).
% cos_minus_converges
thf(fact_3893_termdiff__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,K5: real,C2: nat > A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K5 )
=> ( ! [X4: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ K5 )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X4 @ N3 ) ) ) )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) ) ) ) ) ).
% termdiff_converges
thf(fact_3894_monoseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X9: nat > A] :
( ! [M3: nat,N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
=> ( ord_less_eq @ A @ ( X9 @ M3 ) @ ( X9 @ N3 ) ) )
| ! [M3: nat,N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
=> ( ord_less_eq @ A @ ( X9 @ N3 ) @ ( X9 @ M3 ) ) ) ) ) ) ) ).
% monoseq_def
thf(fact_3895_monoI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [M4: nat,N2: nat] :
( ( ord_less_eq @ nat @ M4 @ N2 )
=> ( ord_less_eq @ A @ ( X8 @ N2 ) @ ( X8 @ M4 ) ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% monoI2
thf(fact_3896_monoI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [M4: nat,N2: nat] :
( ( ord_less_eq @ nat @ M4 @ N2 )
=> ( ord_less_eq @ A @ ( X8 @ M4 ) @ ( X8 @ N2 ) ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% monoI1
thf(fact_3897_mono__SucI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% mono_SucI1
thf(fact_3898_monoseq__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X9: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
| ! [N3: nat] : ( ord_less_eq @ A @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) ) ) ) ) ) ).
% monoseq_Suc
thf(fact_3899_mono__SucI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% mono_SucI2
thf(fact_3900_exp__first__two__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X: A] :
( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X )
@ ( suminf @ A
@ ^ [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 @ X @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% exp_first_two_terms
thf(fact_3901_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_3902_Maclaurin__sin__bound,axiom,
! [X3: real,N: nat] :
( ord_less_eq @ real
@ ( abs_abs @ real
@ ( minus_minus @ real @ ( sin @ real @ X3 )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( sin_coeff @ M3 ) @ ( power_power @ real @ X3 @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) )
@ ( times_times @ real @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( abs_abs @ real @ X3 ) @ N ) ) ) ).
% Maclaurin_sin_bound
thf(fact_3903_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_3904_inverse__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% inverse_zero
thf(fact_3905_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_3906_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_3907_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_3908_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_3909_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_3910_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_3911_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_3912_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_3913_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_3914_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_3915_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_3916_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_3917_inverse__eq__divide__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num] :
( ( inverse_inverse @ A @ ( numeral_numeral @ A @ W ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_3918_pred__numeral__simps_I2_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit0 @ K ) )
= ( numeral_numeral @ nat @ ( bitM @ K ) ) ) ).
% pred_numeral_simps(2)
thf(fact_3919_inverse__eq__divide__neg__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_3920_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_3921_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_3922_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_3923_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_3924_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_3925_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_3926_nonzero__of__real__inverse,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [X3: real] :
( ( X3
!= ( zero_zero @ real ) )
=> ( ( real_Vector_of_real @ A @ ( inverse_inverse @ real @ X3 ) )
= ( inverse_inverse @ A @ ( real_Vector_of_real @ A @ X3 ) ) ) ) ) ).
% nonzero_of_real_inverse
thf(fact_3927_real__sqrt__inverse,axiom,
! [X3: real] :
( ( sqrt @ ( inverse_inverse @ real @ X3 ) )
= ( inverse_inverse @ real @ ( sqrt @ X3 ) ) ) ).
% real_sqrt_inverse
thf(fact_3928_semiring__norm_I26_J,axiom,
( ( bitM @ one2 )
= one2 ) ).
% semiring_norm(26)
thf(fact_3929_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_3930_nonzero__inverse__scaleR__distrib,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [A2: real,X3: A] :
( ( A2
!= ( zero_zero @ real ) )
=> ( ( X3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) )
= ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ A2 ) @ ( inverse_inverse @ A @ X3 ) ) ) ) ) ) ).
% nonzero_inverse_scaleR_distrib
thf(fact_3931_norm__inverse__le__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [R2: real,X3: A] :
( ( ord_less_eq @ real @ R2 @ ( real_V7770717601297561774m_norm @ A @ X3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ X3 ) ) @ ( inverse_inverse @ real @ R2 ) ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_3932_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_3933_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_3934_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_3935_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_3936_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_3937_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_3938_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_3939_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_3940_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_3941_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_3942_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_3943_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ N ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ N ) @ ( power_power @ A @ X3 @ M ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_3944_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: nat] :
( ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( inverse_inverse @ A @ X3 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X3 ) @ ( power_power @ A @ X3 @ M ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_3945_mult__inverse__of__nat__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: nat,X3: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) @ X3 )
= ( times_times @ A @ X3 @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_3946_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_3947_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: int,X3: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) @ X3 )
= ( times_times @ A @ X3 @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_3948_divide__real__def,axiom,
( ( divide_divide @ real )
= ( ^ [X: real,Y: real] : ( times_times @ real @ X @ ( inverse_inverse @ real @ Y ) ) ) ) ).
% divide_real_def
thf(fact_3949_semiring__norm_I28_J,axiom,
! [N: num] :
( ( bitM @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ N ) ) ) ).
% semiring_norm(28)
thf(fact_3950_semiring__norm_I27_J,axiom,
! [N: num] :
( ( bitM @ ( bit0 @ N ) )
= ( bit1 @ ( bitM @ N ) ) ) ).
% semiring_norm(27)
thf(fact_3951_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_3952_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_3953_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_3954_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_3955_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_3956_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_3957_inverse__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X3 ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ) ).
% inverse_le_1_iff
thf(fact_3958_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_3959_one__less__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X3 ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
& ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ) ).
% one_less_inverse_iff
thf(fact_3960_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_3961_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_3962_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_3963_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_3964_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_3965_inverse__powr,axiom,
! [Y3: real,A2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( powr @ real @ ( inverse_inverse @ real @ Y3 ) @ A2 )
= ( inverse_inverse @ real @ ( powr @ real @ Y3 @ A2 ) ) ) ) ).
% inverse_powr
thf(fact_3966_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_3967_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_3968_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_3969_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_3970_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_3971_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_3972_inverse__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ X3 ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
| ( ord_less @ A @ ( one_one @ A ) @ X3 ) ) ) ) ).
% inverse_less_1_iff
thf(fact_3973_one__le__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X3 ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
& ( ord_less_eq @ A @ X3 @ ( one_one @ A ) ) ) ) ) ).
% one_le_inverse_iff
thf(fact_3974_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus @ num @ ( bitM @ N ) @ one2 )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_3975_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_3976_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_3977_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ? [N2: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) ) @ X3 ) ) ) ).
% reals_Archimedean
thf(fact_3978_real__vector__affinity__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M: real,X3: A,C2: A,Y3: A] :
( ( M
!= ( zero_zero @ real ) )
=> ( ( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X3 ) @ C2 )
= Y3 )
= ( X3
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y3 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) ) ) ) ) ) ).
% real_vector_affinity_eq
thf(fact_3979_real__vector__eq__affinity,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M: real,Y3: A,X3: A,C2: A] :
( ( M
!= ( zero_zero @ real ) )
=> ( ( Y3
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X3 ) @ C2 ) )
= ( ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y3 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) )
= X3 ) ) ) ) ).
% real_vector_eq_affinity
thf(fact_3980_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_3981_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_3982_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_3983_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_3984_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_3985_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_3986_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_3987_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_3988_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_3989_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_3990_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_3991_sqrt__divide__self__eq,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( divide_divide @ real @ ( sqrt @ X3 ) @ X3 )
= ( inverse_inverse @ real @ ( sqrt @ X3 ) ) ) ) ).
% sqrt_divide_self_eq
thf(fact_3992_ln__inverse,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ln_ln @ real @ ( inverse_inverse @ real @ X3 ) )
= ( uminus_uminus @ real @ ( ln_ln @ real @ X3 ) ) ) ) ).
% ln_inverse
thf(fact_3993_summable__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N3 ) ) @ ( power_power @ A @ X3 @ N3 ) ) ) ) ).
% summable_exp
thf(fact_3994_summable__exp__generic,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( summable @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X3 @ N3 ) ) ) ) ).
% summable_exp_generic
thf(fact_3995_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_3996_odd__numeral__BitM,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [W: num] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bitM @ W ) ) ) ) ).
% odd_numeral_BitM
thf(fact_3997_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ X3 ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_3998_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: nat,N: nat] :
( ( X3
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( power_power @ A @ X3 @ ( minus_minus @ nat @ N @ M ) )
= ( times_times @ A @ ( power_power @ A @ X3 @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ M ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_3999_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_4000_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_4001_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_4002_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_4003_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_4004_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_4005_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_4006_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_4007_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_4008_log__inverse,axiom,
! [A2: real,X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( log @ A2 @ ( inverse_inverse @ real @ X3 ) )
= ( uminus_uminus @ real @ ( log @ A2 @ X3 ) ) ) ) ) ) ).
% log_inverse
thf(fact_4009_exp__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X3 @ N3 ) )
@ ( exp @ A @ X3 ) ) ) ).
% exp_converges
thf(fact_4010_exp__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X: A] :
( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ) ).
% exp_def
thf(fact_4011_summable__norm__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X3 @ N3 ) ) ) ) ) ).
% summable_norm_exp
thf(fact_4012_exp__plus__inverse__exp,axiom,
! [X3: real] : ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ ( exp @ real @ X3 ) @ ( inverse_inverse @ real @ ( exp @ real @ X3 ) ) ) ) ).
% exp_plus_inverse_exp
thf(fact_4013_plus__inverse__ge__2,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ X3 @ ( inverse_inverse @ real @ X3 ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_4014_real__inv__sqrt__pow2,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( power_power @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( inverse_inverse @ real @ X3 ) ) ) ).
% real_inv_sqrt_pow2
thf(fact_4015_tan__cot,axiom,
! [X3: real] :
( ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 ) )
= ( inverse_inverse @ real @ ( tan @ real @ X3 ) ) ) ).
% tan_cot
thf(fact_4016_real__le__x__sinh,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X3 ) @ ( inverse_inverse @ real @ ( exp @ real @ X3 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_x_sinh
thf(fact_4017_real__le__abs__sinh,axiom,
! [X3: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( abs_abs @ real @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X3 ) @ ( inverse_inverse @ real @ ( exp @ real @ X3 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_abs_sinh
thf(fact_4018_exp__series__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A,Y3: A,N: nat] :
( ( ( times_times @ A @ X3 @ Y3 )
= ( times_times @ A @ Y3 @ X3 ) )
=> ( ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ ( plus_plus @ A @ X3 @ Y3 ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ I4 ) ) @ ( power_power @ A @ X3 @ I4 ) ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ I4 ) ) ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ N @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% exp_series_add_commuting
thf(fact_4019_exp__first__term,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X: A] :
( plus_plus @ A @ ( one_one @ A )
@ ( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( suc @ N3 ) ) ) @ ( power_power @ A @ X @ ( suc @ N3 ) ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_4020_tan__sec,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ A @ ( inverse_inverse @ A @ ( cos @ A @ X3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% tan_sec
thf(fact_4021_powr__real__of__int,axiom,
! [X3: real,N: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ N ) )
= ( power_power @ real @ X3 @ ( nat2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ N ) )
= ( inverse_inverse @ real @ ( power_power @ real @ X3 @ ( nat2 @ ( uminus_uminus @ int @ N ) ) ) ) ) ) ) ) ).
% powr_real_of_int
thf(fact_4022_exp__first__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [K: nat] :
( ( exp @ A )
= ( ^ [X: A] :
( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X @ 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 @ X @ ( plus_plus @ nat @ N3 @ K ) ) ) ) ) ) ) ) ).
% exp_first_terms
thf(fact_4023_sinh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( sums @ A
@ ^ [N3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X3 @ N3 ) ) )
@ ( sinh @ A @ X3 ) ) ) ).
% sinh_converges
thf(fact_4024_cosh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X3: A] :
( sums @ A
@ ^ [N3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X3 @ N3 ) ) @ ( zero_zero @ A ) )
@ ( cosh @ A @ X3 ) ) ) ).
% cosh_converges
thf(fact_4025_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_4026_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_4027_sinh__real__zero__iff,axiom,
! [X3: real] :
( ( ( sinh @ real @ X3 )
= ( zero_zero @ real ) )
= ( X3
= ( zero_zero @ real ) ) ) ).
% sinh_real_zero_iff
thf(fact_4028_sinh__real__less__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ ( sinh @ real @ X3 ) @ ( sinh @ real @ Y3 ) )
= ( ord_less @ real @ X3 @ Y3 ) ) ).
% sinh_real_less_iff
thf(fact_4029_sinh__real__le__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( sinh @ real @ X3 ) @ ( sinh @ real @ Y3 ) )
= ( ord_less_eq @ real @ X3 @ Y3 ) ) ).
% sinh_real_le_iff
thf(fact_4030_sinh__real__pos__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X3 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ).
% sinh_real_pos_iff
thf(fact_4031_sinh__real__neg__iff,axiom,
! [X3: real] :
( ( ord_less @ real @ ( sinh @ real @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ).
% sinh_real_neg_iff
thf(fact_4032_sinh__real__nonpos__iff,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( sinh @ real @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) ) ) ).
% sinh_real_nonpos_iff
thf(fact_4033_sinh__real__nonneg__iff,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X3 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) ) ).
% sinh_real_nonneg_iff
thf(fact_4034_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_4035_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_4036_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_4037_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_4038_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_4039_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_4040_or__not__num__neg_Osimps_I1_J,axiom,
( ( bit_or_not_num_neg @ one2 @ one2 )
= one2 ) ).
% or_not_num_neg.simps(1)
thf(fact_4041_sinh__le__cosh__real,axiom,
! [X3: real] : ( ord_less_eq @ real @ ( sinh @ real @ X3 ) @ ( cosh @ real @ X3 ) ) ).
% sinh_le_cosh_real
thf(fact_4042_sinh__less__cosh__real,axiom,
! [X3: real] : ( ord_less @ real @ ( sinh @ real @ X3 ) @ ( cosh @ real @ X3 ) ) ).
% sinh_less_cosh_real
thf(fact_4043_cosh__real__nonzero,axiom,
! [X3: real] :
( ( cosh @ real @ X3 )
!= ( zero_zero @ real ) ) ).
% cosh_real_nonzero
thf(fact_4044_cosh__real__pos,axiom,
! [X3: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X3 ) ) ).
% cosh_real_pos
thf(fact_4045_cosh__real__nonpos__le__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( cosh @ real @ X3 ) @ ( cosh @ real @ Y3 ) )
= ( ord_less_eq @ real @ Y3 @ X3 ) ) ) ) ).
% cosh_real_nonpos_le_iff
thf(fact_4046_cosh__real__nonneg__le__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less_eq @ real @ ( cosh @ real @ X3 ) @ ( cosh @ real @ Y3 ) )
= ( ord_less_eq @ real @ X3 @ Y3 ) ) ) ) ).
% cosh_real_nonneg_le_iff
thf(fact_4047_cosh__real__nonneg,axiom,
! [X3: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X3 ) ) ).
% cosh_real_nonneg
thf(fact_4048_cosh__real__ge__1,axiom,
! [X3: real] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( cosh @ real @ X3 ) ) ).
% cosh_real_ge_1
thf(fact_4049_sinh__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( sinh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sinh @ A @ X3 ) ) @ ( cosh @ A @ X3 ) ) ) ) ).
% sinh_double
thf(fact_4050_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_4051_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_4052_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_4053_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_4054_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_4055_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_4056_cosh__real__strict__mono,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ Y3 )
=> ( ord_less @ real @ ( cosh @ real @ X3 ) @ ( cosh @ real @ Y3 ) ) ) ) ).
% cosh_real_strict_mono
thf(fact_4057_cosh__real__nonneg__less__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ord_less @ real @ ( cosh @ real @ X3 ) @ ( cosh @ real @ Y3 ) )
= ( ord_less @ real @ X3 @ Y3 ) ) ) ) ).
% cosh_real_nonneg_less_iff
thf(fact_4058_cosh__real__nonpos__less__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( cosh @ real @ X3 ) @ ( cosh @ real @ Y3 ) )
= ( ord_less @ real @ Y3 @ X3 ) ) ) ) ).
% cosh_real_nonpos_less_iff
thf(fact_4059_cosh__square__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( power_power @ A @ ( cosh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( power_power @ A @ ( sinh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% cosh_square_eq
thf(fact_4060_hyperbolic__pythagoras,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( minus_minus @ A @ ( power_power @ A @ ( cosh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% hyperbolic_pythagoras
thf(fact_4061_sinh__square__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( power_power @ A @ ( sinh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( cosh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% sinh_square_eq
thf(fact_4062_arcosh__cosh__real,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( arcosh @ real @ ( cosh @ real @ X3 ) )
= X3 ) ) ).
% arcosh_cosh_real
thf(fact_4063_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_4064_cosh__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( cosh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) )
= ( plus_plus @ A @ ( power_power @ A @ ( cosh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_double
thf(fact_4065_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_4066_or__not__num__neg_Oelims,axiom,
! [X3: num,Xa: num,Y3: num] :
( ( ( bit_or_not_num_neg @ X3 @ Xa )
= Y3 )
=> ( ( ( X3 = one2 )
=> ( ( Xa = one2 )
=> ( Y3 != one2 ) ) )
=> ( ( ( X3 = one2 )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( Y3
!= ( bit1 @ M4 ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( Y3
!= ( bit1 @ M4 ) ) ) )
=> ( ( ? [N2: num] :
( X3
= ( bit0 @ N2 ) )
=> ( ( Xa = one2 )
=> ( Y3
!= ( bit0 @ one2 ) ) ) )
=> ( ! [N2: num] :
( ( X3
= ( bit0 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( Y3
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
=> ( ! [N2: num] :
( ( X3
= ( bit0 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( Y3
!= ( bit0 @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
=> ( ( ? [N2: num] :
( X3
= ( bit1 @ N2 ) )
=> ( ( Xa = one2 )
=> ( Y3 != one2 ) ) )
=> ( ! [N2: num] :
( ( X3
= ( bit1 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( Y3
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
=> ~ ! [N2: num] :
( ( X3
= ( bit1 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( Y3
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.elims
thf(fact_4067_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_4068_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_4069_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_4070_tanh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,Y3: A] :
( ( ( cosh @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( ( ( cosh @ A @ Y3 )
!= ( zero_zero @ A ) )
=> ( ( tanh @ A @ ( plus_plus @ A @ X3 @ Y3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tanh @ A @ X3 ) @ ( tanh @ A @ Y3 ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tanh @ A @ X3 ) @ ( tanh @ A @ Y3 ) ) ) ) ) ) ) ) ).
% tanh_add
thf(fact_4071_sinh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( sinh @ A @ X3 )
= ( zero_zero @ A ) )
= ( member @ A @ ( exp @ A @ X3 ) @ ( insert @ A @ ( one_one @ A ) @ ( insert @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% sinh_zero_iff
thf(fact_4072_cosh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cosh @ A )
= ( ^ [Z6: A] : ( divide_divide @ A @ ( plus_plus @ A @ ( exp @ A @ Z6 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z6 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_field_def
thf(fact_4073_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_4074_sinh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( sinh @ A )
= ( ^ [Z6: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z6 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z6 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sinh_field_def
thf(fact_4075_cosh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( cosh @ A @ X3 )
= ( zero_zero @ A ) )
= ( ( power_power @ A @ ( exp @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% cosh_zero_iff
thf(fact_4076_cosh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cosh @ A )
= ( ^ [X: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) ) ) ) ) ).
% cosh_def
thf(fact_4077_cosh__ln__real,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( cosh @ real @ ( ln_ln @ real @ X3 ) )
= ( divide_divide @ real @ ( plus_plus @ real @ X3 @ ( inverse_inverse @ real @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% cosh_ln_real
thf(fact_4078_sinh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sinh @ A )
= ( ^ [X: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) ) ) ) ) ).
% sinh_def
thf(fact_4079_sinh__ln__real,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( sinh @ real @ ( ln_ln @ real @ X3 ) )
= ( divide_divide @ real @ ( minus_minus @ real @ X3 @ ( inverse_inverse @ real @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% sinh_ln_real
thf(fact_4080_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_4081_cot__less__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cot @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ).
% cot_less_zero
thf(fact_4082_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 )
@ ^ [Q5: A,R5: A] : ( product_Pair @ A @ A @ Q5 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_4083_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_4084_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_4085_concat__bit__0,axiom,
! [K: int,L: int] :
( ( bit_concat_bit @ ( zero_zero @ nat ) @ K @ L )
= L ) ).
% concat_bit_0
thf(fact_4086_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_4087_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_4088_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_4089_cot__pi,axiom,
( ( cot @ real @ pi )
= ( zero_zero @ real ) ) ).
% cot_pi
thf(fact_4090_cot__npi,axiom,
! [N: nat] :
( ( cot @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% cot_npi
thf(fact_4091_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 )
@ ^ [Q5: A,R5: A] : ( product_Pair @ A @ A @ Q5 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_4092_cot__periodic,axiom,
! [X3: real] :
( ( cot @ real @ ( plus_plus @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( cot @ real @ X3 ) ) ).
% cot_periodic
thf(fact_4093_power2__i,axiom,
( ( power_power @ complex @ imaginary_unit @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% power2_i
thf(fact_4094_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_4095_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_4096_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q3: product_prod @ A @ B,F2: A > B > C,G: A > B > C,P6: product_prod @ A @ B] :
( ! [X4: A,Y4: B] :
( ( ( product_Pair @ A @ B @ X4 @ Y4 )
= Q3 )
=> ( ( F2 @ X4 @ Y4 )
= ( G @ X4 @ Y4 ) ) )
=> ( ( P6 = Q3 )
=> ( ( product_case_prod @ A @ B @ C @ F2 @ P6 )
= ( product_case_prod @ A @ B @ C @ G @ Q3 ) ) ) ) ).
% split_cong
thf(fact_4097_complex__i__not__zero,axiom,
( imaginary_unit
!= ( zero_zero @ complex ) ) ).
% complex_i_not_zero
thf(fact_4098_concat__bit__assoc,axiom,
! [N: nat,K: int,M: nat,L: int,R2: int] :
( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
= ( bit_concat_bit @ ( plus_plus @ nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R2 ) ) ).
% concat_bit_assoc
thf(fact_4099_concat__bit__eq__iff,axiom,
! [N: nat,K: int,L: int,R2: int,S: int] :
( ( ( bit_concat_bit @ N @ K @ L )
= ( bit_concat_bit @ N @ R2 @ S ) )
= ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2584673776208193580ke_bit @ int @ N @ R2 ) )
& ( L = S ) ) ) ).
% concat_bit_eq_iff
thf(fact_4100_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_4101_imaginary__unit_Ocode,axiom,
( imaginary_unit
= ( complex2 @ ( zero_zero @ real ) @ ( one_one @ real ) ) ) ).
% imaginary_unit.code
thf(fact_4102_Complex__eq__i,axiom,
! [X3: real,Y3: real] :
( ( ( complex2 @ X3 @ Y3 )
= imaginary_unit )
= ( ( X3
= ( zero_zero @ real ) )
& ( Y3
= ( one_one @ real ) ) ) ) ).
% Complex_eq_i
thf(fact_4103_i__complex__of__real,axiom,
! [R2: real] :
( ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ R2 ) )
= ( complex2 @ ( zero_zero @ real ) @ R2 ) ) ).
% i_complex_of_real
thf(fact_4104_complex__of__real__i,axiom,
! [R2: real] :
( ( times_times @ complex @ ( real_Vector_of_real @ complex @ R2 ) @ imaginary_unit )
= ( complex2 @ ( zero_zero @ real ) @ R2 ) ) ).
% complex_of_real_i
thf(fact_4105_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L2: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q5: nat,R5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L2 ) @ R5 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L2 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_4106_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L2: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q5: int,R5: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L2 ) @ R5 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L2 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_4107_cot__gt__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cot @ real @ X3 ) ) ) ) ).
% cot_gt_zero
thf(fact_4108_tan__cot_H,axiom,
! [X3: real] :
( ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 ) )
= ( cot @ real @ X3 ) ) ).
% tan_cot'
thf(fact_4109_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L2: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R5: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R5 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L2 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_4110_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_4111_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_4112_Arg__ii,axiom,
( ( arg @ imaginary_unit )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% Arg_ii
thf(fact_4113_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_4114_csqrt__eq__0,axiom,
! [Z: complex] :
( ( ( csqrt @ Z )
= ( zero_zero @ complex ) )
= ( Z
= ( zero_zero @ complex ) ) ) ).
% csqrt_eq_0
thf(fact_4115_csqrt__0,axiom,
( ( csqrt @ ( zero_zero @ complex ) )
= ( zero_zero @ complex ) ) ).
% csqrt_0
thf(fact_4116_cis__zero,axiom,
( ( cis @ ( zero_zero @ real ) )
= ( one_one @ complex ) ) ).
% cis_zero
thf(fact_4117_power2__csqrt,axiom,
! [Z: complex] :
( ( power_power @ complex @ ( csqrt @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= Z ) ).
% power2_csqrt
thf(fact_4118_cis__pi__half,axiom,
( ( cis @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= imaginary_unit ) ).
% cis_pi_half
thf(fact_4119_cis__2pi,axiom,
( ( cis @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ complex ) ) ).
% cis_2pi
thf(fact_4120_cis__neq__zero,axiom,
! [A2: real] :
( ( cis @ A2 )
!= ( zero_zero @ complex ) ) ).
% cis_neq_zero
thf(fact_4121_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_4122_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_4123_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_4124_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_4125_Arg__zero,axiom,
( ( arg @ ( zero_zero @ complex ) )
= ( zero_zero @ real ) ) ).
% Arg_zero
thf(fact_4126_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_4127_of__real__sqrt,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( real_Vector_of_real @ complex @ ( sqrt @ X3 ) )
= ( csqrt @ ( real_Vector_of_real @ complex @ X3 ) ) ) ) ).
% of_real_sqrt
thf(fact_4128_Arg__bounded,axiom,
! [Z: complex] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ ( arg @ Z ) )
& ( ord_less_eq @ real @ ( arg @ Z ) @ pi ) ) ).
% Arg_bounded
thf(fact_4129_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,Q5: nat] :
( if @ A
@ ( Q5
= ( 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_4130_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 )
@ ^ [Q5: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q5 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M3 @ N3 ) @ N3 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_4131_bij__betw__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( bij_betw @ nat @ complex
@ ^ [K3: nat] : ( cis @ ( divide_divide @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( semiring_1_of_nat @ real @ K3 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
@ ( set_ord_lessThan @ nat @ N )
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= ( one_one @ complex ) ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_4132_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_4133_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_4134_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_4135_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_4136_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_4137_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_4138_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_4139_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_4140_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_4141_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_4142_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_4143_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_4144_Divides_Oadjust__div__eq,axiom,
! [Q3: int,R2: int] :
( ( adjust_div @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
= ( plus_plus @ int @ Q3
@ ( zero_neq_one_of_bool @ int
@ ( R2
!= ( zero_zero @ int ) ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_4145_odd__of__bool__self,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [P6: $o] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_neq_one_of_bool @ A @ P6 ) ) )
= P6 ) ) ).
% odd_of_bool_self
thf(fact_4146_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_4147_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_4148_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_4149_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_4150_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_4151_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_4152_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_4153_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_4154_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_4155_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P6: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P6 ) )
= ( ~ ( ( P6
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P6
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_4156_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P6: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P6 ) )
= ( ( P6
=> ( P @ ( one_one @ A ) ) )
& ( ~ P6
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_4157_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P5: $o] : ( if @ A @ P5 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_4158_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( product_case_prod @ int @ int @ int
@ ^ [Q5: int,R5: int] :
( plus_plus @ int @ Q5
@ ( zero_neq_one_of_bool @ int
@ ( R5
!= ( zero_zero @ int ) ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_4159_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S4: set @ B,T5: set @ C,H: B > C,S2: set @ B,T6: set @ C,G: C > A] :
( ( finite_finite @ B @ S4 )
=> ( ( finite_finite @ C @ T5 )
=> ( ( bij_betw @ B @ C @ H @ ( minus_minus @ ( set @ B ) @ S2 @ S4 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S4 )
=> ( ( G @ ( H @ A6 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T5 )
=> ( ( G @ B5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( G @ ( H @ X ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T6 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_4160_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_4161_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,B5: $o] :
( ( P @ A6 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B5 ) @ ( 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 @ B5 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% bits_induct
thf(fact_4162_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_4163_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_4164_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_4165_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_4166_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_4167_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_4168_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_4169_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_4170_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_4171_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_4172_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_4173_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_4174_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_4175_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_4176_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_4177_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_4178_and__int_Oelims,axiom,
! [X3: int,Xa: int,Y3: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X3 @ Xa )
= Y3 )
=> ( ( ( ( member @ int @ X3 @ ( 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 ) ) ) ) ) )
=> ( Y3
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X3 @ ( 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 ) ) ) ) ) )
=> ( Y3
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_4179_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D4: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z7: int,Z6: int] :
( ( ord_less_eq @ int @ D4 @ Z6 )
& ( ord_less @ int @ Z7 @ Z6 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_4180_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D4: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z7: int,Z6: int] :
( ( ord_less_eq @ int @ D4 @ Z7 )
& ( ord_less @ int @ Z7 @ Z6 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_4181_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_4182_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_4183_and_Oidem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ A2 )
= A2 ) ) ).
% and.idem
thf(fact_4184_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_4185_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_4186_bit_Oconj__zero__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ X3 )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_left
thf(fact_4187_bit_Oconj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se5824344872417868541ns_and @ A @ X3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_right
thf(fact_4188_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_4189_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_4190_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_4191_bit_Oconj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se5824344872417868541ns_and @ A @ X3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= X3 ) ) ).
% bit.conj_one_right
thf(fact_4192_bit_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X3 ) @ X3 )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_left
thf(fact_4193_bit_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A] :
( ( bit_se5824344872417868541ns_and @ A @ X3 @ ( bit_ri4277139882892585799ns_not @ A @ X3 ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_right
thf(fact_4194_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_4195_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_4196_bit_Ode__Morgan__conj,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A,Y3: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344872417868541ns_and @ A @ X3 @ Y3 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X3 ) @ ( bit_ri4277139882892585799ns_not @ A @ Y3 ) ) ) ) ).
% bit.de_Morgan_conj
thf(fact_4197_bit_Ode__Morgan__disj,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A,Y3: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( bit_se1065995026697491101ons_or @ A @ X3 @ Y3 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X3 ) @ ( bit_ri4277139882892585799ns_not @ A @ Y3 ) ) ) ) ).
% bit.de_Morgan_disj
thf(fact_4198_and__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( one_one @ A ) ) ) ).
% and_numerals(2)
thf(fact_4199_and__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% and_numerals(8)
thf(fact_4200_and__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(1)
thf(fact_4201_and__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(5)
thf(fact_4202_and__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% and_numerals(3)
thf(fact_4203_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_4204_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_4205_and__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% and_numerals(4)
thf(fact_4206_and__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% and_numerals(6)
thf(fact_4207_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_4208_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_4209_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_4210_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_4211_and__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ) ).
% and_numerals(7)
thf(fact_4212_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_4213_bit_Oconj__disj__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( bit_se5824344872417868541ns_and @ A @ X3 @ ( bit_se1065995026697491101ons_or @ A @ Y3 @ Z ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ X3 @ Y3 ) @ ( bit_se5824344872417868541ns_and @ A @ X3 @ Z ) ) ) ) ).
% bit.conj_disj_distrib
thf(fact_4214_bit_Odisj__conj__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( bit_se1065995026697491101ons_or @ A @ X3 @ ( bit_se5824344872417868541ns_and @ A @ Y3 @ Z ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ X3 @ Y3 ) @ ( bit_se1065995026697491101ons_or @ A @ X3 @ Z ) ) ) ) ).
% bit.disj_conj_distrib
thf(fact_4215_bit_Oconj__disj__distrib2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [Y3: A,Z: A,X3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ Y3 @ Z ) @ X3 )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ Y3 @ X3 ) @ ( bit_se5824344872417868541ns_and @ A @ Z @ X3 ) ) ) ) ).
% bit.conj_disj_distrib2
thf(fact_4216_bit_Odisj__conj__distrib2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [Y3: A,Z: A,X3: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ Y3 @ Z ) @ X3 )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ Y3 @ X3 ) @ ( bit_se1065995026697491101ons_or @ A @ Z @ X3 ) ) ) ) ).
% bit.disj_conj_distrib2
thf(fact_4217_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_4218_and_Ocommute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5824344872417868541ns_and @ A )
= ( ^ [A5: A,B4: A] : ( bit_se5824344872417868541ns_and @ A @ B4 @ A5 ) ) ) ) ).
% and.commute
thf(fact_4219_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_4220_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_4221_bit_Oconj__xor__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( bit_se5824344872417868541ns_and @ A @ X3 @ ( bit_se5824344971392196577ns_xor @ A @ Y3 @ Z ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se5824344872417868541ns_and @ A @ X3 @ Y3 ) @ ( bit_se5824344872417868541ns_and @ A @ X3 @ Z ) ) ) ) ).
% bit.conj_xor_distrib
thf(fact_4222_bit_Oconj__xor__distrib2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [Y3: A,Z: A,X3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se5824344971392196577ns_xor @ A @ Y3 @ Z ) @ X3 )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se5824344872417868541ns_and @ A @ Y3 @ X3 ) @ ( bit_se5824344872417868541ns_and @ A @ Z @ X3 ) ) ) ) ).
% bit.conj_xor_distrib2
thf(fact_4223_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_4224_AND__upper2_H,axiom,
! [Y3: int,Z: int,X3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( ord_less_eq @ int @ Y3 @ Z )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X3 @ Y3 ) @ Z ) ) ) ).
% AND_upper2'
thf(fact_4225_AND__upper1_H,axiom,
! [Y3: int,Z: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( ord_less_eq @ int @ Y3 @ Z )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ Y3 @ Ya ) @ Z ) ) ) ).
% AND_upper1'
thf(fact_4226_AND__upper2,axiom,
! [Y3: int,X3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X3 @ Y3 ) @ Y3 ) ) ).
% AND_upper2
thf(fact_4227_AND__upper1,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X3 @ Y3 ) @ X3 ) ) ).
% AND_upper1
thf(fact_4228_AND__lower,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ X3 @ Y3 ) ) ) ).
% AND_lower
thf(fact_4229_and__eq__not__not__or,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se5824344872417868541ns_and @ A )
= ( ^ [A5: A,B4: A] : ( bit_ri4277139882892585799ns_not @ A @ ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ A5 ) @ ( bit_ri4277139882892585799ns_not @ A @ B4 ) ) ) ) ) ) ).
% and_eq_not_not_or
thf(fact_4230_or__eq__not__not__and,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se1065995026697491101ons_or @ A )
= ( ^ [A5: A,B4: A] : ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ A5 ) @ ( bit_ri4277139882892585799ns_not @ A @ B4 ) ) ) ) ) ) ).
% or_eq_not_not_and
thf(fact_4231_plus__and__or,axiom,
! [X3: int,Y3: int] :
( ( plus_plus @ int @ ( bit_se5824344872417868541ns_and @ int @ X3 @ Y3 ) @ ( bit_se1065995026697491101ons_or @ int @ X3 @ Y3 ) )
= ( plus_plus @ int @ X3 @ Y3 ) ) ).
% plus_and_or
thf(fact_4232_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_4233_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_4234_AND__upper1_H_H,axiom,
! [Y3: int,Z: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( ord_less @ int @ Y3 @ Z )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y3 @ Ya ) @ Z ) ) ) ).
% AND_upper1''
thf(fact_4235_AND__upper2_H_H,axiom,
! [Y3: int,Z: int,X3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( ord_less @ int @ Y3 @ Z )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X3 @ Y3 ) @ Z ) ) ) ).
% AND_upper2''
thf(fact_4236_bit_Oxor__def,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se5824344971392196577ns_xor @ A )
= ( ^ [X: A,Y: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ Y ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ Y ) ) ) ) ) ).
% bit.xor_def
thf(fact_4237_bit_Oxor__def2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se5824344971392196577ns_xor @ A )
= ( ^ [X: A,Y: A] : ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ X @ Y ) @ ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ ( bit_ri4277139882892585799ns_not @ A @ Y ) ) ) ) ) ) ).
% bit.xor_def2
thf(fact_4238_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_4239_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_4240_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_4241_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,X3: A,Y3: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ X3 )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ X3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ Y3 )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ Y3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_4242_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_4243_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_4244_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_4245_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_4246_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_4247_bit_Ocompl__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X3: A,Y3: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ X3 @ Y3 )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ X3 @ Y3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( bit_ri4277139882892585799ns_not @ A @ X3 )
= Y3 ) ) ) ) ).
% bit.compl_unique
thf(fact_4248_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_4249_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_4250_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_4251_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_4252_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_4253_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_4254_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_4255_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_4256_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
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= ( one_one @ complex ) ) )
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= C2 ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_4257_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_4258_and__int_Opelims,axiom,
! [X3: int,Xa: int,Y3: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X3 @ Xa ) )
=> ~ ( ( ( ( ( member @ int @ X3 @ ( 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 ) ) ) ) ) )
=> ( Y3
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X3 @ ( 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 ) ) ) ) ) )
=> ( Y3
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X3 @ ( 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 @ X3 @ Xa ) ) ) ) ) ).
% and_int.pelims
thf(fact_4259_vebt__buildup_Opelims,axiom,
! [X3: nat,Y3: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X3 )
= Y3 )
=> ( ( accp @ nat @ vEBT_v4011308405150292612up_rel @ X3 )
=> ( ( ( X3
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X3
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y3
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va: nat] :
( ( X3
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y3
= ( 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 ) ) )
=> ( Y3
= ( 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_4260_real__root__zero,axiom,
! [N: nat] :
( ( root @ N @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% real_root_zero
thf(fact_4261_real__root__Suc__0,axiom,
! [X3: real] :
( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X3 )
= X3 ) ).
% real_root_Suc_0
thf(fact_4262_real__root__eq__iff,axiom,
! [N: nat,X3: real,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X3 )
= ( root @ N @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% real_root_eq_iff
thf(fact_4263_root__0,axiom,
! [X3: real] :
( ( root @ ( zero_zero @ nat ) @ X3 )
= ( zero_zero @ real ) ) ).
% root_0
thf(fact_4264_real__root__eq__0__iff,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X3 )
= ( zero_zero @ real ) )
= ( X3
= ( zero_zero @ real ) ) ) ) ).
% real_root_eq_0_iff
thf(fact_4265_real__root__less__iff,axiom,
! [N: nat,X3: real,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) )
= ( ord_less @ real @ X3 @ Y3 ) ) ) ).
% real_root_less_iff
thf(fact_4266_real__root__le__iff,axiom,
! [N: nat,X3: real,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) )
= ( ord_less_eq @ real @ X3 @ Y3 ) ) ) ).
% real_root_le_iff
thf(fact_4267_real__root__eq__1__iff,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X3 )
= ( one_one @ real ) )
= ( X3
= ( one_one @ real ) ) ) ) ).
% real_root_eq_1_iff
thf(fact_4268_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_4269_real__root__lt__0__iff,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ) ).
% real_root_lt_0_iff
thf(fact_4270_real__root__gt__0__iff,axiom,
! [N: nat,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ Y3 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y3 ) ) ) ).
% real_root_gt_0_iff
thf(fact_4271_real__root__le__0__iff,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) ) ) ) ).
% real_root_le_0_iff
thf(fact_4272_real__root__ge__0__iff,axiom,
! [N: nat,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ Y3 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 ) ) ) ).
% real_root_ge_0_iff
thf(fact_4273_real__root__lt__1__iff,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X3 ) @ ( one_one @ real ) )
= ( ord_less @ real @ X3 @ ( one_one @ real ) ) ) ) ).
% real_root_lt_1_iff
thf(fact_4274_real__root__gt__1__iff,axiom,
! [N: nat,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( root @ N @ Y3 ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y3 ) ) ) ).
% real_root_gt_1_iff
thf(fact_4275_real__root__le__1__iff,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X3 ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X3 @ ( one_one @ real ) ) ) ) ).
% real_root_le_1_iff
thf(fact_4276_real__root__ge__1__iff,axiom,
! [N: nat,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( root @ N @ Y3 ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y3 ) ) ) ).
% real_root_ge_1_iff
thf(fact_4277_and__nat__numerals_I3_J,axiom,
! [X3: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(3)
thf(fact_4278_and__nat__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(1)
thf(fact_4279_real__root__pow__pos2,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( power_power @ real @ ( root @ N @ X3 ) @ N )
= X3 ) ) ) ).
% real_root_pow_pos2
thf(fact_4280_and__nat__numerals_I4_J,axiom,
! [X3: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(4)
thf(fact_4281_and__nat__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(2)
thf(fact_4282_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_4283_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_4284_real__root__mult,axiom,
! [N: nat,X3: real,Y3: real] :
( ( root @ N @ ( times_times @ real @ X3 @ Y3 ) )
= ( times_times @ real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) ) ) ).
% real_root_mult
thf(fact_4285_real__root__divide,axiom,
! [N: nat,X3: real,Y3: real] :
( ( root @ N @ ( divide_divide @ real @ X3 @ Y3 ) )
= ( divide_divide @ real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) ) ) ).
% real_root_divide
thf(fact_4286_real__root__mult__exp,axiom,
! [M: nat,N: nat,X3: real] :
( ( root @ ( times_times @ nat @ M @ N ) @ X3 )
= ( root @ M @ ( root @ N @ X3 ) ) ) ).
% real_root_mult_exp
thf(fact_4287_real__root__minus,axiom,
! [N: nat,X3: real] :
( ( root @ N @ ( uminus_uminus @ real @ X3 ) )
= ( uminus_uminus @ real @ ( root @ N @ X3 ) ) ) ).
% real_root_minus
thf(fact_4288_real__root__commute,axiom,
! [M: nat,N: nat,X3: real] :
( ( root @ M @ ( root @ N @ X3 ) )
= ( root @ N @ ( root @ M @ X3 ) ) ) ).
% real_root_commute
thf(fact_4289_real__root__inverse,axiom,
! [N: nat,X3: real] :
( ( root @ N @ ( inverse_inverse @ real @ X3 ) )
= ( inverse_inverse @ real @ ( root @ N @ X3 ) ) ) ).
% real_root_inverse
thf(fact_4290_real__root__pos__pos__le,axiom,
! [X3: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ X3 ) ) ) ).
% real_root_pos_pos_le
thf(fact_4291_real__root__less__mono,axiom,
! [N: nat,X3: real,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X3 @ Y3 )
=> ( ord_less @ real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) ) ) ) ).
% real_root_less_mono
thf(fact_4292_real__root__le__mono,axiom,
! [N: nat,X3: real,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ X3 @ Y3 )
=> ( ord_less_eq @ real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) ) ) ) ).
% real_root_le_mono
thf(fact_4293_real__root__power,axiom,
! [N: nat,X3: real,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X3 @ K ) )
= ( power_power @ real @ ( root @ N @ X3 ) @ K ) ) ) ).
% real_root_power
thf(fact_4294_real__root__abs,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( abs_abs @ real @ X3 ) )
= ( abs_abs @ real @ ( root @ N @ X3 ) ) ) ) ).
% real_root_abs
thf(fact_4295_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_4296_real__root__gt__zero,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ X3 ) ) ) ) ).
% real_root_gt_zero
thf(fact_4297_real__root__strict__decreasing,axiom,
! [N: nat,N4: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( ord_less @ real @ ( root @ N4 @ X3 ) @ ( root @ N @ X3 ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_4298_sqrt__def,axiom,
( sqrt
= ( root @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% sqrt_def
thf(fact_4299_root__abs__power,axiom,
! [N: nat,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( abs_abs @ real @ ( root @ N @ ( power_power @ real @ Y3 @ N ) ) )
= ( abs_abs @ real @ Y3 ) ) ) ).
% root_abs_power
thf(fact_4300_real__root__pos__pos,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ X3 ) ) ) ) ).
% real_root_pos_pos
thf(fact_4301_real__root__strict__increasing,axiom,
! [N: nat,N4: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( ord_less @ real @ ( root @ N @ X3 ) @ ( root @ N4 @ X3 ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_4302_real__root__decreasing,axiom,
! [N: nat,N4: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X3 )
=> ( ord_less_eq @ real @ ( root @ N4 @ X3 ) @ ( root @ N @ X3 ) ) ) ) ) ).
% real_root_decreasing
thf(fact_4303_real__root__pow__pos,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( power_power @ real @ ( root @ N @ X3 ) @ N )
= X3 ) ) ) ).
% real_root_pow_pos
thf(fact_4304_odd__real__root__pow,axiom,
! [N: nat,X3: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ real @ ( root @ N @ X3 ) @ N )
= X3 ) ) ).
% odd_real_root_pow
thf(fact_4305_odd__real__root__unique,axiom,
! [N: nat,Y3: real,X3: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ( power_power @ real @ Y3 @ N )
= X3 )
=> ( ( root @ N @ X3 )
= Y3 ) ) ) ).
% odd_real_root_unique
thf(fact_4306_odd__real__root__power__cancel,axiom,
! [N: nat,X3: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X3 @ N ) )
= X3 ) ) ).
% odd_real_root_power_cancel
thf(fact_4307_real__root__power__cancel,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( root @ N @ ( power_power @ real @ X3 @ N ) )
= X3 ) ) ) ).
% real_root_power_cancel
thf(fact_4308_real__root__pos__unique,axiom,
! [N: nat,Y3: real,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
=> ( ( ( power_power @ real @ Y3 @ N )
= X3 )
=> ( ( root @ N @ X3 )
= Y3 ) ) ) ) ).
% real_root_pos_unique
thf(fact_4309_real__root__increasing,axiom,
! [N: nat,N4: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( root @ N @ X3 ) @ ( root @ N4 @ X3 ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_4310_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_4311_log__base__root,axiom,
! [N: nat,B2: real,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( log @ ( root @ N @ B2 ) @ X3 )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ X3 ) ) ) ) ) ).
% log_base_root
thf(fact_4312_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_4313_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_4314_root__powr__inverse,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( root @ N @ X3 )
= ( powr @ real @ X3 @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_4315_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_4316_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [K2: int,L4: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L4 ) )
=> ( ( ~ ( ( member @ int @ K2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L4 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( P @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K2 @ L4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_4317_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [I3: int,J2: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I3 @ J2 ) )
=> ( ( ( ord_less_eq @ int @ I3 @ J2 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) )
=> ( P @ I3 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_4318_option_Osize__gen_I2_J,axiom,
! [A: $tType,X3: A > nat,X2: A] :
( ( size_option @ A @ X3 @ ( some @ A @ X2 ) )
= ( plus_plus @ nat @ ( X3 @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% option.size_gen(2)
thf(fact_4319_set__decode__0,axiom,
! [X3: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ ( nat_set_decode @ X3 ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X3 ) ) ) ).
% set_decode_0
thf(fact_4320_set__decode__Suc,axiom,
! [N: nat,X3: nat] :
( ( member @ nat @ ( suc @ N ) @ ( nat_set_decode @ X3 ) )
= ( member @ nat @ N @ ( nat_set_decode @ ( divide_divide @ nat @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_4321_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_4322_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_4323_option_Osize__gen_I1_J,axiom,
! [A: $tType,X3: A > nat] :
( ( size_option @ A @ X3 @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size_gen(1)
thf(fact_4324_set__decode__plus__power__2,axiom,
! [N: nat,Z: nat] :
( ~ ( member @ nat @ N @ ( nat_set_decode @ Z ) )
=> ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Z ) )
= ( insert @ nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_4325_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X: nat] :
( collect @ nat
@ ^ [N3: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_4326_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_4327_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R2: A,A2: A,B2: A,C2: A,D3: A] :
( ( R2
!= ( zero_zero @ A ) )
=> ( ( ( A2 = B2 )
& ( C2 != D3 ) )
=> ( ( plus_plus @ A @ A2 @ ( times_times @ A @ R2 @ C2 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R2 @ D3 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4328_arctan__def,axiom,
( arctan
= ( ^ [Y: real] :
( the @ real
@ ^ [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
& ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X )
= Y ) ) ) ) ) ).
% arctan_def
thf(fact_4329_arcsin__def,axiom,
( arcsin
= ( ^ [Y: real] :
( the @ real
@ ^ [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 ) ) ) )
& ( ( sin @ real @ X )
= Y ) ) ) ) ) ).
% arcsin_def
thf(fact_4330_ln__neg__is__const,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) )
=> ( ( ln_ln @ real @ X3 )
= ( the @ real
@ ^ [X: real] : $false ) ) ) ).
% ln_neg_is_const
thf(fact_4331_arccos__def,axiom,
( arccos
= ( ^ [Y: real] :
( the @ real
@ ^ [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
& ( ord_less_eq @ real @ X @ pi )
& ( ( cos @ real @ X )
= Y ) ) ) ) ) ).
% arccos_def
thf(fact_4332_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_4333_pi__half,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
= ( the @ real
@ ^ [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
& ( ord_less_eq @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X )
= ( zero_zero @ real ) ) ) ) ) ).
% pi_half
thf(fact_4334_pi__def,axiom,
( pi
= ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) )
@ ( the @ real
@ ^ [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
& ( ord_less_eq @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X )
= ( zero_zero @ real ) ) ) ) ) ) ).
% pi_def
thf(fact_4335_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_4336_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_4337_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_4338_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_4339_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_4340_sgn__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_0
thf(fact_4341_sgn__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_zero
thf(fact_4342_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_4343_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_4344_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_4345_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_4346_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_4347_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_4348_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_4349_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_4350_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_4351_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_4352_dvd__mult__sgn__iff,axiom,
! [L: int,K: int,R2: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ K @ ( sgn_sgn @ int @ R2 ) ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( R2
= ( zero_zero @ int ) ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_4353_dvd__sgn__mult__iff,axiom,
! [L: int,R2: int,K: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ ( sgn_sgn @ int @ R2 ) @ K ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( R2
= ( zero_zero @ int ) ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_4354_mult__sgn__dvd__iff,axiom,
! [L: int,R2: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ L @ ( sgn_sgn @ int @ R2 ) ) @ K )
= ( ( dvd_dvd @ int @ L @ K )
& ( ( R2
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_4355_sgn__mult__dvd__iff,axiom,
! [R2: int,L: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ ( sgn_sgn @ int @ R2 ) @ L ) @ K )
= ( ( dvd_dvd @ int @ L @ K )
& ( ( R2
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_4356_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_4357_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_4358_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_4359_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_4360_bit__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ W ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_numeral_simps(2)
thf(fact_4361_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ N ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_4362_bit__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ W ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_numeral_simps(3)
thf(fact_4363_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ N ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_4364_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_4365_bit__minus__numeral__int_I1_J,axiom,
! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% bit_minus_numeral_int(1)
thf(fact_4366_bit__minus__numeral__int_I2_J,axiom,
! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) @ ( numeral_numeral @ nat @ N ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_minus_numeral_int(2)
thf(fact_4367_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_4368_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_4369_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_4370_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_4371_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_4372_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_4373_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_4374_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_4375_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_4376_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_4377_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_4378_sgn__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A] :
( ( ( sgn_sgn @ A @ X3 )
= ( zero_zero @ A ) )
= ( X3
= ( zero_zero @ A ) ) ) ) ).
% sgn_zero_iff
thf(fact_4379_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_4380_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_4381_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_4382_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_4383_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_4384_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_4385_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_4386_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_4387_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_4388_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_4389_int__sgnE,axiom,
! [K: int] :
~ ! [N2: nat,L4: int] :
( K
!= ( times_times @ int @ ( sgn_sgn @ int @ L4 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% int_sgnE
thf(fact_4390_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_4391_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_4392_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_4393_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_4394_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_4395_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_4396_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_4397_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_4398_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X: A] :
( if @ A
@ ( X
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_4399_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_4400_norm__sgn,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A] :
( ( ( X3
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X3 ) )
= ( zero_zero @ real ) ) )
& ( ( X3
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X3 ) )
= ( one_one @ real ) ) ) ) ) ).
% norm_sgn
thf(fact_4401_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_4402_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_4403_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_4404_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_4405_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_4406_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_4407_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_4408_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_4409_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_4410_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_4411_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_4412_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_4413_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_4414_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_4415_eucl__rel__int__remainderI,axiom,
! [R2: int,L: int,K: int,Q3: int] :
( ( ( sgn_sgn @ int @ R2 )
= ( sgn_sgn @ int @ L ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R2 ) @ ( abs_abs @ int @ L ) )
=> ( ( K
= ( plus_plus @ int @ ( times_times @ int @ Q3 @ L ) @ R2 ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R2 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_4416_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_4417_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_4418_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A12: int,A23: int,A32: product_prod @ int @ int] :
( ? [K3: int] :
( ( A12 = K3 )
& ( A23
= ( zero_zero @ int ) )
& ( A32
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) )
| ? [L2: int,K3: int,Q5: int] :
( ( A12 = K3 )
& ( A23 = L2 )
& ( A32
= ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) )
& ( L2
!= ( zero_zero @ int ) )
& ( K3
= ( times_times @ int @ Q5 @ L2 ) ) )
| ? [R5: int,L2: int,K3: int,Q5: int] :
( ( A12 = K3 )
& ( A23 = L2 )
& ( A32
= ( product_Pair @ int @ int @ Q5 @ R5 ) )
& ( ( sgn_sgn @ int @ R5 )
= ( sgn_sgn @ int @ L2 ) )
& ( ord_less @ int @ ( abs_abs @ int @ R5 ) @ ( abs_abs @ int @ L2 ) )
& ( K3
= ( plus_plus @ int @ ( times_times @ int @ Q5 @ L2 ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_4419_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A33: product_prod @ int @ int] :
( ( eucl_rel_int @ A1 @ A22 @ A33 )
=> ( ( ( A22
= ( zero_zero @ int ) )
=> ( A33
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A1 ) ) )
=> ( ! [Q4: int] :
( ( A33
= ( product_Pair @ int @ int @ Q4 @ ( zero_zero @ int ) ) )
=> ( ( A22
!= ( zero_zero @ int ) )
=> ( A1
!= ( times_times @ int @ Q4 @ A22 ) ) ) )
=> ~ ! [R: int,Q4: int] :
( ( A33
= ( product_Pair @ int @ int @ Q4 @ R ) )
=> ( ( ( sgn_sgn @ int @ R )
= ( sgn_sgn @ int @ A22 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R ) @ ( abs_abs @ int @ A22 ) )
=> ( A1
!= ( plus_plus @ int @ ( times_times @ int @ Q4 @ A22 ) @ R ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_4420_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_4421_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_4422_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_4423_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_4424_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_4425_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_4426_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_4427_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_4428_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_4429_num_Osize__gen_I2_J,axiom,
! [X2: num] :
( ( size_num @ ( bit0 @ X2 ) )
= ( plus_plus @ nat @ ( size_num @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(2)
thf(fact_4430_arctan__inverse,axiom,
! [X3: real] :
( ( X3
!= ( zero_zero @ real ) )
=> ( ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ X3 ) )
= ( minus_minus @ real @ ( divide_divide @ real @ ( times_times @ real @ ( sgn_sgn @ real @ X3 ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( arctan @ X3 ) ) ) ) ).
% arctan_inverse
thf(fact_4431_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_4432_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_4433_mask__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% mask_0
thf(fact_4434_sgn__le__0__iff,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( sgn_sgn @ real @ X3 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) ) ) ).
% sgn_le_0_iff
thf(fact_4435_zero__le__sgn__iff,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sgn_sgn @ real @ X3 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) ) ).
% zero_le_sgn_iff
thf(fact_4436_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_4437_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_4438_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_4439_nat__mask__eq,axiom,
! [N: nat] :
( ( nat2 @ ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% nat_mask_eq
thf(fact_4440_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% less_eq_mask
thf(fact_4441_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_4442_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_4443_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_4444_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2239418461657761734s_mask @ int @ N ) ) ).
% mask_nonnegative_int
thf(fact_4445_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( zero_zero @ int ) ) ).
% not_mask_negative_int
thf(fact_4446_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_4447_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_4448_sgn__root,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( sgn_sgn @ real @ ( root @ N @ X3 ) )
= ( sgn_sgn @ real @ X3 ) ) ) ).
% sgn_root
thf(fact_4449_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_4450_cis__Arg,axiom,
! [Z: complex] :
( ( Z
!= ( zero_zero @ complex ) )
=> ( ( cis @ ( arg @ Z ) )
= ( sgn_sgn @ complex @ Z ) ) ) ).
% cis_Arg
thf(fact_4451_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_4452_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_4453_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_4454_sgn__power__injE,axiom,
! [A2: real,N: nat,X3: real,B2: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ A2 ) @ ( power_power @ real @ ( abs_abs @ real @ A2 ) @ N ) )
= X3 )
=> ( ( X3
= ( times_times @ real @ ( sgn_sgn @ real @ B2 ) @ ( power_power @ real @ ( abs_abs @ real @ B2 ) @ N ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A2 = B2 ) ) ) ) ).
% sgn_power_injE
thf(fact_4455_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_4456_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_4457_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_4458_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_4459_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_4460_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_4461_sgn__power__root,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ real @ ( sgn_sgn @ real @ ( root @ N @ X3 ) ) @ ( power_power @ real @ ( abs_abs @ real @ ( root @ N @ X3 ) ) @ N ) )
= X3 ) ) ).
% sgn_power_root
thf(fact_4462_root__sgn__power,axiom,
! [N: nat,Y3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( times_times @ real @ ( sgn_sgn @ real @ Y3 ) @ ( power_power @ real @ ( abs_abs @ real @ Y3 ) @ N ) ) )
= Y3 ) ) ).
% root_sgn_power
thf(fact_4463_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_4464_cis__Arg__unique,axiom,
! [Z: complex,X3: real] :
( ( ( sgn_sgn @ complex @ Z )
= ( cis @ X3 ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ pi )
=> ( ( arg @ Z )
= X3 ) ) ) ) ).
% cis_Arg_unique
thf(fact_4465_split__root,axiom,
! [P: real > $o,N: nat,X3: real] :
( ( P @ ( root @ N @ X3 ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ real ) ) )
& ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ! [Y: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N ) )
= X3 )
=> ( P @ Y ) ) ) ) ) ).
% split_root
thf(fact_4466_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_4467_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_4468_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_4469_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_4470_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_4471_Arg__correct,axiom,
! [Z: complex] :
( ( Z
!= ( zero_zero @ complex ) )
=> ( ( ( sgn_sgn @ complex @ Z )
= ( cis @ ( arg @ Z ) ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ ( arg @ Z ) )
& ( ord_less_eq @ real @ ( arg @ Z ) @ pi ) ) ) ).
% Arg_correct
thf(fact_4472_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_4473_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_4474_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_4475_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_4476_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_4477_Arg__def,axiom,
( arg
= ( ^ [Z6: complex] :
( if @ real
@ ( Z6
= ( zero_zero @ complex ) )
@ ( zero_zero @ real )
@ ( fChoice @ real
@ ^ [A5: real] :
( ( ( sgn_sgn @ complex @ Z6 )
= ( cis @ A5 ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ A5 )
& ( ord_less_eq @ real @ A5 @ pi ) ) ) ) ) ) ).
% Arg_def
thf(fact_4478_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_4479_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_4480_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_4481_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_4482_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_4483_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_4484_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_4485_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_4486_Ints__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ B )
=> ! [A4: set @ A,F2: A > B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( member @ B @ ( F2 @ X4 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A4 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_sum
thf(fact_4487_Ints__prod,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ B )
& ( ring_1 @ B ) )
=> ! [A4: set @ A,F2: A > B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( member @ B @ ( F2 @ X4 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A4 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_prod
thf(fact_4488_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_4489_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_4490_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_4491_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_4492_frac__in__Ints__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( member @ A @ ( archimedean_frac @ A @ X3 ) @ ( ring_1_Ints @ A ) )
= ( member @ A @ X3 @ ( ring_1_Ints @ A ) ) ) ) ).
% frac_in_Ints_iff
thf(fact_4493_frac__eq__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ( archimedean_frac @ A @ X3 )
= ( zero_zero @ A ) )
= ( member @ A @ X3 @ ( ring_1_Ints @ A ) ) ) ) ).
% frac_eq_0_iff
thf(fact_4494_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_4495_floor__add2,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y3: A] :
( ( ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
| ( member @ A @ Y3 @ ( ring_1_Ints @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y3 ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) ) ) ) ).
% floor_add2
thf(fact_4496_frac__gt__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X3 ) )
= ( ~ ( member @ A @ X3 @ ( ring_1_Ints @ A ) ) ) ) ) ).
% frac_gt_0_iff
thf(fact_4497_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_4498_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_4499_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_4500_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_4501_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_4502_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_4503_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_4504_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_4505_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_4506_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_4507_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_4508_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_4509_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_4510_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_4511_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_4512_verit__sko__ex_H,axiom,
! [A: $tType,P: A > $o,A4: $o] :
( ( ( P @ ( fChoice @ A @ P ) )
= A4 )
=> ( ( ? [X9: A] : ( P @ X9 ) )
= A4 ) ) ).
% verit_sko_ex'
thf(fact_4513_verit__sko__forall,axiom,
! [A: $tType] :
( ( ^ [P2: A > $o] :
! [X5: A] : ( P2 @ X5 ) )
= ( ^ [P3: A > $o] :
( P3
@ ( fChoice @ A
@ ^ [X: A] :
~ ( P3 @ X ) ) ) ) ) ).
% verit_sko_forall
thf(fact_4514_verit__sko__forall_H,axiom,
! [A: $tType,P: A > $o,A4: $o] :
( ( ( P
@ ( fChoice @ A
@ ^ [X: A] :
~ ( P @ X ) ) )
= A4 )
=> ( ( ! [X9: A] : ( P @ X9 ) )
= A4 ) ) ).
% verit_sko_forall'
thf(fact_4515_verit__sko__forall_H_H,axiom,
! [A: $tType,B6: A,A4: A,P: A > $o] :
( ( B6 = A4 )
=> ( ( ( fChoice @ A @ P )
= A4 )
= ( ( fChoice @ A @ P )
= B6 ) ) ) ).
% verit_sko_forall''
thf(fact_4516_verit__sko__ex__indirect,axiom,
! [A: $tType,X3: A,P: A > $o] :
( ( X3
= ( fChoice @ A @ P ) )
=> ( ( ? [X9: A] : ( P @ X9 ) )
= ( P @ X3 ) ) ) ).
% verit_sko_ex_indirect
thf(fact_4517_verit__sko__ex__indirect2,axiom,
! [A: $tType,X3: A,P: A > $o,P4: A > $o] :
( ( X3
= ( fChoice @ A @ P ) )
=> ( ! [X4: A] :
( ( P @ X4 )
= ( P4 @ X4 ) )
=> ( ( ? [X9: A] : ( P4 @ X9 ) )
= ( P @ X3 ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_4518_verit__sko__forall__indirect,axiom,
! [A: $tType,X3: A,P: A > $o] :
( ( X3
= ( fChoice @ A
@ ^ [X: A] :
~ ( P @ X ) ) )
=> ( ( ! [X9: A] : ( P @ X9 ) )
= ( P @ X3 ) ) ) ).
% verit_sko_forall_indirect
thf(fact_4519_verit__sko__forall__indirect2,axiom,
! [A: $tType,X3: A,P: A > $o,P4: A > $o] :
( ( X3
= ( fChoice @ A
@ ^ [X: A] :
~ ( P @ X ) ) )
=> ( ! [X4: A] :
( ( P @ X4 )
= ( P4 @ X4 ) )
=> ( ( ! [X9: A] : ( P4 @ X9 ) )
= ( P @ X3 ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_4520_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_4521_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_4522_Ints__cases,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q3: A] :
( ( member @ A @ Q3 @ ( ring_1_Ints @ A ) )
=> ~ ! [Z3: int] :
( Q3
!= ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% Ints_cases
thf(fact_4523_Ints__induct,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q3: A,P: A > $o] :
( ( member @ A @ Q3 @ ( ring_1_Ints @ A ) )
=> ( ! [Z3: int] : ( P @ ( ring_1_of_int @ A @ Z3 ) )
=> ( P @ Q3 ) ) ) ) ).
% Ints_induct
thf(fact_4524_Ints__of__int,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int] : ( member @ A @ ( ring_1_of_int @ A @ Z ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_of_int
thf(fact_4525_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_4526_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_4527_Ints__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: num] : ( member @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_numeral
thf(fact_4528_Ints__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_0
thf(fact_4529_minus__in__Ints__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: A] :
( ( member @ A @ ( uminus_uminus @ A @ X3 ) @ ( ring_1_Ints @ A ) )
= ( member @ A @ X3 @ ( ring_1_Ints @ A ) ) ) ) ).
% minus_in_Ints_iff
thf(fact_4530_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_4531_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_4532_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_4533_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_4534_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_4535_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_4536_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_4537_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_4538_finite__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( finite_finite @ A
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ A2 @ X )
& ( ord_less_eq @ A @ X @ B2 ) ) ) ) ) ).
% finite_int_segment
thf(fact_4539_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_4540_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_4541_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_4542_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_4543_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_4544_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_4545_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_4546_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_4547_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_4548_finite__abs__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A] :
( finite_finite @ 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_4549_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_4550_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_4551_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_4552_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
=> ( ( X3
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X3 ) ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_4553_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( abs_abs @ A @ X3 ) @ ( one_one @ A ) )
=> ( X3
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_4554_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ Y3 @ ( ring_1_Ints @ A ) )
=> ( ( X3 = Y3 )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ Y3 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_4555_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_4556_sin__times__pi__eq__0,axiom,
! [X3: real] :
( ( ( sin @ real @ ( times_times @ real @ X3 @ pi ) )
= ( zero_zero @ real ) )
= ( member @ real @ X3 @ ( ring_1_Ints @ real ) ) ) ).
% sin_times_pi_eq_0
thf(fact_4557_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_4558_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_4559_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_4560_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_4561_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_4562_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_4563_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 )
=> ~ ! [B5: A] :
( A2
!= ( bit_se4730199178511100633sh_bit @ A @ N @ B5 ) ) ) ) ).
% exp_dvdE
thf(fact_4564_frac__neg,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X3 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X3 ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X3 ) ) ) ) ) ) ).
% frac_neg
thf(fact_4565_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_4566_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,A2: A] :
( ( ( archimedean_frac @ A @ X3 )
= A2 )
= ( ( member @ A @ ( minus_minus @ A @ X3 @ A2 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ A2 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_4567_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_4568_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_4569_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_4570_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_4571_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_4572_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_4573_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_finite @ 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_4574_int__of__nat__def,axiom,
( code_T6385005292777649522of_nat
= ( semiring_1_of_nat @ int ) ) ).
% int_of_nat_def
thf(fact_4575_Cauchy__iff2,axiom,
( ( topolo3814608138187158403Cauchy @ real )
= ( ^ [X9: 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 @ ( X9 @ M3 ) @ ( X9 @ N3 ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_4576_Sum__Ico__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ ( 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_4577_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% atLeastLessThan_iff
thf(fact_4578_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_4579_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_4580_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_4581_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_4582_infinite__Ico__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ico_iff
thf(fact_4583_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_4584_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P6: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P6 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_4585_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,P6: B > A,I: B] :
( ( finite_finite @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I6 )
& ( ( P6 @ X )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P6 @ ( insert @ B @ I @ I6 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P6 @ I6 ) ) )
& ( ~ ( member @ B @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P6 @ ( insert @ B @ I @ I6 ) )
= ( plus_plus @ A @ ( P6 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P6 @ I6 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_4586_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_4587_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_4588_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( B2 = D3 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_4589_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( A2 = C2 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_4590_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( ( A2 = C2 )
& ( B2 = D3 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_4591_sum_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,I6: set @ B] :
( ( groups1027152243600224163dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I6 )
& ( ( G @ X )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) ) ) ).
% sum.non_neutral'
thf(fact_4592_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_4593_infinite__Ico,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ico
thf(fact_4594_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N )
& ( P @ M3 ) ) )
= ( ? [X: nat] :
( ( member @ nat @ X @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X ) ) ) ) ).
% ex_nat_less_eq
thf(fact_4595_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N )
=> ( P @ M3 ) ) )
= ( ! [X: nat] :
( ( member @ nat @ X @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X ) ) ) ) ).
% all_nat_less_eq
thf(fact_4596_lessThan__atLeast0,axiom,
( ( set_ord_lessThan @ nat )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% lessThan_atLeast0
thf(fact_4597_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_4598_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A2: B,C2: B,B2: B,D3: B,G: B > A,H: B > A] :
( ( A2 = C2 )
=> ( ( B2 = D3 )
=> ( ! [X4: B] :
( ( ord_less_eq @ B @ C2 @ X4 )
=> ( ( ord_less @ B @ X4 @ D3 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ ( set_or7035219750837199246ssThan @ B @ C2 @ D3 ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_4599_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A2: B,C2: B,B2: B,D3: B,G: B > A,H: B > A] :
( ( A2 = C2 )
=> ( ( B2 = D3 )
=> ( ! [X4: B] :
( ( ord_less_eq @ B @ C2 @ X4 )
=> ( ( ord_less @ B @ X4 @ D3 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H @ ( set_or7035219750837199246ssThan @ B @ C2 @ D3 ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_4600_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,P6: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P6 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P6 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P6 ) ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_4601_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,P6: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P6 )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M @ P6 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ N @ P6 ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_4602_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,P6: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P6 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P6 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P6 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_4603_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_4604_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_finite @ nat @ N4 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_4605_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T6: set @ B,G: B > A,H: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 )
= ( groups1027152243600224163dd_sum @ B @ A @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_4606_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T6: set @ B,H: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( H @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S2 )
= ( groups1027152243600224163dd_sum @ B @ A @ H @ T6 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_4607_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_4608_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S2 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_4609_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4610_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_4611_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_4612_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_4613_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_4614_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_4615_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_4616_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G: B > A,H: B > A] :
( ( finite_finite @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I6 )
& ( ( G @ X )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I6 )
& ( ( H @ X )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I4: B] : ( plus_plus @ A @ ( G @ I4 ) @ ( H @ I4 ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H @ I6 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_4617_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_4618_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_4619_sum_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ B @ A )
= ( ^ [P5: B > A,I7: set @ B] :
( if @ A
@ ( finite_finite @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I7 )
& ( ( P5 @ X )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P5
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I7 )
& ( ( P5 @ X )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_4620_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_4621_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_4622_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_4623_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_4624_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_4625_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_4626_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_4627_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_4628_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_4629_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_4630_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_4631_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_4632_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_4633_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_4634_summable__Cauchy,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ( ( summable @ A )
= ( ^ [F3: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [N6: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ N6 @ 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_4635_CauchyD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,E2: real] :
( ( topolo3814608138187158403Cauchy @ A @ X8 )
=> ( ( 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 @ ( X8 @ M2 ) @ ( X8 @ N9 ) ) ) @ E2 ) ) ) ) ) ) ).
% CauchyD
thf(fact_4636_CauchyI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: 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 @ ( X8 @ M4 ) @ ( X8 @ N2 ) ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ).
% CauchyI
thf(fact_4637_Cauchy__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X9: 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 @ ( X9 @ M3 ) @ ( X9 @ N3 ) ) ) @ E3 ) ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_4638_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_4639_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_4640_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_4641_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_4642_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_4643_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_4644_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_4645_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_4646_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_4647_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I6: set @ A,F2: A > B,I: A] :
( ( finite_finite @ 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_4648_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_4649_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F3: B > A,A5: A,Xs: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ ( nth @ B @ Xs @ N3 ) ) @ ( power_power @ A @ A5 @ N3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_4650_Chebyshev__sum__upper,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: nat > A,B2: nat > A] :
( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( A2 @ I3 ) @ ( A2 @ J2 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( B2 @ J2 ) @ ( B2 @ I3 ) ) ) )
=> ( ord_less_eq @ A
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( 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_4651_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A2: nat > nat,B2: nat > nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( A2 @ I3 ) @ ( A2 @ J2 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( B2 @ J2 ) @ ( B2 @ I3 ) ) ) )
=> ( ord_less_eq @ nat
@ ( times_times @ nat @ N
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ ( 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_4652_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_4653_VEBT_Osize_I3_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] :
( ( size_size @ vEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_list @ vEBT_VEBT @ ( size_size @ vEBT_VEBT ) @ X13 ) @ ( size_size @ vEBT_VEBT @ X14 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% VEBT.size(3)
thf(fact_4654_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_4655_valid__eq1,axiom,
! [T2: vEBT_VEBT,D3: nat] :
( ( vEBT_invar_vebt @ T2 @ D3 )
=> ( vEBT_VEBT_valid @ T2 @ D3 ) ) ).
% valid_eq1
thf(fact_4656_valid__eq2,axiom,
! [T2: vEBT_VEBT,D3: nat] :
( ( vEBT_VEBT_valid @ T2 @ D3 )
=> ( vEBT_invar_vebt @ T2 @ D3 ) ) ).
% valid_eq2
thf(fact_4657_size__list__estimation,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Y3: nat,F2: A > nat] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( ord_less @ nat @ Y3 @ ( F2 @ X3 ) )
=> ( ord_less @ nat @ Y3 @ ( size_list @ A @ F2 @ Xs2 ) ) ) ) ).
% size_list_estimation
thf(fact_4658_size__list__pointwise,axiom,
! [A: $tType,Xs2: list @ A,F2: A > nat,G: A > nat] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ nat @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq @ nat @ ( size_list @ A @ F2 @ Xs2 ) @ ( size_list @ A @ G @ Xs2 ) ) ) ).
% size_list_pointwise
thf(fact_4659_size__list__estimation_H,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Y3: nat,F2: A > nat] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( ord_less_eq @ nat @ Y3 @ ( F2 @ X3 ) )
=> ( ord_less_eq @ nat @ Y3 @ ( size_list @ A @ F2 @ Xs2 ) ) ) ) ).
% size_list_estimation'
thf(fact_4660_VEBT_Osize__gen_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] :
( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_list @ vEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% VEBT.size_gen(1)
thf(fact_4661_length__subseqs,axiom,
! [A: $tType,Xs2: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( subseqs @ A @ Xs2 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_subseqs
thf(fact_4662_Code__Target__Int_Opositive__def,axiom,
( code_Target_positive
= ( numeral_numeral @ int ) ) ).
% Code_Target_Int.positive_def
thf(fact_4663_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_4664_length__mul__elem,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),N: nat] :
( ! [X4: list @ A] :
( ( member @ ( list @ A ) @ X4 @ ( set2 @ ( list @ A ) @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ X4 )
= N ) )
=> ( ( size_size @ ( list @ A ) @ ( concat @ A @ Xs2 ) )
= ( times_times @ nat @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) @ N ) ) ) ).
% length_mul_elem
thf(fact_4665_csqrt_Osimps_I1_J,axiom,
! [Z: complex] :
( ( re @ ( csqrt @ Z ) )
= ( sqrt @ ( divide_divide @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% csqrt.simps(1)
thf(fact_4666_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L2: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q5: code_integer,R5: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L2 ) @ R5 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R5 @ ( numeral_numeral @ code_integer @ L2 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_4667_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_4668_complex__Re__numeral,axiom,
! [V2: num] :
( ( re @ ( numeral_numeral @ complex @ V2 ) )
= ( numeral_numeral @ real @ V2 ) ) ).
% complex_Re_numeral
thf(fact_4669_Re__divide__of__nat,axiom,
! [Z: complex,N: nat] :
( ( re @ ( divide_divide @ complex @ Z @ ( semiring_1_of_nat @ complex @ N ) ) )
= ( divide_divide @ real @ ( re @ Z ) @ ( semiring_1_of_nat @ real @ N ) ) ) ).
% Re_divide_of_nat
thf(fact_4670_Re__divide__numeral,axiom,
! [Z: complex,W: num] :
( ( re @ ( divide_divide @ complex @ Z @ ( numeral_numeral @ complex @ W ) ) )
= ( divide_divide @ real @ ( re @ Z ) @ ( numeral_numeral @ real @ W ) ) ) ).
% Re_divide_numeral
thf(fact_4671_cos__Arg__i__mult__zero,axiom,
! [Y3: complex] :
( ( Y3
!= ( zero_zero @ complex ) )
=> ( ( ( re @ Y3 )
= ( zero_zero @ real ) )
=> ( ( cos @ real @ ( arg @ Y3 ) )
= ( zero_zero @ real ) ) ) ) ).
% cos_Arg_i_mult_zero
thf(fact_4672_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_4673_times__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( times_times @ code_integer @ ( zero_zero @ code_integer ) @ L )
= ( zero_zero @ code_integer ) ) ).
% times_integer_code(2)
thf(fact_4674_times__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( times_times @ code_integer @ K @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% times_integer_code(1)
thf(fact_4675_minus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( minus_minus @ code_integer @ ( zero_zero @ code_integer ) @ L )
= ( uminus_uminus @ code_integer @ L ) ) ).
% minus_integer_code(2)
thf(fact_4676_minus__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( minus_minus @ code_integer @ K @ ( zero_zero @ code_integer ) )
= K ) ).
% minus_integer_code(1)
thf(fact_4677_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_4678_plus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( plus_plus @ code_integer @ ( zero_zero @ code_integer ) @ L )
= L ) ).
% plus_integer_code(2)
thf(fact_4679_plus__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( plus_plus @ code_integer @ K @ ( zero_zero @ code_integer ) )
= K ) ).
% plus_integer_code(1)
thf(fact_4680_imaginary__unit_Osimps_I1_J,axiom,
( ( re @ imaginary_unit )
= ( zero_zero @ real ) ) ).
% imaginary_unit.simps(1)
thf(fact_4681_complex__Re__le__cmod,axiom,
! [X3: complex] : ( ord_less_eq @ real @ ( re @ X3 ) @ ( real_V7770717601297561774m_norm @ complex @ X3 ) ) ).
% complex_Re_le_cmod
thf(fact_4682_zero__complex_Osimps_I1_J,axiom,
( ( re @ ( zero_zero @ complex ) )
= ( zero_zero @ real ) ) ).
% zero_complex.simps(1)
thf(fact_4683_abs__Re__le__cmod,axiom,
! [X3: complex] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( re @ X3 ) ) @ ( real_V7770717601297561774m_norm @ complex @ X3 ) ) ).
% abs_Re_le_cmod
thf(fact_4684_Re__csqrt,axiom,
! [Z: complex] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( csqrt @ Z ) ) ) ).
% Re_csqrt
thf(fact_4685_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_4686_zero__integer_Orsp,axiom,
( ( zero_zero @ int )
= ( zero_zero @ int ) ) ).
% zero_integer.rsp
thf(fact_4687_cmod__plus__Re__le__0__iff,axiom,
! [Z: complex] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( re @ Z ) ) @ ( zero_zero @ real ) )
= ( ( re @ Z )
= ( uminus_uminus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) ) ) ) ).
% cmod_plus_Re_le_0_iff
thf(fact_4688_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_4689_csqrt_Ocode,axiom,
( csqrt
= ( ^ [Z6: complex] :
( complex2 @ ( sqrt @ ( divide_divide @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z6 ) @ ( re @ Z6 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( times_times @ real
@ ( if @ real
@ ( ( im @ Z6 )
= ( zero_zero @ real ) )
@ ( one_one @ real )
@ ( sgn_sgn @ real @ ( im @ Z6 ) ) )
@ ( sqrt @ ( divide_divide @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z6 ) @ ( re @ Z6 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% csqrt.code
thf(fact_4690_csqrt_Osimps_I2_J,axiom,
! [Z: complex] :
( ( im @ ( csqrt @ Z ) )
= ( times_times @ real
@ ( if @ real
@ ( ( im @ Z )
= ( zero_zero @ real ) )
@ ( one_one @ real )
@ ( sgn_sgn @ real @ ( im @ Z ) ) )
@ ( sqrt @ ( divide_divide @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% csqrt.simps(2)
thf(fact_4691_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_4692_complex__Im__fact,axiom,
! [N: nat] :
( ( im @ ( semiring_char_0_fact @ complex @ N ) )
= ( zero_zero @ real ) ) ).
% complex_Im_fact
thf(fact_4693_complex__Im__of__int,axiom,
! [Z: int] :
( ( im @ ( ring_1_of_int @ complex @ Z ) )
= ( zero_zero @ real ) ) ).
% complex_Im_of_int
thf(fact_4694_Im__complex__of__real,axiom,
! [Z: real] :
( ( im @ ( real_Vector_of_real @ complex @ Z ) )
= ( zero_zero @ real ) ) ).
% Im_complex_of_real
thf(fact_4695_Im__power__real,axiom,
! [X3: complex,N: nat] :
( ( ( im @ X3 )
= ( zero_zero @ real ) )
=> ( ( im @ ( power_power @ complex @ X3 @ N ) )
= ( zero_zero @ real ) ) ) ).
% Im_power_real
thf(fact_4696_complex__Im__numeral,axiom,
! [V2: num] :
( ( im @ ( numeral_numeral @ complex @ V2 ) )
= ( zero_zero @ real ) ) ).
% complex_Im_numeral
thf(fact_4697_complex__Im__of__nat,axiom,
! [N: nat] :
( ( im @ ( semiring_1_of_nat @ complex @ N ) )
= ( zero_zero @ real ) ) ).
% complex_Im_of_nat
thf(fact_4698_Re__power__real,axiom,
! [X3: complex,N: nat] :
( ( ( im @ X3 )
= ( zero_zero @ real ) )
=> ( ( re @ ( power_power @ complex @ X3 @ N ) )
= ( power_power @ real @ ( re @ X3 ) @ N ) ) ) ).
% Re_power_real
thf(fact_4699_Im__divide__numeral,axiom,
! [Z: complex,W: num] :
( ( im @ ( divide_divide @ complex @ Z @ ( numeral_numeral @ complex @ W ) ) )
= ( divide_divide @ real @ ( im @ Z ) @ ( numeral_numeral @ real @ W ) ) ) ).
% Im_divide_numeral
thf(fact_4700_Im__divide__of__nat,axiom,
! [Z: complex,N: nat] :
( ( im @ ( divide_divide @ complex @ Z @ ( semiring_1_of_nat @ complex @ N ) ) )
= ( divide_divide @ real @ ( im @ Z ) @ ( semiring_1_of_nat @ real @ N ) ) ) ).
% Im_divide_of_nat
thf(fact_4701_csqrt__of__real__nonneg,axiom,
! [X3: complex] :
( ( ( im @ X3 )
= ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ X3 ) )
=> ( ( csqrt @ X3 )
= ( real_Vector_of_real @ complex @ ( sqrt @ ( re @ X3 ) ) ) ) ) ) ).
% csqrt_of_real_nonneg
thf(fact_4702_csqrt__minus,axiom,
! [X3: complex] :
( ( ( ord_less @ real @ ( im @ X3 ) @ ( zero_zero @ real ) )
| ( ( ( im @ X3 )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ X3 ) ) ) )
=> ( ( csqrt @ ( uminus_uminus @ complex @ X3 ) )
= ( times_times @ complex @ imaginary_unit @ ( csqrt @ X3 ) ) ) ) ).
% csqrt_minus
thf(fact_4703_csqrt__of__real__nonpos,axiom,
! [X3: complex] :
( ( ( im @ X3 )
= ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( re @ X3 ) @ ( zero_zero @ real ) )
=> ( ( csqrt @ X3 )
= ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sqrt @ ( abs_abs @ real @ ( re @ X3 ) ) ) ) ) ) ) ) ).
% csqrt_of_real_nonpos
thf(fact_4704_zero__integer__def,axiom,
( ( zero_zero @ code_integer )
= ( code_integer_of_int @ ( zero_zero @ int ) ) ) ).
% zero_integer_def
thf(fact_4705_uminus__integer__code_I1_J,axiom,
( ( uminus_uminus @ code_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% uminus_integer_code(1)
thf(fact_4706_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_4707_less__integer__code_I1_J,axiom,
~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(1)
thf(fact_4708_less__integer_Oabs__eq,axiom,
! [Xa: int,X3: int] :
( ( ord_less @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X3 ) )
= ( ord_less @ int @ Xa @ X3 ) ) ).
% less_integer.abs_eq
thf(fact_4709_zero__complex_Osimps_I2_J,axiom,
( ( im @ ( zero_zero @ complex ) )
= ( zero_zero @ real ) ) ).
% zero_complex.simps(2)
thf(fact_4710_one__complex_Osimps_I2_J,axiom,
( ( im @ ( one_one @ complex ) )
= ( zero_zero @ real ) ) ).
% one_complex.simps(2)
thf(fact_4711_less__eq__integer_Oabs__eq,axiom,
! [Xa: int,X3: int] :
( ( ord_less_eq @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X3 ) )
= ( ord_less_eq @ int @ Xa @ X3 ) ) ).
% less_eq_integer.abs_eq
thf(fact_4712_complex__is__Int__iff,axiom,
! [Z: complex] :
( ( member @ complex @ Z @ ( ring_1_Ints @ complex ) )
= ( ( ( im @ Z )
= ( zero_zero @ real ) )
& ? [I4: int] :
( ( re @ Z )
= ( ring_1_of_int @ real @ I4 ) ) ) ) ).
% complex_is_Int_iff
thf(fact_4713_abs__Im__le__cmod,axiom,
! [X3: complex] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( im @ X3 ) ) @ ( real_V7770717601297561774m_norm @ complex @ X3 ) ) ).
% abs_Im_le_cmod
thf(fact_4714_Im__eq__0,axiom,
! [Z: complex] :
( ( ( abs_abs @ real @ ( re @ Z ) )
= ( real_V7770717601297561774m_norm @ complex @ Z ) )
=> ( ( im @ Z )
= ( zero_zero @ real ) ) ) ).
% Im_eq_0
thf(fact_4715_cmod__eq__Im,axiom,
! [Z: complex] :
( ( ( re @ Z )
= ( zero_zero @ real ) )
=> ( ( real_V7770717601297561774m_norm @ complex @ Z )
= ( abs_abs @ real @ ( im @ Z ) ) ) ) ).
% cmod_eq_Im
thf(fact_4716_cmod__eq__Re,axiom,
! [Z: complex] :
( ( ( im @ Z )
= ( zero_zero @ real ) )
=> ( ( real_V7770717601297561774m_norm @ complex @ Z )
= ( abs_abs @ real @ ( re @ Z ) ) ) ) ).
% cmod_eq_Re
thf(fact_4717_cmod__Re__le__iff,axiom,
! [X3: complex,Y3: complex] :
( ( ( im @ X3 )
= ( im @ Y3 ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ complex @ X3 ) @ ( real_V7770717601297561774m_norm @ complex @ Y3 ) )
= ( ord_less_eq @ real @ ( abs_abs @ real @ ( re @ X3 ) ) @ ( abs_abs @ real @ ( re @ Y3 ) ) ) ) ) ).
% cmod_Re_le_iff
thf(fact_4718_cmod__Im__le__iff,axiom,
! [X3: complex,Y3: complex] :
( ( ( re @ X3 )
= ( re @ Y3 ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ complex @ X3 ) @ ( real_V7770717601297561774m_norm @ complex @ Y3 ) )
= ( ord_less_eq @ real @ ( abs_abs @ real @ ( im @ X3 ) ) @ ( abs_abs @ real @ ( im @ Y3 ) ) ) ) ) ).
% cmod_Im_le_iff
thf(fact_4719_csqrt__principal,axiom,
! [Z: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( csqrt @ Z ) ) )
| ( ( ( re @ ( csqrt @ Z ) )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% csqrt_principal
thf(fact_4720_cmod__le,axiom,
! [Z: complex] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( plus_plus @ real @ ( abs_abs @ real @ ( re @ Z ) ) @ ( abs_abs @ real @ ( im @ Z ) ) ) ) ).
% cmod_le
thf(fact_4721_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_4722_cmod__power2,axiom,
! [Z: complex] :
( ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ real @ ( power_power @ real @ ( re @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% cmod_power2
thf(fact_4723_Im__power2,axiom,
! [X3: complex] :
( ( im @ ( power_power @ complex @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( re @ X3 ) ) @ ( im @ X3 ) ) ) ).
% Im_power2
thf(fact_4724_Re__power2,axiom,
! [X3: complex] :
( ( re @ ( power_power @ complex @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( minus_minus @ real @ ( power_power @ real @ ( re @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% Re_power2
thf(fact_4725_complex__eq__0,axiom,
! [Z: complex] :
( ( Z
= ( zero_zero @ complex ) )
= ( ( plus_plus @ real @ ( power_power @ real @ ( re @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ) ).
% complex_eq_0
thf(fact_4726_norm__complex__def,axiom,
( ( real_V7770717601297561774m_norm @ complex )
= ( ^ [Z6: complex] : ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z6 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z6 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% norm_complex_def
thf(fact_4727_inverse__complex_Osimps_I1_J,axiom,
! [X3: complex] :
( ( re @ ( inverse_inverse @ complex @ X3 ) )
= ( divide_divide @ real @ ( re @ X3 ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% inverse_complex.simps(1)
thf(fact_4728_complex__neq__0,axiom,
! [Z: complex] :
( ( Z
!= ( zero_zero @ complex ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_neq_0
thf(fact_4729_Re__divide,axiom,
! [X3: complex,Y3: complex] :
( ( re @ ( divide_divide @ complex @ X3 @ Y3 ) )
= ( divide_divide @ real @ ( plus_plus @ real @ ( times_times @ real @ ( re @ X3 ) @ ( re @ Y3 ) ) @ ( times_times @ real @ ( im @ X3 ) @ ( im @ Y3 ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Re_divide
thf(fact_4730_csqrt__unique,axiom,
! [W: complex,Z: complex] :
( ( ( power_power @ complex @ W @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= Z )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ W ) )
| ( ( ( re @ W )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ W ) ) ) )
=> ( ( csqrt @ Z )
= W ) ) ) ).
% csqrt_unique
thf(fact_4731_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_4732_inverse__complex_Osimps_I2_J,axiom,
! [X3: complex] :
( ( im @ ( inverse_inverse @ complex @ X3 ) )
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( im @ X3 ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% inverse_complex.simps(2)
thf(fact_4733_Im__divide,axiom,
! [X3: complex,Y3: complex] :
( ( im @ ( divide_divide @ complex @ X3 @ Y3 ) )
= ( divide_divide @ real @ ( minus_minus @ real @ ( times_times @ real @ ( im @ X3 ) @ ( re @ Y3 ) ) @ ( times_times @ real @ ( re @ X3 ) @ ( im @ Y3 ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Im_divide
thf(fact_4734_complex__abs__le__norm,axiom,
! [Z: complex] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( abs_abs @ real @ ( re @ Z ) ) @ ( abs_abs @ real @ ( im @ Z ) ) ) @ ( times_times @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( real_V7770717601297561774m_norm @ complex @ Z ) ) ) ).
% complex_abs_le_norm
thf(fact_4735_complex__unit__circle,axiom,
! [Z: complex] :
( ( Z
!= ( zero_zero @ complex ) )
=> ( ( plus_plus @ real @ ( power_power @ real @ ( divide_divide @ real @ ( re @ Z ) @ ( real_V7770717601297561774m_norm @ complex @ Z ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( divide_divide @ real @ ( im @ Z ) @ ( real_V7770717601297561774m_norm @ complex @ Z ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) ) ) ).
% complex_unit_circle
thf(fact_4736_inverse__complex_Ocode,axiom,
( ( inverse_inverse @ complex )
= ( ^ [X: complex] : ( complex2 @ ( 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 ) ) ) ) ) @ ( 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.code
thf(fact_4737_Complex__divide,axiom,
( ( divide_divide @ complex )
= ( ^ [X: complex,Y: complex] : ( complex2 @ ( 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 ) ) ) ) ) @ ( 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 ) ) ) ) ) ) ) ) ).
% Complex_divide
thf(fact_4738_Im__Reals__divide,axiom,
! [R2: complex,Z: complex] :
( ( member @ complex @ R2 @ ( real_Vector_Reals @ complex ) )
=> ( ( im @ ( divide_divide @ complex @ R2 @ Z ) )
= ( divide_divide @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( re @ R2 ) ) @ ( im @ Z ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Im_Reals_divide
thf(fact_4739_Re__Reals__divide,axiom,
! [R2: complex,Z: complex] :
( ( member @ complex @ R2 @ ( real_Vector_Reals @ complex ) )
=> ( ( re @ ( divide_divide @ complex @ R2 @ Z ) )
= ( divide_divide @ real @ ( times_times @ real @ ( re @ R2 ) @ ( re @ Z ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Re_Reals_divide
thf(fact_4740_complex__mult__cnj,axiom,
! [Z: complex] :
( ( times_times @ complex @ Z @ ( cnj @ Z ) )
= ( real_Vector_of_real @ complex @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_mult_cnj
thf(fact_4741_complex__cnj__zero,axiom,
( ( cnj @ ( zero_zero @ complex ) )
= ( zero_zero @ complex ) ) ).
% complex_cnj_zero
thf(fact_4742_complex__cnj__zero__iff,axiom,
! [Z: complex] :
( ( ( cnj @ Z )
= ( zero_zero @ complex ) )
= ( Z
= ( zero_zero @ complex ) ) ) ).
% complex_cnj_zero_iff
thf(fact_4743_complex__cnj__power,axiom,
! [X3: complex,N: nat] :
( ( cnj @ ( power_power @ complex @ X3 @ N ) )
= ( power_power @ complex @ ( cnj @ X3 ) @ N ) ) ).
% complex_cnj_power
thf(fact_4744_complex__In__mult__cnj__zero,axiom,
! [Z: complex] :
( ( im @ ( times_times @ complex @ Z @ ( cnj @ Z ) ) )
= ( zero_zero @ real ) ) ).
% complex_In_mult_cnj_zero
thf(fact_4745_imaginary__eq__real__iff,axiom,
! [Y3: complex,X3: complex] :
( ( member @ complex @ Y3 @ ( real_Vector_Reals @ complex ) )
=> ( ( member @ complex @ X3 @ ( real_Vector_Reals @ complex ) )
=> ( ( ( times_times @ complex @ imaginary_unit @ Y3 )
= X3 )
= ( ( X3
= ( zero_zero @ complex ) )
& ( Y3
= ( zero_zero @ complex ) ) ) ) ) ) ).
% imaginary_eq_real_iff
thf(fact_4746_real__eq__imaginary__iff,axiom,
! [Y3: complex,X3: complex] :
( ( member @ complex @ Y3 @ ( real_Vector_Reals @ complex ) )
=> ( ( member @ complex @ X3 @ ( real_Vector_Reals @ complex ) )
=> ( ( X3
= ( times_times @ complex @ imaginary_unit @ Y3 ) )
= ( ( X3
= ( zero_zero @ complex ) )
& ( Y3
= ( zero_zero @ complex ) ) ) ) ) ) ).
% real_eq_imaginary_iff
thf(fact_4747_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_4748_Reals__0,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( real_Vector_Reals @ A ) ) ) ).
% Reals_0
thf(fact_4749_Reals__of__int,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [Z: int] : ( member @ A @ ( ring_1_of_int @ A @ Z ) @ ( real_Vector_Reals @ A ) ) ) ).
% Reals_of_int
thf(fact_4750_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_4751_Reals__numeral,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [W: num] : ( member @ A @ ( numeral_numeral @ A @ W ) @ ( real_Vector_Reals @ A ) ) ) ).
% Reals_numeral
thf(fact_4752_complex__is__Real__iff,axiom,
! [Z: complex] :
( ( member @ complex @ Z @ ( real_Vector_Reals @ complex ) )
= ( ( im @ Z )
= ( zero_zero @ real ) ) ) ).
% complex_is_Real_iff
thf(fact_4753_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_4754_Complex__in__Reals,axiom,
! [X3: real] : ( member @ complex @ ( complex2 @ X3 @ ( zero_zero @ real ) ) @ ( real_Vector_Reals @ complex ) ) ).
% Complex_in_Reals
thf(fact_4755_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_4756_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_4757_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_4758_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_4759_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_4760_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_4761_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_4762_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_4763_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_4764_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_4765_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_4766_complex__mod__mult__cnj,axiom,
! [Z: complex] :
( ( real_V7770717601297561774m_norm @ complex @ ( times_times @ complex @ Z @ ( cnj @ Z ) ) )
= ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% complex_mod_mult_cnj
thf(fact_4767_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_4768_complex__norm__square,axiom,
! [Z: complex] :
( ( real_Vector_of_real @ complex @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( times_times @ complex @ Z @ ( cnj @ Z ) ) ) ).
% complex_norm_square
thf(fact_4769_complex__add__cnj,axiom,
! [Z: complex] :
( ( plus_plus @ complex @ Z @ ( cnj @ Z ) )
= ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( re @ Z ) ) ) ) ).
% complex_add_cnj
thf(fact_4770_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_4771_complex__diff__cnj,axiom,
! [Z: complex] :
( ( minus_minus @ complex @ Z @ ( cnj @ Z ) )
= ( times_times @ complex @ ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% complex_diff_cnj
thf(fact_4772_complex__div__cnj,axiom,
( ( divide_divide @ complex )
= ( ^ [A5: complex,B4: complex] : ( divide_divide @ complex @ ( times_times @ complex @ A5 @ ( cnj @ B4 ) ) @ ( real_Vector_of_real @ complex @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ B4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% complex_div_cnj
thf(fact_4773_cnj__add__mult__eq__Re,axiom,
! [Z: complex,W: complex] :
( ( plus_plus @ complex @ ( times_times @ complex @ Z @ ( cnj @ W ) ) @ ( times_times @ complex @ ( cnj @ Z ) @ W ) )
= ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( re @ ( times_times @ complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% cnj_add_mult_eq_Re
thf(fact_4774_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_4775_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_4776_setceilmax,axiom,
! [S: vEBT_VEBT,M: nat,Listy: list @ vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ S @ M )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ Listy ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( M
= ( suc @ N ) )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ Listy ) )
=> ( ( semiring_1_of_nat @ int @ ( vEBT_VEBT_height @ X4 ) )
= ( 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 ) ) ) ) ) ) ) ) ) ).
% setceilmax
thf(fact_4777_height__compose__list,axiom,
! [T2: vEBT_VEBT,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ T2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( 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 @ TreeList2 ) ) ) ) ) ) ).
% height_compose_list
thf(fact_4778_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_4779_max__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 ) ) ) ) ) ) ) ).
% max_ins_scaled
thf(fact_4780_height__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 ) ) ) ) ) ) ).
% height_i_max
thf(fact_4781_max__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 ) ) ) ) ) ) ) ) ) ).
% max_idx_list
thf(fact_4782_Max__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A] :
( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% Max_singleton
thf(fact_4783_zero__integer_Orep__eq,axiom,
( ( code_int_of_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ int ) ) ).
% zero_integer.rep_eq
thf(fact_4784_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_4785_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D4: nat] : ( dvd_dvd @ nat @ D4 @ N ) ) )
= N ) ) ).
% Max_divisors_self_nat
thf(fact_4786_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ X3 )
= ( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_4787_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ X3 )
= ( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( ord_less @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_4788_Max__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ B,C2: A] :
( ( finite_finite @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image @ B @ A
@ ^ [Uu3: B] : C2
@ A4 ) )
= C2 ) ) ) ) ).
% Max_const
thf(fact_4789_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
=> ( ( image @ A @ A @ ( times_times @ A @ D3 ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D3 @ A2 ) @ ( times_times @ A @ D3 @ B2 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_4790_Max__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X3 @ A4 ) )
= ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ) ).
% Max_insert
thf(fact_4791_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
=> ( ( image @ A @ A
@ ^ [C4: A] : ( divide_divide @ A @ C4 @ D3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A2 @ D3 ) @ ( divide_divide @ A @ B2 @ D3 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_4792_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S2: set @ B,F2: B > A,K: A] :
( ( finite_finite @ B @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image @ B @ A
@ ^ [X: B] : ( plus_plus @ A @ ( F2 @ X ) @ K )
@ S2 ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image @ B @ A @ F2 @ S2 ) ) @ K ) ) ) ) ) ).
% Max_add_commute
thf(fact_4793_hom__Max__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H: A > A,N4: set @ A] :
( ! [X4: A,Y4: A] :
( ( H @ ( ord_max @ A @ X4 @ Y4 ) )
= ( ord_max @ A @ ( H @ X4 ) @ ( H @ Y4 ) ) )
=> ( ( finite_finite @ A @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic643756798349783984er_Max @ A @ N4 ) )
= ( lattic643756798349783984er_Max @ A @ ( image @ A @ A @ H @ N4 ) ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_4794_Max__ge,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ).
% Max_ge
thf(fact_4795_Max__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ A4 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= X3 ) ) ) ) ) ).
% Max_eqI
thf(fact_4796_Max__eq__if,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( finite_finite @ A @ B6 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ B6 )
& ( ord_less_eq @ A @ X4 @ Xa2 ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ B6 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ A4 )
& ( ord_less_eq @ A @ X4 @ Xa2 ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= ( lattic643756798349783984er_Max @ A @ B6 ) ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_4797_Max_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,A2: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ A2 @ A4 )
=> ( ord_less_eq @ A @ A2 @ ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ).
% Max.coboundedI
thf(fact_4798_Max__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ A4 ) ) ) ) ).
% Max_in
thf(fact_4799_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_4800_less__integer_Orep__eq,axiom,
( ( ord_less @ code_integer )
= ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_integer.rep_eq
thf(fact_4801_Max_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ).
% Max.in_idem
thf(fact_4802_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_4803_less__eq__integer_Orep__eq,axiom,
( ( ord_less_eq @ code_integer )
= ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_eq_integer.rep_eq
thf(fact_4804_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,M: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798349783984er_Max @ A @ A4 )
= M )
= ( ( member @ A @ M @ A4 )
& ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ X @ M ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_4805_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( ? [X: A] :
( ( member @ A @ X @ A4 )
& ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_4806_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,M: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( ( member @ A @ M @ A4 )
& ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ X @ M ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_4807_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ X3 )
=> ! [A8: A] :
( ( member @ A @ A8 @ A4 )
=> ( ord_less_eq @ A @ A8 @ X3 ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_4808_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ( ord_less_eq @ A @ A6 @ X3 ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ X3 ) ) ) ) ) ).
% Max.boundedI
thf(fact_4809_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( ? [X: A] :
( ( member @ A @ X @ A4 )
& ( ord_less @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_4810_Max__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,A2: A] :
( ( finite_finite @ A @ A4 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ A4 )
=> ( ord_less_eq @ A @ B5 @ A2 ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ A2 @ A4 ) )
= A2 ) ) ) ) ).
% Max_insert2
thf(fact_4811_Max_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite @ A @ A4 )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Max.infinite
thf(fact_4812_VEBT__internal_Oheight_Oelims,axiom,
! [X3: vEBT_VEBT,Y3: nat] :
( ( ( vEBT_VEBT_height @ X3 )
= Y3 )
=> ( ( ? [A6: $o,B5: $o] :
( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( Y3
!= ( zero_zero @ nat ) ) )
=> ~ ! [Uu: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uu @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( Y3
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.height.elims
thf(fact_4813_scaleR__image__atLeastAtMost,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,X3: A,Y3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( image @ A @ A @ ( real_V8093663219630862766scaleR @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ Y3 ) ) ) ) ) ).
% scaleR_image_atLeastAtMost
thf(fact_4814_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B6 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ ( lattic643756798349783984er_Max @ A @ B6 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_4815_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_finite @ A @ N4 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M7 ) @ ( lattic643756798349783984er_Max @ A @ N4 ) ) ) ) ) ) ).
% Max_mono
thf(fact_4816_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B6 ) @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ) ).
% Max.subset
thf(fact_4817_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ~ ( member @ A @ X3 @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X3 @ A4 ) )
= ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_4818_Max_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y4: A] : ( member @ A @ ( ord_max @ A @ X4 @ Y4 ) @ ( insert @ A @ X4 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ A4 ) ) ) ) ) ).
% Max.closed
thf(fact_4819_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_4820_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X3 @ A4 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X3 @ A4 ) )
= ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_4821_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_4822_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,X3: A,Y3: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X3 ) @ ( times_times @ A @ C2 @ Y3 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y3 ) @ ( times_times @ A @ C2 @ X3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_4823_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A,C2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A
@ ^ [X: A] : ( times_times @ A @ X @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X3 @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X3 @ C2 ) @ ( times_times @ A @ Y3 @ C2 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A
@ ^ [X: A] : ( times_times @ A @ X @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X3 @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y3 @ C2 ) @ ( times_times @ A @ X3 @ C2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( image @ A @ A
@ ^ [X: A] : ( times_times @ A @ X @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X3 @ Y3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_4824_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_4825_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_4826_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_4827_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_4828_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_4829_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_4830_VEBT__internal_Oheight_Opelims,axiom,
! [X3: vEBT_VEBT,Y3: nat] :
( ( ( vEBT_VEBT_height @ X3 )
= Y3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ X3 )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Y3
= ( zero_zero @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A6 @ B5 ) ) ) )
=> ~ ! [Uu: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uu @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( Y3
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Node @ Uu @ Deg2 @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ).
% VEBT_internal.height.pelims
thf(fact_4831_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_4832_bij__betw__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N4: set @ nat,A4: set @ A] :
( ( bij_betw @ nat @ A @ ( semiring_1_of_nat @ A ) @ N4 @ A4 )
= ( ( image @ nat @ A @ ( semiring_1_of_nat @ A ) @ N4 )
= A4 ) ) ) ).
% bij_betw_of_nat
thf(fact_4833_Max__divisors__self__int,axiom,
! [N: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D4: int] : ( dvd_dvd @ int @ D4 @ N ) ) )
= ( abs_abs @ int @ N ) ) ) ).
% Max_divisors_self_int
thf(fact_4834_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_4835_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_4836_zero__notin__Suc__image,axiom,
! [A4: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ A4 ) ) ).
% zero_notin_Suc_image
thf(fact_4837_None__notin__image__Some,axiom,
! [A: $tType,A4: set @ A] :
~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ) ).
% None_notin_image_Some
thf(fact_4838_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite @ A )
= ( ^ [A9: set @ A] :
? [N3: nat,F3: nat > A] :
( A9
= ( image @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N3 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_4839_nat__seg__image__imp__finite,axiom,
! [A: $tType,A4: set @ A,F2: nat > A,N: nat] :
( ( A4
= ( image @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N ) ) ) )
=> ( finite_finite @ A @ A4 ) ) ).
% nat_seg_image_imp_finite
thf(fact_4840_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_4841_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_4842_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_4843_nat__of__integer_Orep__eq,axiom,
( code_nat_of_integer
= ( ^ [X: code_integer] : ( nat2 @ ( code_int_of_integer @ X ) ) ) ) ).
% nat_of_integer.rep_eq
thf(fact_4844_nat__of__integer_Oabs__eq,axiom,
! [X3: int] :
( ( code_nat_of_integer @ ( code_integer_of_int @ X3 ) )
= ( nat2 @ X3 ) ) ).
% nat_of_integer.abs_eq
thf(fact_4845_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_4846_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_4847_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_4848_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_4849_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_4850_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image @ int @ int
@ ^ [X: int] : ( plus_plus @ int @ X @ L )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_4851_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ U )
=> ( ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U )
= ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_ord_lessThan @ nat @ ( nat2 @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_int
thf(fact_4852_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y3: nat,X3: nat] :
( ( ( ord_less @ nat @ C2 @ Y3 )
=> ( ( image @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X3 @ Y3 ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X3 @ C2 ) @ ( minus_minus @ nat @ Y3 @ C2 ) ) ) )
& ( ~ ( ord_less @ nat @ C2 @ Y3 )
=> ( ( ( ord_less @ nat @ X3 @ Y3 )
=> ( ( image @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X3 @ Y3 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X3 @ Y3 )
=> ( ( image @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X3 @ Y3 ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_4853_vebt__maxt_Opelims,axiom,
! [X3: vEBT_VEBT,Y3: option @ nat] :
( ( ( vEBT_vebt_maxt @ X3 )
= Y3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ X3 )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( ( B5
=> ( Y3
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B5
=> ( ( A6
=> ( Y3
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( Y3
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A6 @ B5 ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y3
= ( none @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( some @ nat @ Ma2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_maxt.pelims
thf(fact_4854_vebt__mint_Opelims,axiom,
! [X3: vEBT_VEBT,Y3: option @ nat] :
( ( ( vEBT_vebt_mint @ X3 )
= Y3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ X3 )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( ( A6
=> ( Y3
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( ( B5
=> ( Y3
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B5
=> ( Y3
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A6 @ B5 ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y3
= ( none @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( some @ nat @ Mi2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_mint.pelims
thf(fact_4855_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
! [X3: vEBT_VEBT,Y3: nat] :
( ( ( vEBT_T_m_i_n_t @ X3 )
= Y3 )
=> ( ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ X3 )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Y3
= ( 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 @ B5 ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y3
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y3
= ( 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 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
thf(fact_4856_even__sum__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( finite_card @ B
@ ( collect @ B
@ ^ [A5: B] :
( ( member @ B @ A5 @ A4 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ A5 ) ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_4857_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_4858_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_4859_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_4860_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_4861_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_4862_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_4863_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_4864_card_Oinfinite,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( finite_finite @ A @ A4 )
=> ( ( finite_card @ A @ A4 )
= ( zero_zero @ nat ) ) ) ).
% card.infinite
thf(fact_4865_prod__constant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y3: A,A4: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : Y3
@ A4 )
= ( power_power @ A @ Y3 @ ( finite_card @ B @ A4 ) ) ) ) ).
% prod_constant
thf(fact_4866_card__atLeastLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ L ) ) ) ).
% card_atLeastLessThan_int
thf(fact_4867_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_4868_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_4869_card__0__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( ( finite_card @ A @ A4 )
= ( zero_zero @ nat ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_4870_sum__constant,axiom,
! [B: $tType,A: $tType] :
( ( semiring_1 @ A )
=> ! [Y3: A,A4: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : Y3
@ A4 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) @ Y3 ) ) ) ).
% sum_constant
thf(fact_4871_card__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ U @ L ) @ ( one_one @ int ) ) ) ) ).
% card_atLeastAtMost_int
thf(fact_4872_nat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H: A > B,F1: A,F22: nat > A,Nat: nat] :
( ( H @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( case_nat @ B @ ( H @ F1 )
@ ^ [X: nat] : ( H @ ( F22 @ X ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_4873_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F22: nat > A,X2: nat] :
( ( case_nat @ A @ F1 @ F22 @ ( suc @ X2 ) )
= ( F22 @ X2 ) ) ).
% old.nat.simps(5)
thf(fact_4874_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_4875_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B6: set @ A,A4: set @ B,R2: B > A > $o] :
( ( finite_finite @ A @ B6 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ A4 )
=> ? [B8: A] :
( ( member @ A @ B8 @ B6 )
& ( R2 @ A6 @ B8 ) ) )
=> ( ! [A13: B,A24: B,B5: A] :
( ( member @ B @ A13 @ A4 )
=> ( ( member @ B @ A24 @ A4 )
=> ( ( member @ A @ B5 @ B6 )
=> ( ( R2 @ A13 @ B5 )
=> ( ( R2 @ A24 @ B5 )
=> ( A13 = A24 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A4 ) @ ( finite_card @ A @ B6 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_4876_card__insert__le,axiom,
! [A: $tType,A4: set @ A,X3: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ ( insert @ A @ X3 @ A4 ) ) ) ).
% card_insert_le
thf(fact_4877_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat
!= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $false
@ ^ [Uu3: nat] : $true
@ Nat ) ) ).
% nat.disc_eq_case(2)
thf(fact_4878_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat
= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $true
@ ^ [Uu3: nat] : $false
@ Nat ) ) ).
% nat.disc_eq_case(1)
thf(fact_4879_card__lists__length__eq,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( finite_finite @ A @ A4 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A4 )
& ( ( size_size @ ( list @ A ) @ Xs )
= N ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A4 ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_4880_card__2__iff_H,axiom,
! [A: $tType,S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X: A] :
( ( member @ A @ X @ S2 )
& ? [Y: A] :
( ( member @ A @ Y @ S2 )
& ( X != Y )
& ! [Z6: A] :
( ( member @ A @ Z6 @ S2 )
=> ( ( Z6 = X )
| ( Z6 = Y ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_4881_card__eq__0__iff,axiom,
! [A: $tType,A4: set @ A] :
( ( ( finite_card @ A @ A4 )
= ( zero_zero @ nat ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite @ A @ A4 ) ) ) ).
% card_eq_0_iff
thf(fact_4882_card__ge__0__finite,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A4 ) )
=> ( finite_finite @ A @ A4 ) ) ).
% card_ge_0_finite
thf(fact_4883_card__image__le,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > B] :
( ( finite_finite @ A @ A4 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( image @ A @ B @ F2 @ A4 ) ) @ ( finite_card @ A @ A4 ) ) ) ).
% card_image_le
thf(fact_4884_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_finite @ A @ T7 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_4885_card__mono,axiom,
! [A: $tType,B6: set @ A,A4: set @ A] :
( ( finite_finite @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) ) ) ) ).
% card_mono
thf(fact_4886_card__seteq,axiom,
! [A: $tType,B6: set @ A,A4: set @ A] :
( ( finite_finite @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B6 ) @ ( finite_card @ A @ A4 ) )
=> ( A4 = B6 ) ) ) ) ).
% card_seteq
thf(fact_4887_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F4: set @ A,C5: nat] :
( ! [G3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G3 @ F4 )
=> ( ( finite_finite @ A @ G3 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G3 ) @ C5 ) ) )
=> ( ( finite_finite @ A @ F4 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F4 ) @ C5 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_4888_card__less__sym__Diff,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( finite_finite @ A @ B6 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B6 @ A4 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_4889_card__length,axiom,
! [A: $tType,Xs2: list @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( set2 @ A @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% card_length
thf(fact_4890_card__le__sym__Diff,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( finite_finite @ A @ B6 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B6 @ A4 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_4891_psubset__card__mono,axiom,
! [A: $tType,B6: set @ A,A4: set @ A] :
( ( finite_finite @ A @ B6 )
=> ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
=> ( ord_less @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) ) ) ) ).
% psubset_card_mono
thf(fact_4892_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_4893_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_4894_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_4895_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) )
= ( nat2 @ U ) ) ).
% card_atLeastZeroLessThan_int
thf(fact_4896_sum__constant__scaleR,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Y3: A,A4: set @ C] :
( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X: C] : Y3
@ A4 )
= ( real_V8093663219630862766scaleR @ A @ ( semiring_1_of_nat @ real @ ( finite_card @ C @ A4 ) ) @ Y3 ) ) ) ).
% sum_constant_scaleR
thf(fact_4897_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_4898_real__of__card,axiom,
! [A: $tType,A4: set @ A] :
( ( semiring_1_of_nat @ real @ ( finite_card @ A @ A4 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X: A] : ( one_one @ real )
@ A4 ) ) ).
% real_of_card
thf(fact_4899_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_4900_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_4901_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A4: set @ B,K5: A,F2: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ord_less_eq @ A @ K5 @ ( F2 @ I3 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) ) ) ) ).
% sum_bounded_below
thf(fact_4902_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A4: set @ B,F2: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ K5 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above
thf(fact_4903_card__le__Suc0__iff__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ! [Y: A] :
( ( member @ A @ Y @ A4 )
=> ( X = Y ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_4904_card__1__singleton__iff,axiom,
! [A: $tType,A4: set @ A] :
( ( ( finite_card @ A @ A4 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X: A] :
( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_4905_card__eq__SucD,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( ( finite_card @ A @ A4 )
= ( suc @ K ) )
=> ? [B5: A,B7: set @ A] :
( ( A4
= ( insert @ A @ B5 @ B7 ) )
& ~ ( member @ A @ B5 @ B7 )
& ( ( finite_card @ A @ B7 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B7
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_4906_card__Suc__eq,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( ( finite_card @ A @ A4 )
= ( suc @ K ) )
= ( ? [B4: A,B9: set @ A] :
( ( A4
= ( insert @ A @ B4 @ B9 ) )
& ~ ( member @ A @ B4 @ B9 )
& ( ( finite_card @ A @ B9 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B9
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_4907_card__gt__0__iff,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A4 ) )
= ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite @ A @ A4 ) ) ) ).
% card_gt_0_iff
thf(fact_4908_surj__card__le,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ B,F2: A > B] :
( ( finite_finite @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image @ A @ B @ F2 @ A4 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ B6 ) @ ( finite_card @ A @ A4 ) ) ) ) ).
% surj_card_le
thf(fact_4909_card__le__Suc__iff,axiom,
! [A: $tType,N: nat,A4: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( finite_card @ A @ A4 ) )
= ( ? [A5: A,B9: set @ A] :
( ( A4
= ( insert @ A @ A5 @ B9 ) )
& ~ ( member @ A @ A5 @ B9 )
& ( ord_less_eq @ nat @ N @ ( finite_card @ A @ B9 ) )
& ( finite_finite @ A @ B9 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_4910_card__Diff1__le,axiom,
! [A: $tType,A4: set @ A,X3: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) ) ).
% card_Diff1_le
thf(fact_4911_card__psubset,axiom,
! [A: $tType,B6: set @ A,A4: set @ A] :
( ( finite_finite @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) )
=> ( ord_less @ ( set @ A ) @ A4 @ B6 ) ) ) ) ).
% card_psubset
thf(fact_4912_diff__card__le__card__Diff,axiom,
! [A: $tType,B6: set @ A,A4: set @ A] :
( ( finite_finite @ A @ B6 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_4913_card__lists__length__le,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( finite_finite @ A @ A4 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A4 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A4 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_4914_ex__bij__betw__nat__finite,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite @ 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_4915_card__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( power_power @ A @ Z6 @ N )
= ( one_one @ A ) ) ) )
@ N ) ) ) ).
% card_roots_unity
thf(fact_4916_ex__bij__betw__finite__nat,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite @ 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_4917_card__le__Suc__Max,axiom,
! [S2: set @ nat] :
( ( finite_finite @ nat @ S2 )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ S2 ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S2 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_4918_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_4919_card__sum__le__nat__sum,axiom,
! [S2: set @ nat] :
( ord_less_eq @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S2 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ S2 ) ) ).
% card_sum_le_nat_sum
thf(fact_4920_card__nth__roots,axiom,
! [C2: complex,N: nat] :
( ( C2
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= C2 ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_4921_card__roots__unity__eq,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= ( one_one @ complex ) ) ) )
= N ) ) ).
% card_roots_unity_eq
thf(fact_4922_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_4923_card__2__iff,axiom,
! [A: $tType,S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X: A,Y: A] :
( ( S2
= ( insert @ A @ X @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X != Y ) ) ) ) ).
% card_2_iff
thf(fact_4924_card__3__iff,axiom,
! [A: $tType,S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X: A,Y: A,Z6: A] :
( ( S2
= ( insert @ A @ X @ ( insert @ A @ Y @ ( insert @ A @ Z6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X != Y )
& ( Y != Z6 )
& ( X != Z6 ) ) ) ) ).
% card_3_iff
thf(fact_4925_odd__card__imp__not__empty,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A4 ) )
=> ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% odd_card_imp_not_empty
thf(fact_4926_card__Diff1__less__iff,axiom,
! [A: $tType,A4: set @ A,X3: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) )
= ( ( finite_finite @ A @ A4 )
& ( member @ A @ X3 @ A4 ) ) ) ).
% card_Diff1_less_iff
thf(fact_4927_card__Diff2__less,axiom,
! [A: $tType,A4: set @ A,X3: A,Y3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( member @ A @ Y3 @ A4 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_4928_card__Diff1__less,axiom,
! [A: $tType,A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) ) ) ) ).
% card_Diff1_less
thf(fact_4929_bit__numeral__rec_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ W ) ) @ N )
= ( case_nat @ $o @ $false @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) ) @ N ) ) ) ).
% bit_numeral_rec(1)
thf(fact_4930_bit__numeral__rec_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ W ) ) @ N )
= ( case_nat @ $o @ $true @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) ) @ N ) ) ) ).
% bit_numeral_rec(2)
thf(fact_4931_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType] :
( ( case_nat @ A )
= ( ^ [X: A,F3: nat > A,N3: nat] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ X
@ ( F3 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_4932_sum__norm__bound,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [S2: set @ B,F2: B > A,K5: real] :
( ! [X4: B] :
( ( member @ B @ X4 @ S2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X4 ) ) @ K5 ) )
=> ( 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 ) ) @ K5 ) ) ) ) ).
% sum_norm_bound
thf(fact_4933_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F2: B > A,N: A,K: nat] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) )
& ( ord_less_eq @ A @ ( F2 @ I3 ) @ N ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A4 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( power_power @ A @ N @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_4934_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A4: set @ B,F2: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ord_less @ A @ ( F2 @ I3 ) @ K5 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A4 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_4935_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A4: set @ B,F2: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ ( divide_divide @ A @ K5 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) ) ) )
=> ( ( finite_finite @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_4936_card__insert__le__m1,axiom,
! [A: $tType,N: nat,Y3: set @ A,X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y3 ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert @ A @ X3 @ Y3 ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_4937_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S2: set @ A,R3: set @ B,G: A > B,F2: B > C] :
( ( finite_finite @ A @ S2 )
=> ( ( finite_finite @ B @ R3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ G @ S2 ) @ R3 )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X: A] : ( F2 @ ( G @ X ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ S2 )
& ( ( G @ X )
= Y ) ) ) ) )
@ ( F2 @ Y ) )
@ R3 ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_4938_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A2: B,B2: B > A,C2: A] :
( ( finite_finite @ 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_4939_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
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z6 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ).
% polyfun_roots_card
thf(fact_4940_sum__le__card__Max,axiom,
! [A: $tType,A4: set @ A,F2: A > nat] :
( ( finite_finite @ A @ A4 )
=> ( ord_less_eq @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) @ ( times_times @ nat @ ( finite_card @ A @ A4 ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ A @ nat @ F2 @ A4 ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_4941_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_finite @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z6 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
& ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z6 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ) ).
% polyfun_rootbound
thf(fact_4942_old_Orec__nat__def,axiom,
! [T: $tType] :
( ( rec_nat @ T )
= ( ^ [F12: T,F23: nat > T > T,X: nat] : ( the @ T @ ( rec_set_nat @ T @ F12 @ F23 @ X ) ) ) ) ).
% old.rec_nat_def
thf(fact_4943_card__lists__distinct__length__eq,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( finite_finite @ A @ A4 )
=> ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A4 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= K )
& ( distinct @ A @ Xs )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A4 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_4944_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K: nat,A4: set @ A] :
( ( ord_less @ nat @ K @ ( finite_card @ A @ A4 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= K )
& ( distinct @ A @ Xs )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A4 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_4945_distinct__swap,axiom,
! [A: $tType,I: nat,Xs2: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs2 @ I @ ( nth @ A @ Xs2 @ J ) ) @ J @ ( nth @ A @ Xs2 @ I ) ) )
= ( distinct @ A @ Xs2 ) ) ) ) ).
% distinct_swap
thf(fact_4946_nth__eq__iff__index__eq,axiom,
! [A: $tType,Xs2: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Xs2 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( nth @ A @ Xs2 @ I )
= ( nth @ A @ Xs2 @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_4947_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs: list @ A] :
! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( I4 != J3 )
=> ( ( nth @ A @ Xs @ I4 )
!= ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_4948_distinct__Ex1,axiom,
! [A: $tType,Xs2: list @ A,X3: A] :
( ( distinct @ A @ Xs2 )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ? [X4: nat] :
( ( ord_less @ nat @ X4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ( nth @ A @ Xs2 @ X4 )
= X3 )
& ! [Y5: nat] :
( ( ( ord_less @ nat @ Y5 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ( nth @ A @ Xs2 @ Y5 )
= X3 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_4949_floor__real__def,axiom,
( ( archim6421214686448440834_floor @ real )
= ( ^ [X: real] :
( the @ int
@ ^ [Z6: int] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ Z6 ) @ X )
& ( ord_less @ real @ X @ ( ring_1_of_int @ real @ ( plus_plus @ int @ Z6 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_4950_set__update__distinct,axiom,
! [A: $tType,Xs2: list @ A,N: nat,X3: A] :
( ( distinct @ A @ Xs2 )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( set2 @ A @ ( list_update @ A @ Xs2 @ N @ X3 ) )
= ( insert @ A @ X3 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( insert @ A @ ( nth @ A @ Xs2 @ N ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_4951_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_4952_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_4953_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_4954_pred__def,axiom,
( pred
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [X23: nat] : X23 ) ) ).
% pred_def
thf(fact_4955_dual__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Min @ A
@ ^ [X: A,Y: A] : ( ord_less_eq @ A @ Y @ X ) )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% dual_Min
thf(fact_4956_bezw__0,axiom,
! [X3: nat] :
( ( bezw @ X3 @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_4957_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_4958_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_4959_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_4960_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_4961_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_4962_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_4963_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_4964_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_4965_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_4966_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_4967_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_4968_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_4969_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_4970_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_4971_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_4972_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_4973_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_4974_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_4975_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_4976_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_4977_linorder_OMin_Ocong,axiom,
! [A: $tType] :
( ( lattices_Min @ A )
= ( lattices_Min @ A ) ) ).
% linorder.Min.cong
thf(fact_4978_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_4979_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_4980_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_4981_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_4982_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_4983_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_4984_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_4985_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_4986_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_4987_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_4988_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_4989_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_4990_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_4991_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_4992_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_4993_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_4994_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_4995_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_4996_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_4997_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_4998_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_4999_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_5000_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_5001_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_5002_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_5003_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_5004_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_5005_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_5006_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_5007_prod__decode__aux_Oelims,axiom,
! [X3: nat,Xa: nat,Y3: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X3 @ Xa )
= Y3 )
=> ( ( ( ord_less_eq @ nat @ Xa @ X3 )
=> ( Y3
= ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X3 @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa @ X3 )
=> ( Y3
= ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus @ nat @ Xa @ ( suc @ X3 ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_5008_size__prod__simp,axiom,
! [B: $tType,A: $tType] :
( ( basic_BNF_size_prod @ A @ B )
= ( ^ [F3: A > nat,G2: B > nat,P5: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F3 @ ( product_fst @ A @ B @ P5 ) ) @ ( G2 @ ( product_snd @ A @ B @ P5 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% size_prod_simp
thf(fact_5009_in__set__enumerate__eq,axiom,
! [A: $tType,P6: product_prod @ nat @ A,N: nat,Xs2: list @ A] :
( ( member @ ( product_prod @ nat @ A ) @ P6 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs2 ) ) )
= ( ( ord_less_eq @ nat @ N @ ( product_fst @ nat @ A @ P6 ) )
& ( ord_less @ nat @ ( product_fst @ nat @ A @ P6 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) )
& ( ( nth @ A @ Xs2 @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P6 ) @ N ) )
= ( product_snd @ nat @ A @ P6 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_5010_finite__enumerate,axiom,
! [S2: set @ nat] :
( ( finite_finite @ 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_5011_bezw__non__0,axiom,
! [Y3: nat,X3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y3 )
=> ( ( bezw @ X3 @ Y3 )
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X3 @ Y3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X3 @ Y3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X3 @ Y3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_5012_bezw_Oelims,axiom,
! [X3: nat,Xa: nat,Y3: product_prod @ int @ int] :
( ( ( bezw @ X3 @ Xa )
= Y3 )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y3
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X3 @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X3 @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X3 @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Xa ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_5013_bezw_Osimps,axiom,
( bezw
= ( ^ [X: nat,Y: nat] :
( if @ ( product_prod @ int @ int )
@ ( Y
= ( zero_zero @ nat ) )
@ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
@ ( 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.simps
thf(fact_5014_nth__enumerate__eq,axiom,
! [A: $tType,M: nat,Xs2: list @ A,N: nat] :
( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs2 ) @ M )
= ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N @ M ) @ ( nth @ A @ Xs2 @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_5015_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_5016_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_5017_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_5018_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_5019_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_5020_bezw_Opelims,axiom,
! [X3: nat,Xa: nat,Y3: product_prod @ int @ int] :
( ( ( bezw @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa ) )
=> ~ ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y3
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X3 @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X3 @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X3 @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Xa ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa ) ) ) ) ) ).
% bezw.pelims
thf(fact_5021_strict__mono__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( strict_mono_on @ A @ B )
= ( ^ [F3: A > B,A9: set @ A] :
! [R5: A,S6: A] :
( ( ( member @ A @ R5 @ A9 )
& ( member @ A @ S6 @ A9 )
& ( ord_less @ A @ R5 @ S6 ) )
=> ( ord_less @ B @ ( F3 @ R5 ) @ ( F3 @ S6 ) ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_5022_strict__mono__onI,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [A4: set @ A,F2: A > B] :
( ! [R: A,S3: A] :
( ( member @ A @ R @ A4 )
=> ( ( member @ A @ S3 @ A4 )
=> ( ( ord_less @ A @ R @ S3 )
=> ( ord_less @ B @ ( F2 @ R ) @ ( F2 @ S3 ) ) ) ) )
=> ( strict_mono_on @ A @ B @ F2 @ A4 ) ) ) ).
% strict_mono_onI
thf(fact_5023_strict__mono__on__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( preorder @ B ) )
=> ! [F2: A > B,A4: set @ A,X3: A,Y3: A] :
( ( strict_mono_on @ A @ B @ F2 @ A4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( member @ A @ Y3 @ A4 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_5024_strict__mono__onD,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [F2: A > B,A4: set @ A,R2: A,S: A] :
( ( strict_mono_on @ A @ B @ F2 @ A4 )
=> ( ( member @ A @ R2 @ A4 )
=> ( ( member @ A @ S @ A4 )
=> ( ( ord_less @ A @ R2 @ S )
=> ( ord_less @ B @ ( F2 @ R2 ) @ ( F2 @ S ) ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_5025_prod__decode__aux_Opelims,axiom,
! [X3: nat,Xa: nat,Y3: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ nat @ Xa @ X3 )
=> ( Y3
= ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X3 @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa @ X3 )
=> ( Y3
= ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus @ nat @ Xa @ ( suc @ X3 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_5026_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,S6: 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 ) @ S6 ) )
@ ( S6
= ( one_one @ code_integer ) ) )
@ ( code_divmod_abs @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_5027_nth__rotate1,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( rotate1 @ A @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( modulo_modulo @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ).
% nth_rotate1
thf(fact_5028_rotate1__length01,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) )
=> ( ( rotate1 @ A @ Xs2 )
= Xs2 ) ) ).
% rotate1_length01
thf(fact_5029_divmod__abs__code_I6_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ ( zero_zero @ code_integer ) @ J )
= ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ) ).
% divmod_abs_code(6)
thf(fact_5030_divmod__abs__code_I5_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ J @ ( zero_zero @ code_integer ) )
= ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( abs_abs @ code_integer @ J ) ) ) ).
% divmod_abs_code(5)
thf(fact_5031_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_5032_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,S6: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S6
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ 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 @ S6 ) ) )
@ ( 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,S6: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S6
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ 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 ) @ S6 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_5033_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ ( plus_plus @ int @ L @ ( one_one @ int ) ) ) ) ) ).
% card_greaterThanLessThan_int
thf(fact_5034_xor__minus__numerals_I1_J,axiom,
! [N: num,K: int] :
( ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) @ K )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344971392196577ns_xor @ int @ ( neg_numeral_sub @ int @ N @ one2 ) @ K ) ) ) ).
% xor_minus_numerals(1)
thf(fact_5035_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_5036_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_5037_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_5038_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_5039_infinite__Ioo__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ioo_iff
thf(fact_5040_sub__num__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_sub @ A @ one2 @ one2 )
= ( zero_zero @ A ) ) ) ).
% sub_num_simps(1)
thf(fact_5041_diff__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ M @ N ) ) ) ).
% diff_numeral_simps(1)
thf(fact_5042_sub__num__simps_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ ( bit0 @ L ) )
= ( neg_numeral_dbl @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(6)
thf(fact_5043_sub__num__simps_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ ( bit1 @ L ) )
= ( neg_numeral_dbl @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(9)
thf(fact_5044_add__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ M @ N ) ) ) ).
% add_neg_numeral_simps(1)
thf(fact_5045_add__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ N @ M ) ) ) ).
% add_neg_numeral_simps(2)
thf(fact_5046_semiring__norm_I166_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W: num,Y3: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y3 ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ V2 @ W ) @ Y3 ) ) ) ).
% semiring_norm(166)
thf(fact_5047_semiring__norm_I167_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W: num,Y3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Y3 ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ W @ V2 ) @ Y3 ) ) ) ).
% semiring_norm(167)
thf(fact_5048_diff__numeral__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ N @ M ) ) ) ).
% diff_numeral_simps(4)
thf(fact_5049_sub__num__simps_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ ( bit1 @ L ) )
= ( neg_numeral_dbl_dec @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(7)
thf(fact_5050_sub__num__simps_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ ( bit0 @ L ) )
= ( neg_numeral_dbl_inc @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(8)
thf(fact_5051_diff__numeral__special_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ one2 @ N ) ) ) ).
% diff_numeral_special(1)
thf(fact_5052_diff__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ M @ one2 ) ) ) ).
% diff_numeral_special(2)
thf(fact_5053_sub__num__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ one2 )
= ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% sub_num_simps(5)
thf(fact_5054_not__minus__numeral__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_ri4277139882892585799ns_not @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% not_minus_numeral_eq
thf(fact_5055_sub__num__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ one2 )
= ( numeral_numeral @ A @ ( bitM @ K ) ) ) ) ).
% sub_num_simps(4)
thf(fact_5056_add__neg__numeral__special_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) )
= ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% add_neg_numeral_special(1)
thf(fact_5057_add__neg__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% add_neg_numeral_special(2)
thf(fact_5058_add__neg__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ M @ one2 ) ) ) ).
% add_neg_numeral_special(3)
thf(fact_5059_add__neg__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% add_neg_numeral_special(4)
thf(fact_5060_minus__sub__one__diff__one,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( neg_numeral_sub @ A @ M @ one2 ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% minus_sub_one_diff_one
thf(fact_5061_diff__numeral__special_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% diff_numeral_special(7)
thf(fact_5062_diff__numeral__special_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% diff_numeral_special(8)
thf(fact_5063_sub__num__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [L: num] :
( ( neg_numeral_sub @ A @ one2 @ ( bit1 @ L ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ L ) ) ) ) ) ).
% sub_num_simps(3)
thf(fact_5064_sub__num__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [L: num] :
( ( neg_numeral_sub @ A @ one2 @ ( bit0 @ L ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bitM @ L ) ) ) ) ) ).
% sub_num_simps(2)
thf(fact_5065_xor__minus__numerals_I2_J,axiom,
! [K: int,N: num] :
( ( bit_se5824344971392196577ns_xor @ int @ K @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ ( neg_numeral_sub @ int @ N @ one2 ) ) ) ) ).
% xor_minus_numerals(2)
thf(fact_5066_infinite__Ioo,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ioo
thf(fact_5067_neg__numeral__class_Osub__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_sub @ A )
= ( ^ [K3: num,L2: num] : ( minus_minus @ A @ ( numeral_numeral @ A @ K3 ) @ ( numeral_numeral @ A @ L2 ) ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_5068_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_5069_sub__non__negative,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N @ M ) )
= ( ord_less_eq @ num @ M @ N ) ) ) ).
% sub_non_negative
thf(fact_5070_sub__non__positive,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M: num] :
( ( ord_less_eq @ A @ ( neg_numeral_sub @ A @ N @ M ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ num @ N @ M ) ) ) ).
% sub_non_positive
thf(fact_5071_sub__negative,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M: num] :
( ( ord_less @ A @ ( neg_numeral_sub @ A @ N @ M ) @ ( zero_zero @ A ) )
= ( ord_less @ num @ N @ M ) ) ) ).
% sub_negative
thf(fact_5072_sub__positive,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N @ M ) )
= ( ord_less @ num @ M @ N ) ) ) ).
% sub_positive
thf(fact_5073_sub__inc__One__eq,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( neg_numeral_sub @ A @ ( inc @ N ) @ one2 )
= ( numeral_numeral @ A @ N ) ) ) ).
% sub_inc_One_eq
thf(fact_5074_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_5075_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_5076_minus__numeral__eq__not__sub__one,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( neg_numeral_sub @ A @ N @ one2 ) ) ) ) ).
% minus_numeral_eq_not_sub_one
thf(fact_5077_sub__BitM__One__eq,axiom,
! [N: num] :
( ( neg_numeral_sub @ int @ ( bitM @ N ) @ one2 )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( neg_numeral_sub @ int @ N @ one2 ) ) ) ).
% sub_BitM_One_eq
thf(fact_5078_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,S6: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S6
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ 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 ) @ S6 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_5079_bounded__linear__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( real_V4916620083959148203axioms @ A @ B )
= ( ^ [F3: A > B] :
? [K6: real] :
! [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ K6 ) ) ) ) ) ).
% bounded_linear_axioms_def
thf(fact_5080_bounded__linear__axioms_Ointro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B] :
( ? [K8: real] :
! [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X4 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ K8 ) )
=> ( real_V4916620083959148203axioms @ A @ B @ F2 ) ) ) ).
% bounded_linear_axioms.intro
thf(fact_5081_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E4: $tType,F2: B > A,G: C > B,X3: C,F6: D > A,G4: E4 > D,X7: E4] :
( ( ( F2 @ ( G @ X3 ) )
= ( F6 @ ( G4 @ X7 ) ) )
=> ( ( comp @ B @ A @ C @ F2 @ G @ X3 )
= ( comp @ D @ A @ E4 @ F6 @ G4 @ X7 ) ) ) ).
% comp_cong
thf(fact_5082_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( comm_monoid_add @ B )
& ( comm_monoid_add @ A ) )
=> ! [H: B > A,G: C > B,A4: set @ C] :
( ( ( H @ ( zero_zero @ B ) )
= ( zero_zero @ A ) )
=> ( ! [X4: B,Y4: B] :
( ( H @ ( plus_plus @ B @ X4 @ Y4 ) )
= ( plus_plus @ A @ ( H @ X4 ) @ ( H @ Y4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H @ G ) @ A4 )
= ( H @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A4 ) ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_5083_sum_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,H: B > C,G: C > A] :
( ( finite_finite @ B @ A4 )
=> ( ! [X4: B,Y4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ( member @ B @ Y4 @ A4 )
=> ( ( X4 != Y4 )
=> ( ( ( H @ X4 )
= ( H @ Y4 ) )
=> ( ( G @ ( H @ X4 ) )
= ( zero_zero @ A ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image @ B @ C @ H @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H ) @ A4 ) ) ) ) ) ).
% sum.reindex_nontrivial
thf(fact_5084_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [I6: set @ C,G: A > B,F2: C > A] :
( ( finite_finite @ C @ I6 )
=> ( ! [I3: C] :
( ( member @ C @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F2 @ I3 ) ) ) )
=> ( 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_5085_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F3: B > A,A5: A,Xs: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( compow @ ( A > A ) @ N3 @ ( times_times @ A @ A5 ) @ ( F3 @ ( nth @ B @ Xs @ N3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_5086_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat )
@ ^ [X: nat,Y: nat] : ( ord_less_eq @ nat @ Y @ X )
@ ^ [X: nat,Y: nat] : ( ord_less @ nat @ Y @ X ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_5087_bit_Oabstract__boolean__algebra__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( boolea2506097494486148201lgebra @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.abstract_boolean_algebra_axioms
thf(fact_5088_Suc__funpow,axiom,
! [N: nat] :
( ( compow @ ( nat > nat ) @ N @ suc )
= ( plus_plus @ nat @ N ) ) ).
% Suc_funpow
thf(fact_5089_funpow__0,axiom,
! [A: $tType,F2: A > A,X3: A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F2 @ X3 )
= X3 ) ).
% funpow_0
thf(fact_5090_comp__funpow,axiom,
! [B: $tType,A: $tType,N: nat,F2: A > A] :
( ( compow @ ( ( B > A ) > B > A ) @ N @ ( comp @ A @ A @ B @ F2 ) )
= ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N @ F2 ) ) ) ).
% comp_funpow
thf(fact_5091_funpow__mult,axiom,
! [A: $tType,N: nat,M: nat,F2: A > A] :
( ( compow @ ( A > A ) @ N @ ( compow @ ( A > A ) @ M @ F2 ) )
= ( compow @ ( A > A ) @ ( times_times @ nat @ M @ N ) @ F2 ) ) ).
% funpow_mult
thf(fact_5092_funpow__swap1,axiom,
! [A: $tType,F2: A > A,N: nat,X3: A] :
( ( F2 @ ( compow @ ( A > A ) @ N @ F2 @ X3 ) )
= ( compow @ ( A > A ) @ N @ F2 @ ( F2 @ X3 ) ) ) ).
% funpow_swap1
thf(fact_5093_bij__betw__funpow,axiom,
! [A: $tType,F2: A > A,S2: set @ A,N: nat] :
( ( bij_betw @ A @ A @ F2 @ S2 @ S2 )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F2 ) @ S2 @ S2 ) ) ).
% bij_betw_funpow
thf(fact_5094_funpow__times__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [F2: A > nat,X3: A] :
( ( compow @ ( A > A ) @ ( F2 @ X3 ) @ ( times_times @ A @ X3 ) )
= ( times_times @ A @ ( power_power @ A @ X3 @ ( F2 @ X3 ) ) ) ) ) ).
% funpow_times_power
thf(fact_5095_funpow_Osimps_I2_J,axiom,
! [A: $tType,N: nat,F2: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N ) @ F2 )
= ( comp @ A @ A @ A @ F2 @ ( compow @ ( A > A ) @ N @ F2 ) ) ) ).
% funpow.simps(2)
thf(fact_5096_funpow__Suc__right,axiom,
! [A: $tType,N: nat,F2: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N ) @ F2 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F2 ) @ F2 ) ) ).
% funpow_Suc_right
thf(fact_5097_funpow__add,axiom,
! [A: $tType,M: nat,N: nat,F2: A > A] :
( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M @ N ) @ F2 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M @ F2 ) @ ( compow @ ( A > A ) @ N @ F2 ) ) ) ).
% funpow_add
thf(fact_5098_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_5099_numeral__add__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num,A2: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ K ) @ A2 )
= ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A2 ) ) ) ).
% numeral_add_unfold_funpow
thf(fact_5100_of__nat__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N3: nat] : ( compow @ ( A > A ) @ N3 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_def
thf(fact_5101_infinite__int__iff__infinite__nat__abs,axiom,
! [S2: set @ int] :
( ( ~ ( finite_finite @ int @ S2 ) )
= ( ~ ( finite_finite @ nat @ ( image @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ S2 ) ) ) ) ).
% infinite_int_iff_infinite_nat_abs
thf(fact_5102_numeral__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( numeral_numeral @ A )
= ( ^ [K3: num] : ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K3 ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% numeral_unfold_funpow
thf(fact_5103_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_5104_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_5105_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_5106_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_5107_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_5108_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_5109_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_5110_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_5111_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_5112_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_5113_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_5114_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_5115_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_5116_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_5117_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_5118_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_5119_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_5120_relpowp__fun__conv,axiom,
! [A: $tType] :
( ( compow @ ( A > A > $o ) )
= ( ^ [N3: nat,P3: A > A > $o,X: A,Y: A] :
? [F3: nat > A] :
( ( ( F3 @ ( zero_zero @ nat ) )
= X )
& ( ( F3 @ N3 )
= Y )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N3 )
=> ( P3 @ ( F3 @ I4 ) @ ( F3 @ ( suc @ I4 ) ) ) ) ) ) ) ).
% relpowp_fun_conv
thf(fact_5121_Nat_Ofunpow__code__def,axiom,
! [A: $tType] :
( ( funpow @ A )
= ( compow @ ( A > A ) ) ) ).
% Nat.funpow_code_def
thf(fact_5122_relpowp__0__I,axiom,
! [A: $tType,P: A > A > $o,X3: A] : ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X3 @ X3 ) ).
% relpowp_0_I
thf(fact_5123_relpowp__0__E,axiom,
! [A: $tType,P: A > A > $o,X3: A,Y3: A] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X3 @ Y3 )
=> ( X3 = Y3 ) ) ).
% relpowp_0_E
thf(fact_5124_relpowp_Osimps_I1_J,axiom,
! [A: $tType,R3: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ R3 )
= ( ^ [Y6: A,Z2: A] : Y6 = Z2 ) ) ).
% relpowp.simps(1)
thf(fact_5125_relpowp__E,axiom,
! [A: $tType,N: nat,P: A > A > $o,X3: A,Z: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X3 @ Z )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X3 != Z ) )
=> ~ ! [Y4: A,M4: nat] :
( ( N
= ( suc @ M4 ) )
=> ( ( compow @ ( A > A > $o ) @ M4 @ P @ X3 @ Y4 )
=> ~ ( P @ Y4 @ Z ) ) ) ) ) ).
% relpowp_E
thf(fact_5126_relpowp__E2,axiom,
! [A: $tType,N: nat,P: A > A > $o,X3: A,Z: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X3 @ Z )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X3 != Z ) )
=> ~ ! [Y4: A,M4: nat] :
( ( N
= ( suc @ M4 ) )
=> ( ( P @ X3 @ Y4 )
=> ~ ( compow @ ( A > A > $o ) @ M4 @ P @ Y4 @ Z ) ) ) ) ) ).
% relpowp_E2
thf(fact_5127_relpowp__bot,axiom,
! [A: $tType,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( compow @ ( A > A > $o ) @ N @ ( bot_bot @ ( A > A > $o ) ) )
= ( bot_bot @ ( A > A > $o ) ) ) ) ).
% relpowp_bot
thf(fact_5128_measure__function__int,axiom,
fun_is_measure @ int @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) ).
% measure_function_int
thf(fact_5129_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S2: set @ A,F2: A > B] :
( ( finite_finite @ 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_5130_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S2: set @ A,Y3: A,F2: A > B] :
( ( finite_finite @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ Y3 @ S2 )
=> ( ord_less_eq @ B @ ( F2 @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S2 ) ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% arg_min_least
thf(fact_5131_measure__fst,axiom,
! [B: $tType,A: $tType,F2: A > nat] :
( ( fun_is_measure @ A @ F2 )
=> ( fun_is_measure @ ( product_prod @ A @ B )
@ ^ [P5: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ P5 ) ) ) ) ).
% measure_fst
thf(fact_5132_measure__snd,axiom,
! [A: $tType,B: $tType,F2: A > nat] :
( ( fun_is_measure @ A @ F2 )
=> ( fun_is_measure @ ( product_prod @ B @ A )
@ ^ [P5: product_prod @ B @ A] : ( F2 @ ( product_snd @ B @ A @ P5 ) ) ) ) ).
% measure_snd
thf(fact_5133_bot2E,axiom,
! [A: $tType,B: $tType,X3: A,Y3: B] :
~ ( bot_bot @ ( A > B > $o ) @ X3 @ Y3 ) ).
% bot2E
thf(fact_5134_is__measure__trivial,axiom,
! [A: $tType,F2: A > nat] : ( fun_is_measure @ A @ F2 ) ).
% is_measure_trivial
thf(fact_5135_is__measure_Osimps,axiom,
! [A: $tType] :
( ( fun_is_measure @ A )
= ( ^ [A5: A > nat] :
? [X9: A > nat] :
( ^ [Y6: A > nat,Z2: A > nat] : Y6 = Z2
@ A5
@ X9 ) ) ) ).
% is_measure.simps
thf(fact_5136_measure__size,axiom,
! [A: $tType] :
( ( size @ A )
=> ( fun_is_measure @ A @ ( size_size @ A ) ) ) ).
% measure_size
thf(fact_5137_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S2: set @ A,F2: A > B] :
( ( finite_finite @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S2 ) @ S2 ) ) ) ) ).
% arg_min_if_finite(1)
thf(fact_5138_times__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( times_times @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ U2 ) @ ( times_times @ nat @ Y @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X @ V5 ) @ ( times_times @ nat @ Y @ U2 ) ) ) )
@ Xa
@ X3 ) ) ) ).
% times_int.abs_eq
thf(fact_5139_eq__numeral__iff__iszero_I7_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X3 @ one2 ) ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_5140_eq__numeral__iff__iszero_I8_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y3: num] :
( ( ( one_one @ A )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y3 ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ Y3 ) ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_5141_iszero__neg__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: num] :
( ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W ) ) ) ) ).
% iszero_neg_numeral
thf(fact_5142_eq__iff__iszero__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [X: A,Y: A] : ( ring_1_iszero @ A @ ( minus_minus @ A @ X @ Y ) ) ) ) ) ).
% eq_iff_iszero_diff
thf(fact_5143_iszero__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ring_1_iszero @ A @ ( zero_zero @ A ) ) ) ).
% iszero_0
thf(fact_5144_iszero__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_iszero @ A )
= ( ^ [Z6: A] :
( Z6
= ( zero_zero @ A ) ) ) ) ) ).
% iszero_def
thf(fact_5145_int_Oabs__induct,axiom,
! [P: int > $o,X3: int] :
( ! [Y4: product_prod @ nat @ nat] : ( P @ ( abs_Integ @ Y4 ) )
=> ( P @ X3 ) ) ).
% int.abs_induct
thf(fact_5146_not__iszero__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( one_one @ A ) ) ) ).
% not_iszero_1
thf(fact_5147_not__iszero__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W: num] :
~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W ) ) ) ).
% not_iszero_numeral
thf(fact_5148_eq__Abs__Integ,axiom,
! [Z: int] :
~ ! [X4: nat,Y4: nat] :
( Z
!= ( abs_Integ @ ( product_Pair @ nat @ nat @ X4 @ Y4 ) ) ) ).
% eq_Abs_Integ
thf(fact_5149_nat_Oabs__eq,axiom,
! [X3: product_prod @ nat @ nat] :
( ( nat2 @ ( abs_Integ @ X3 ) )
= ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ X3 ) ) ).
% nat.abs_eq
thf(fact_5150_eq__numeral__iff__iszero_I9_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: num] :
( ( ( numeral_numeral @ A @ X3 )
= ( zero_zero @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X3 ) ) ) ) ).
% eq_numeral_iff_iszero(9)
thf(fact_5151_eq__numeral__iff__iszero_I10_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y3: num] :
( ( ( zero_zero @ A )
= ( numeral_numeral @ A @ Y3 ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(10)
thf(fact_5152_not__iszero__Numeral1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ one2 ) ) ) ).
% not_iszero_Numeral1
thf(fact_5153_not__iszero__neg__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_iszero_neg_1
thf(fact_5154_eq__numeral__iff__iszero_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: num,Y3: num] :
( ( ( numeral_numeral @ A @ X3 )
= ( numeral_numeral @ A @ Y3 ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X3 @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(1)
thf(fact_5155_zero__int__def,axiom,
( ( zero_zero @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% zero_int_def
thf(fact_5156_int__def,axiom,
( ( semiring_1_of_nat @ int )
= ( ^ [N3: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N3 @ ( zero_zero @ nat ) ) ) ) ) ).
% int_def
thf(fact_5157_eq__numeral__iff__iszero_I12_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y3: num] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y3 ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_5158_eq__numeral__iff__iszero_I11_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) )
= ( zero_zero @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X3 ) ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_5159_not__iszero__neg__Numeral1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_5160_eq__numeral__iff__iszero_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: num,Y3: num] :
( ( ( numeral_numeral @ A @ X3 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y3 ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X3 @ Y3 ) ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_5161_eq__numeral__iff__iszero_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: num,Y3: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) )
= ( numeral_numeral @ A @ Y3 ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X3 @ Y3 ) ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_5162_eq__numeral__iff__iszero_I4_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: num,Y3: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y3 ) ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ Y3 @ X3 ) ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_5163_uminus__int_Oabs__eq,axiom,
! [X3: product_prod @ nat @ nat] :
( ( uminus_uminus @ int @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X: nat,Y: nat] : ( product_Pair @ nat @ nat @ Y @ X )
@ X3 ) ) ) ).
% uminus_int.abs_eq
thf(fact_5164_one__int__def,axiom,
( ( one_one @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% one_int_def
thf(fact_5165_of__int_Oabs__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: product_prod @ nat @ nat] :
( ( ring_1_of_int @ A @ ( abs_Integ @ X3 ) )
= ( product_case_prod @ nat @ nat @ A
@ ^ [I4: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J3 ) )
@ X3 ) ) ) ).
% of_int.abs_eq
thf(fact_5166_less__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( ord_less @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X3 ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y ) ) )
@ Xa
@ X3 ) ) ).
% less_int.abs_eq
thf(fact_5167_less__eq__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( ord_less_eq @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X3 ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y ) ) )
@ Xa
@ X3 ) ) ).
% less_eq_int.abs_eq
thf(fact_5168_eq__numeral__iff__iszero_I5_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: num] :
( ( ( numeral_numeral @ A @ X3 )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X3 @ one2 ) ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_5169_eq__numeral__iff__iszero_I6_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y3: num] :
( ( ( one_one @ A )
= ( numeral_numeral @ A @ Y3 ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ one2 @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_5170_plus__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( plus_plus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ U2 ) @ ( plus_plus @ nat @ Y @ V5 ) ) )
@ Xa
@ X3 ) ) ) ).
% plus_int.abs_eq
thf(fact_5171_minus__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( minus_minus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ Y @ U2 ) ) )
@ Xa
@ X3 ) ) ) ).
% minus_int.abs_eq
thf(fact_5172_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_5173_Gcd__remove0__nat,axiom,
! [M7: set @ nat] :
( ( finite_finite @ 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_5174_pow_Osimps_I3_J,axiom,
! [X3: num,Y3: num] :
( ( pow @ X3 @ ( bit1 @ Y3 ) )
= ( times_times @ num @ ( sqr @ ( pow @ X3 @ Y3 ) ) @ X3 ) ) ).
% pow.simps(3)
thf(fact_5175_num__of__nat__numeral__eq,axiom,
! [Q3: num] :
( ( num_of_nat @ ( numeral_numeral @ nat @ Q3 ) )
= Q3 ) ).
% num_of_nat_numeral_eq
thf(fact_5176_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_5177_Gcd__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( ( gcd_Gcd @ A @ A4 )
= ( zero_zero @ A ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Gcd_0_iff
thf(fact_5178_sqr_Osimps_I2_J,axiom,
! [N: num] :
( ( sqr @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% sqr.simps(2)
thf(fact_5179_sqr_Osimps_I1_J,axiom,
( ( sqr @ one2 )
= one2 ) ).
% sqr.simps(1)
thf(fact_5180_sqr__conv__mult,axiom,
( sqr
= ( ^ [X: num] : ( times_times @ num @ X @ X ) ) ) ).
% sqr_conv_mult
thf(fact_5181_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ ( zero_zero @ nat ) )
= one2 ) ).
% num_of_nat.simps(1)
thf(fact_5182_numeral__sqr,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num] :
( ( numeral_numeral @ A @ ( sqr @ K ) )
= ( times_times @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ A @ K ) ) ) ) ).
% numeral_sqr
thf(fact_5183_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_5184_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_5185_pow_Osimps_I2_J,axiom,
! [X3: num,Y3: num] :
( ( pow @ X3 @ ( bit0 @ Y3 ) )
= ( sqr @ ( pow @ X3 @ Y3 ) ) ) ).
% pow.simps(2)
thf(fact_5186_num__of__integer_Orep__eq,axiom,
( code_num_of_integer
= ( ^ [X: code_integer] : ( num_of_nat @ ( nat2 @ ( code_int_of_integer @ X ) ) ) ) ) ).
% num_of_integer.rep_eq
thf(fact_5187_num__of__integer_Oabs__eq,axiom,
! [X3: int] :
( ( code_num_of_integer @ ( code_integer_of_int @ X3 ) )
= ( num_of_nat @ ( nat2 @ X3 ) ) ) ).
% num_of_integer.abs_eq
thf(fact_5188_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_5189_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_5190_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_5191_sqr_Osimps_I3_J,axiom,
! [N: num] :
( ( sqr @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ ( plus_plus @ num @ ( sqr @ N ) @ N ) ) ) ) ).
% sqr.simps(3)
thf(fact_5192_Gcd__int__def,axiom,
( ( gcd_Gcd @ int )
= ( ^ [K6: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ ( image @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K6 ) ) ) ) ) ).
% Gcd_int_def
thf(fact_5193_less__eq__int_Orep__eq,axiom,
( ( ord_less_eq @ int )
= ( ^ [X: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y: nat,Z6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y @ V5 ) @ ( plus_plus @ nat @ U2 @ Z6 ) ) )
@ ( rep_Integ @ X )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_5194_less__int_Orep__eq,axiom,
( ( ord_less @ int )
= ( ^ [X: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y: nat,Z6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y @ V5 ) @ ( plus_plus @ nat @ U2 @ Z6 ) ) )
@ ( rep_Integ @ X )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_5195_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_5196_Gcd__nat__abs__eq,axiom,
! [K5: set @ int] :
( ( gcd_Gcd @ nat
@ ( image @ int @ nat
@ ^ [K3: int] : ( nat2 @ ( abs_abs @ int @ K3 ) )
@ K5 ) )
= ( nat2 @ ( gcd_Gcd @ int @ K5 ) ) ) ).
% Gcd_nat_abs_eq
thf(fact_5197_Gcd__int__greater__eq__0,axiom,
! [K5: set @ int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_Gcd @ int @ K5 ) ) ).
% Gcd_int_greater_eq_0
thf(fact_5198_nat_Orep__eq,axiom,
( nat2
= ( ^ [X: int] : ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ ( rep_Integ @ X ) ) ) ) ).
% nat.rep_eq
thf(fact_5199_of__int_Orep__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [X: int] :
( product_case_prod @ nat @ nat @ A
@ ^ [I4: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J3 ) )
@ ( rep_Integ @ X ) ) ) ) ) ).
% of_int.rep_eq
thf(fact_5200_semiring__char__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiri4206861660011772517g_char @ A )
= ( ^ [Uu4: itself @ A] :
( gcd_Gcd @ nat
@ ( collect @ nat
@ ^ [N3: nat] :
( ( semiring_1_of_nat @ A @ N3 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% semiring_char_def
thf(fact_5201_Gcd__eq__Max,axiom,
! [M7: set @ nat] :
( ( finite_finite @ 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
@ ^ [D4: nat] : ( dvd_dvd @ nat @ D4 @ M3 ) )
@ M7 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_5202_uminus__int__def,axiom,
( ( uminus_uminus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X: nat,Y: nat] : ( product_Pair @ nat @ nat @ Y @ X ) ) ) ) ).
% uminus_int_def
thf(fact_5203_cInf__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ Y3 @ X3 ) )
= Y3 ) ) ) ).
% cInf_atLeastAtMost
thf(fact_5204_Inf__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y3 ) )
= X3 ) ) ) ).
% Inf_atLeastAtMost
thf(fact_5205_cInf__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A] :
( ( complete_Inf_Inf @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% cInf_singleton
thf(fact_5206_cInf__atLeastLessThan,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less @ A @ Y3 @ X3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ Y3 @ X3 ) )
= Y3 ) ) ) ).
% cInf_atLeastLessThan
thf(fact_5207_Inf__atLeastLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X3 @ Y3 ) )
= X3 ) ) ) ).
% Inf_atLeastLessThan
thf(fact_5208_Inf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X3 @ Y3 ) )
= X3 ) ) ) ).
% Inf_greaterThanLessThan
thf(fact_5209_cInf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y3: A,X3: A] :
( ( ord_less @ A @ Y3 @ X3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ Y3 @ X3 ) )
= Y3 ) ) ) ).
% cInf_greaterThanLessThan
thf(fact_5210_cINF__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,C2: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X: B] : C2
@ A4 ) )
= C2 ) ) ) ).
% cINF_const
thf(fact_5211_cInf__eq__minimum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z: A,X8: set @ A] :
( ( member @ A @ Z @ X8 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less_eq @ A @ Z @ X4 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= Z ) ) ) ) ).
% cInf_eq_minimum
thf(fact_5212_cInf__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_top @ A ) )
=> ! [X8: set @ A,A2: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less_eq @ A @ A2 @ X4 ) )
=> ( ! [Y4: A] :
( ! [X6: A] :
( ( member @ A @ X6 @ X8 )
=> ( ord_less_eq @ A @ Y4 @ X6 ) )
=> ( ord_less_eq @ A @ Y4 @ A2 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= A2 ) ) ) ) ).
% cInf_eq
thf(fact_5213_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,Z: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less_eq @ A @ Z @ X4 ) )
=> ( ord_less_eq @ A @ Z @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ).
% cInf_greatest
thf(fact_5214_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less_eq @ A @ A2 @ X4 ) )
=> ( ! [Y4: A] :
( ! [X6: A] :
( ( member @ A @ X6 @ X8 )
=> ( ord_less_eq @ A @ Y4 @ X6 ) )
=> ( ord_less_eq @ A @ Y4 @ A2 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= A2 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_5215_cInf__le__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,X3: A] :
( ( finite_finite @ A @ X8 )
=> ( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X3 ) ) ) ) ).
% cInf_le_finite
thf(fact_5216_cInf__lessD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Z: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Z )
=> ? [X4: A] :
( ( member @ A @ X4 @ X8 )
& ( ord_less @ A @ X4 @ Z ) ) ) ) ) ).
% cInf_lessD
thf(fact_5217_finite__imp__less__Inf,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,X3: A,A2: A] :
( ( finite_finite @ A @ X8 )
=> ( ( member @ A @ X3 @ X8 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less @ A @ A2 @ X4 ) )
=> ( ord_less @ A @ A2 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ) ).
% finite_imp_less_Inf
thf(fact_5218_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,M: A,F2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ M @ ( F2 @ X4 ) ) )
=> ( ord_less_eq @ A @ M @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_5219_finite__less__Inf__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,A2: A] :
( ( finite_finite @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ A2 @ ( complete_Inf_Inf @ A @ X8 ) )
= ( ! [X: A] :
( ( member @ A @ X @ X8 )
=> ( ord_less @ A @ A2 @ X ) ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_5220_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,A2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X4 ) @ A2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S2 ) ) @ A2 ) ) ) ) ).
% cInf_abs_ge
thf(fact_5221_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,L: A,E2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X4 @ L ) ) @ E2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S2 ) @ L ) ) @ E2 ) ) ) ) ).
% cInf_asclose
thf(fact_5222_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A4: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) )
= ( bot_bot @ A ) )
= ( ! [X: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X )
=> ? [Y: B] :
( ( member @ B @ Y @ A4 )
& ( ord_less @ A @ ( F2 @ Y ) @ X ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_5223_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A] :
( ( ( complete_Inf_Inf @ A @ A4 )
= ( bot_bot @ A ) )
= ( ! [X: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X )
=> ? [Y: A] :
( ( member @ A @ Y @ A4 )
& ( ord_less @ A @ Y @ X ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_5224_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: set @ B,A4: set @ B,F2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ B6 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B6 ) ) ) ) ) ) ).
% INF_superset_mono
thf(fact_5225_Inf__nat__def1,axiom,
! [K5: set @ nat] :
( ( K5
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( complete_Inf_Inf @ nat @ K5 ) @ K5 ) ) ).
% Inf_nat_def1
thf(fact_5226_Inf__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,Z: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ Z @ X4 ) )
=> ( ord_less_eq @ A @ Z @ ( complete_Inf_Inf @ A @ A4 ) ) ) ) ).
% Inf_greatest
thf(fact_5227_le__Inf__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: A,A4: set @ A] :
( ( ord_less_eq @ A @ B2 @ ( complete_Inf_Inf @ A @ A4 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ B2 @ X ) ) ) ) ) ).
% le_Inf_iff
thf(fact_5228_Inf__lower2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A4: set @ A,V2: A] :
( ( member @ A @ U @ A4 )
=> ( ( ord_less_eq @ A @ U @ V2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ V2 ) ) ) ) ).
% Inf_lower2
thf(fact_5229_Inf__lower,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A,A4: set @ A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ X3 ) ) ) ).
% Inf_lower
thf(fact_5230_Inf__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: set @ A,A4: set @ A] :
( ! [B5: A] :
( ( member @ A @ B5 @ B6 )
=> ? [X6: A] :
( ( member @ A @ X6 @ A4 )
& ( ord_less_eq @ A @ X6 @ B5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ).
% Inf_mono
thf(fact_5231_Inf__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,X3: A] :
( ! [I3: A] :
( ( member @ A @ I3 @ A4 )
=> ( ord_less_eq @ A @ X3 @ I3 ) )
=> ( ! [Y4: A] :
( ! [I2: A] :
( ( member @ A @ I2 @ A4 )
=> ( ord_less_eq @ A @ Y4 @ I2 ) )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ( complete_Inf_Inf @ A @ A4 )
= X3 ) ) ) ) ).
% Inf_eqI
thf(fact_5232_Inf__less__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [S2: set @ A,A2: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S2 ) @ A2 )
= ( ? [X: A] :
( ( member @ A @ X @ S2 )
& ( ord_less @ A @ X @ A2 ) ) ) ) ) ).
% Inf_less_iff
thf(fact_5233_Inf__le__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A,X3: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ X3 )
= ( ! [Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ? [X: A] :
( ( member @ A @ X @ A4 )
& ( ord_less @ A @ X @ Y ) ) ) ) ) ) ).
% Inf_le_iff
thf(fact_5234_INF__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B6: set @ C,G: C > A,F2: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ? [X6: C] :
( ( member @ C @ X6 @ B6 )
& ( ord_less_eq @ A @ ( G @ X6 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B6 )
=> ? [X6: B] :
( ( member @ B @ X6 @ A4 )
& ( ord_less_eq @ A @ ( F2 @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) )
= ( complete_Inf_Inf @ A @ ( image @ C @ A @ G @ B6 ) ) ) ) ) ) ).
% INF_eq
thf(fact_5235_Inf__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,U: A] :
( ! [V4: A] :
( ( member @ A @ V4 @ A4 )
=> ( ord_less_eq @ A @ V4 @ U ) )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ U ) ) ) ) ).
% Inf_less_eq
thf(fact_5236_Inf__superset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ).
% Inf_superset_mono
thf(fact_5237_INF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,U: A,F2: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ord_less_eq @ A @ U @ ( F2 @ I3 ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ) ).
% INF_greatest
thf(fact_5238_le__INF__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,F2: B > A,A4: set @ B] :
( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A4 )
=> ( ord_less_eq @ A @ U @ ( F2 @ X ) ) ) ) ) ) ).
% le_INF_iff
thf(fact_5239_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A4: set @ B,F2: B > A,U: A] :
( ( member @ B @ I @ A4 )
=> ( ( ord_less_eq @ A @ ( F2 @ I ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ U ) ) ) ) ).
% INF_lower2
thf(fact_5240_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,G: B > A,A4: set @ B] :
( ! [X4: B] : ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ A4 ) ) ) ) ) ).
% INF_mono'
thf(fact_5241_INF__lower,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A4: set @ B,F2: B > A] :
( ( member @ B @ I @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( F2 @ I ) ) ) ) ).
% INF_lower
thf(fact_5242_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: set @ B,A4: set @ C,F2: C > A,G: B > A] :
( ! [M4: B] :
( ( member @ B @ M4 @ B6 )
=> ? [X6: C] :
( ( member @ C @ X6 @ A4 )
& ( ord_less_eq @ A @ ( F2 @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B6 ) ) ) ) ) ).
% INF_mono
thf(fact_5243_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,X3: A,F2: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ord_less_eq @ A @ X3 @ ( F2 @ I3 ) ) )
=> ( ! [Y4: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less_eq @ A @ Y4 @ ( F2 @ I2 ) ) )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) )
= X3 ) ) ) ) ).
% INF_eqI
thf(fact_5244_INF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A4: set @ B,A2: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ A2 )
= ( ? [X: B] :
( ( member @ B @ X @ A4 )
& ( ord_less @ A @ ( F2 @ X ) @ A2 ) ) ) ) ) ).
% INF_less_iff
thf(fact_5245_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y3: A,F2: B > A,A4: set @ B,I: B] :
( ( ord_less @ A @ Y3 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ Y3 @ ( F2 @ I ) ) ) ) ) ).
% less_INF_D
thf(fact_5246_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A4: set @ B,X3: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ X3 )
= ( ! [Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ? [X: B] :
( ( member @ B @ X @ A4 )
& ( ord_less @ A @ ( F2 @ X ) @ Y ) ) ) ) ) ) ).
% INF_le_iff
thf(fact_5247_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F2: B > A,C2: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ C2 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ I6 ) )
= C2 )
= ( ! [X: B] :
( ( member @ B @ X @ I6 )
=> ( ( F2 @ X )
= C2 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_5248_times__int__def,axiom,
( ( times_times @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ U2 ) @ ( times_times @ nat @ Y @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X @ V5 ) @ ( times_times @ nat @ Y @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_5249_card__UNION,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( finite_finite @ ( set @ A ) @ A4 )
=> ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A4 )
=> ( finite_finite @ A @ X4 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) )
= ( nat2
@ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int
@ ^ [I7: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I7 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I7 ) ) ) )
@ ( collect @ ( set @ ( set @ A ) )
@ ^ [I7: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I7 @ A4 )
& ( I7
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% card_UNION
thf(fact_5250_minus__int__def,axiom,
( ( minus_minus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ Y @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_5251_cSup__lessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A )
& ( no_bot @ A ) )
=> ! [X3: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_lessThan @ A @ X3 ) )
= X3 ) ) ).
% cSup_lessThan
thf(fact_5252_cSup__atMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_atMost @ A @ X3 ) )
= X3 ) ) ).
% cSup_atMost
thf(fact_5253_cSup__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ Y3 @ X3 ) )
= X3 ) ) ) ).
% cSup_atLeastAtMost
thf(fact_5254_Sup__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y3 ) )
= Y3 ) ) ) ).
% Sup_atLeastAtMost
thf(fact_5255_cSup__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% cSup_singleton
thf(fact_5256_cSup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y3: A,X3: A] :
( ( ord_less @ A @ Y3 @ X3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ Y3 @ X3 ) )
= X3 ) ) ) ).
% cSup_atLeastLessThan
thf(fact_5257_Sup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X3 @ Y3 ) )
= Y3 ) ) ) ).
% Sup_atLeastLessThan
thf(fact_5258_Sup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X3 @ Y3 ) )
= Y3 ) ) ) ).
% Sup_greaterThanLessThan
thf(fact_5259_cSup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y3: A,X3: A] :
( ( ord_less @ A @ Y3 @ X3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ Y3 @ X3 ) )
= X3 ) ) ) ).
% cSup_greaterThanLessThan
thf(fact_5260_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,C2: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X: B] : C2
@ A4 ) )
= C2 ) ) ) ).
% cSUP_const
thf(fact_5261_less__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A2: A,S2: set @ A] :
( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ S2 ) )
= ( ? [X: A] :
( ( member @ A @ X @ S2 )
& ( ord_less @ A @ A2 @ X ) ) ) ) ) ).
% less_Sup_iff
thf(fact_5262_Sup__upper2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A4: set @ A,V2: A] :
( ( member @ A @ U @ A4 )
=> ( ( ord_less_eq @ A @ V2 @ U )
=> ( ord_less_eq @ A @ V2 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% Sup_upper2
thf(fact_5263_Sup__le__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B2: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ B2 )
= ( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ X @ B2 ) ) ) ) ) ).
% Sup_le_iff
thf(fact_5264_Sup__upper,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A,A4: set @ A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ).
% Sup_upper
thf(fact_5265_Sup__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,Z: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ Z ) ) ) ).
% Sup_least
thf(fact_5266_Sup__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ? [X6: A] :
( ( member @ A @ X6 @ B6 )
& ( ord_less_eq @ A @ A6 @ X6 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ).
% Sup_mono
thf(fact_5267_Sup__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,X3: A] :
( ! [Y4: A] :
( ( member @ A @ Y4 @ A4 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ! [Y4: A] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ A4 )
=> ( ord_less_eq @ A @ Z4 @ Y4 ) )
=> ( ord_less_eq @ A @ X3 @ Y4 ) )
=> ( ( complete_Sup_Sup @ A @ A4 )
= X3 ) ) ) ) ).
% Sup_eqI
thf(fact_5268_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z: A,X8: set @ A] :
( ( member @ A @ Z @ X8 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= Z ) ) ) ) ).
% cSup_eq_maximum
thf(fact_5269_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_bot @ A ) )
=> ! [X8: set @ A,A2: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less_eq @ A @ X4 @ A2 ) )
=> ( ! [Y4: A] :
( ! [X6: A] :
( ( member @ A @ X6 @ X8 )
=> ( ord_less_eq @ A @ X6 @ Y4 ) )
=> ( ord_less_eq @ A @ A2 @ Y4 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= A2 ) ) ) ) ).
% cSup_eq
thf(fact_5270_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X3: A,A4: set @ A] :
( ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ A4 ) )
= ( ! [Y: A] :
( ( ord_less @ A @ Y @ X3 )
=> ? [X: A] :
( ( member @ A @ X @ A4 )
& ( ord_less @ A @ Y @ X ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_5271_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B6: set @ C,F2: B > A,G: C > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ? [X6: C] :
( ( member @ C @ X6 @ B6 )
& ( ord_less_eq @ A @ ( F2 @ I3 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B6 )
=> ? [X6: B] :
( ( member @ B @ X6 @ A4 )
& ( ord_less_eq @ A @ ( G @ J2 ) @ ( F2 @ X6 ) ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) )
= ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B6 ) ) ) ) ) ) ).
% SUP_eq
thf(fact_5272_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,Z: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X8 ) @ Z ) ) ) ) ).
% cSup_least
thf(fact_5273_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less_eq @ A @ X4 @ A2 ) )
=> ( ! [Y4: A] :
( ! [X6: A] :
( ( member @ A @ X6 @ X8 )
=> ( ord_less_eq @ A @ X6 @ Y4 ) )
=> ( ord_less_eq @ A @ A2 @ Y4 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= A2 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_5274_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,U: A] :
( ! [V4: A] :
( ( member @ A @ V4 @ A4 )
=> ( ord_less_eq @ A @ U @ V4 ) )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_5275_le__cSup__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,X3: A] :
( ( finite_finite @ A @ X8 )
=> ( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ).
% le_cSup_finite
thf(fact_5276_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ).
% Sup_subset_mono
thf(fact_5277_less__cSupE,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [Y3: A,X8: set @ A] :
( ( ord_less @ A @ Y3 @ ( complete_Sup_Sup @ A @ X8 ) )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ~ ( ord_less @ A @ Y3 @ X4 ) ) ) ) ) ).
% less_cSupE
thf(fact_5278_less__cSupD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Z: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z @ ( complete_Sup_Sup @ A @ X8 ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ X8 )
& ( ord_less @ A @ Z @ X4 ) ) ) ) ) ).
% less_cSupD
thf(fact_5279_finite__imp__Sup__less,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,X3: A,A2: A] :
( ( finite_finite @ A @ X8 )
=> ( ( member @ A @ X3 @ X8 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less @ A @ X4 @ A2 ) )
=> ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A2 ) ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_5280_card__Union__le__sum__card,axiom,
! [A: $tType,U3: set @ ( set @ A )] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U3 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U3 ) ) ).
% card_Union_le_sum_card
thf(fact_5281_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A4: set @ B,U: A,F2: B > A] :
( ( member @ B @ I @ A4 )
=> ( ( ord_less_eq @ A @ U @ ( F2 @ I ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_5282_SUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A4: set @ B,U: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ U )
= ( ! [X: B] :
( ( member @ B @ X @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X ) @ U ) ) ) ) ) ).
% SUP_le_iff
thf(fact_5283_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A4: set @ B,F2: B > A] :
( ( member @ B @ I @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ) ).
% SUP_upper
thf(fact_5284_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,G: B > A,A4: set @ B] :
( ! [X4: B] : ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A4 ) ) ) ) ) ).
% SUP_mono'
thf(fact_5285_SUP__least,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F2: B > A,U: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ U ) ) ) ).
% SUP_least
thf(fact_5286_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B6: set @ C,F2: B > A,G: C > A] :
( ! [N2: B] :
( ( member @ B @ N2 @ A4 )
=> ? [X6: C] :
( ( member @ C @ X6 @ B6 )
& ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B6 ) ) ) ) ) ).
% SUP_mono
thf(fact_5287_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F2: B > A,X3: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ X3 ) )
=> ( ! [Y4: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ Y4 ) )
=> ( ord_less_eq @ A @ X3 @ Y4 ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) )
= X3 ) ) ) ) ).
% SUP_eqI
thf(fact_5288_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A2: A,F2: B > A,A4: set @ B] :
( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) )
= ( ? [X: B] :
( ( member @ B @ X @ A4 )
& ( ord_less @ A @ A2 @ ( F2 @ X ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_5289_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A4: set @ B,Y3: A,I: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ Y3 )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ ( F2 @ I ) @ Y3 ) ) ) ) ).
% SUP_lessD
thf(fact_5290_le__SUP__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X3: A,F2: B > A,A4: set @ B] :
( ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) )
= ( ! [Y: A] :
( ( ord_less @ A @ Y @ X3 )
=> ? [X: B] :
( ( member @ B @ X @ A4 )
& ( ord_less @ A @ Y @ ( F2 @ X ) ) ) ) ) ) ) ).
% le_SUP_iff
thf(fact_5291_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,M7: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ M7 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ M7 ) ) ) ) ).
% cSUP_least
thf(fact_5292_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,C2: A,F2: B > A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( ord_less_eq @ A @ C2 @ ( F2 @ I3 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ I6 ) )
= C2 )
= ( ! [X: B] :
( ( member @ B @ X @ I6 )
=> ( ( F2 @ X )
= C2 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_5293_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,A2: A] :
( ( finite_finite @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A2 )
= ( ! [X: A] :
( ( member @ A @ X @ X8 )
=> ( ord_less @ A @ X @ A2 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_5294_Inf__le__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ).
% Inf_le_Sup
thf(fact_5295_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,A2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X4 ) @ A2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S2 ) ) @ A2 ) ) ) ) ).
% cSup_abs_le
thf(fact_5296_sum_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B6: set @ ( set @ B ),G: B > A] :
( ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ B6 )
=> ( finite_finite @ B @ X4 ) )
=> ( ! [A14: set @ B] :
( ( member @ ( set @ B ) @ A14 @ B6 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B6 )
=> ( ( A14 != A25 )
=> ! [X4: B] :
( ( member @ B @ X4 @ A14 )
=> ( ( member @ B @ X4 @ A25 )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B6 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ B6 ) ) ) ) ) ).
% sum.Union_comp
thf(fact_5297_Max__Sup,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% Max_Sup
thf(fact_5298_cSup__eq__Max,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A] :
( ( finite_finite @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= ( lattic643756798349783984er_Max @ A @ X8 ) ) ) ) ) ).
% cSup_eq_Max
thf(fact_5299_card__Union__le__sum__card__weak,axiom,
! [A: $tType,U3: set @ ( set @ A )] :
( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ U3 )
=> ( finite_finite @ A @ X4 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U3 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U3 ) ) ) ).
% card_Union_le_sum_card_weak
thf(fact_5300_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B6: set @ B,F2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A4 @ B6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B6 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_5301_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,L: A,E2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X4 @ L ) ) @ E2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S2 ) @ L ) ) @ E2 ) ) ) ) ).
% cSup_asclose
thf(fact_5302_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ) ).
% INF_le_SUP
thf(fact_5303_Sup__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S2: set @ A,X3: A] :
( ( finite_finite @ A @ S2 )
=> ( ( ( S2
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ X3 @ S2 ) )
= X3 ) )
& ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ X3 @ S2 ) )
= ( ord_max @ A @ X3 @ ( complete_Sup_Sup @ A @ S2 ) ) ) ) ) ) ) ).
% Sup_insert_finite
thf(fact_5304_card__UN__le,axiom,
! [B: $tType,A: $tType,I6: set @ A,A4: A > ( set @ B )] :
( ( finite_finite @ A @ I6 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A4 @ I6 ) ) )
@ ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] : ( finite_card @ B @ ( A4 @ I4 ) )
@ I6 ) ) ) ).
% card_UN_le
thf(fact_5305_plus__int__def,axiom,
( ( plus_plus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ U2 ) @ ( plus_plus @ nat @ Y @ V5 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_5306_card__Pow,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( finite_card @ ( set @ A ) @ ( pow2 @ A @ A4 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A4 ) ) ) ) ).
% card_Pow
thf(fact_5307_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_5308_UN__finite2__subset,axiom,
! [A: $tType,A4: nat > ( set @ A ),B6: nat > ( set @ A ),K: nat] :
( ! [N2: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B6 @ ( 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 ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B6 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_5309_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_5310_finite__option__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
= ( finite_finite @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_option_UNIV
thf(fact_5311_max__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X3: A] :
( ( ord_max @ A @ ( top_top @ A ) @ X3 )
= ( top_top @ A ) ) ) ).
% max_top
thf(fact_5312_max__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X3: A] :
( ( ord_max @ A @ X3 @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% max_top2
thf(fact_5313_Sup__nat__empty,axiom,
( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% Sup_nat_empty
thf(fact_5314_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A] :
( ( ( complete_Sup_Sup @ A @ A4 )
= ( top_top @ A ) )
= ( ! [X: A] :
( ( ord_less @ A @ X @ ( top_top @ A ) )
=> ? [Y: A] :
( ( member @ A @ Y @ A4 )
& ( ord_less @ A @ X @ Y ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_5315_surj__fn,axiom,
! [A: $tType,F2: A > A,N: nat] :
( ( ( image @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( compow @ ( A > A ) @ N @ F2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_fn
thf(fact_5316_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A4: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) )
= ( top_top @ A ) )
= ( ! [X: A] :
( ( ord_less @ A @ X @ ( top_top @ A ) )
=> ? [Y: B] :
( ( member @ B @ Y @ A4 )
& ( ord_less @ A @ X @ ( F2 @ Y ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_5317_Inf__atMostLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( top_top @ A ) @ X3 )
=> ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X3 ) )
= ( bot_bot @ A ) ) ) ) ).
% Inf_atMostLessThan
thf(fact_5318_Inf__real__def,axiom,
( ( complete_Inf_Inf @ real )
= ( ^ [X9: set @ real] : ( uminus_uminus @ real @ ( complete_Sup_Sup @ real @ ( image @ real @ real @ ( uminus_uminus @ real ) @ X9 ) ) ) ) ) ).
% Inf_real_def
thf(fact_5319_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_5320_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_5321_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_5322_UNIV__option__conv,axiom,
! [A: $tType] :
( ( top_top @ ( set @ ( option @ A ) ) )
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% UNIV_option_conv
thf(fact_5323_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ).
% top.extremum_strict
thf(fact_5324_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_5325_bij__fn,axiom,
! [A: $tType,F2: A > A,N: nat] :
( ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F2 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_fn
thf(fact_5326_Inf__int__def,axiom,
( ( complete_Inf_Inf @ int )
= ( ^ [X9: set @ int] : ( uminus_uminus @ int @ ( complete_Sup_Sup @ int @ ( image @ int @ int @ ( uminus_uminus @ int ) @ X9 ) ) ) ) ) ).
% Inf_int_def
thf(fact_5327_suminf__eq__SUP__real,axiom,
! [X8: nat > real] :
( ( summable @ real @ X8 )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( X8 @ I3 ) )
=> ( ( suminf @ real @ X8 )
= ( complete_Sup_Sup @ real
@ ( image @ nat @ real
@ ^ [I4: nat] : ( groups7311177749621191930dd_sum @ nat @ real @ X8 @ ( set_ord_lessThan @ nat @ I4 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% suminf_eq_SUP_real
thf(fact_5328_Sup__nat__def,axiom,
( ( complete_Sup_Sup @ nat )
= ( ^ [X9: set @ nat] :
( if @ nat
@ ( X9
= ( bot_bot @ ( set @ nat ) ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat @ X9 ) ) ) ) ).
% Sup_nat_def
thf(fact_5329_finite__range__Some,axiom,
! [A: $tType] :
( ( finite_finite @ ( option @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( finite_finite @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_range_Some
thf(fact_5330_notin__range__Some,axiom,
! [A: $tType,X3: option @ A] :
( ( ~ ( member @ ( option @ A ) @ X3 @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) )
= ( X3
= ( none @ A ) ) ) ).
% notin_range_Some
thf(fact_5331_finite__UNIV__card__ge__0,axiom,
! [A: $tType] :
( ( finite_finite @ 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_5332_UN__UN__finite__eq,axiom,
! [A: $tType,A4: nat > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ nat @ ( set @ A )
@ ^ [N3: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
@ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UN_UN_finite_eq
thf(fact_5333_binomial__def,axiom,
( binomial
= ( ^ [N3: nat,K3: nat] :
( finite_card @ ( set @ nat )
@ ( collect @ ( set @ nat )
@ ^ [K6: set @ nat] :
( ( member @ ( set @ nat ) @ K6 @ ( pow2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
& ( ( finite_card @ nat @ K6 )
= K3 ) ) ) ) ) ) ).
% binomial_def
thf(fact_5334_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F2: B > A] :
( ( finite_finite @ 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_5335_UN__finite__subset,axiom,
! [A: $tType,A4: nat > ( set @ A ),C5: set @ A] :
( ! [N2: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) @ C5 ) ) ).
% UN_finite_subset
thf(fact_5336_UN__finite2__eq,axiom,
! [A: $tType,A4: nat > ( set @ A ),B6: nat > ( set @ A ),K: nat] :
( ! [N2: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N2 @ K ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B6 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_5337_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_5338_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_5339_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F2: nat > ( set @ A ),S2: set @ A] :
( ! [I3: nat] : ( ord_less_eq @ ( set @ A ) @ ( F2 @ I3 ) @ S2 )
=> ( ( finite_finite @ A @ S2 )
=> ( ? [N7: nat] :
( ! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ N7 )
=> ! [M4: nat] :
( ( ord_less_eq @ nat @ M4 @ N7 )
=> ( ( ord_less @ nat @ M4 @ N2 )
=> ( ord_less @ ( set @ A ) @ ( F2 @ M4 ) @ ( F2 @ N2 ) ) ) ) )
& ! [N2: nat] :
( ( ord_less_eq @ nat @ N7 @ N2 )
=> ( ( F2 @ N7 )
= ( 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_5340_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_5341_finite__fun__UNIVD1,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
=> ( ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
!= ( suc @ ( zero_zero @ nat ) ) )
=> ( finite_finite @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_fun_UNIVD1
thf(fact_5342_card__UNIV__bool,axiom,
( ( finite_card @ $o @ ( top_top @ ( set @ $o ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% card_UNIV_bool
thf(fact_5343_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_5344_infinite__UNIV__int,axiom,
~ ( finite_finite @ int @ ( top_top @ ( set @ int ) ) ) ).
% infinite_UNIV_int
thf(fact_5345_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_5346_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_5347_dependent__nat__choice,axiom,
! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
( ? [X_12: A] : ( P @ ( zero_zero @ nat ) @ X_12 )
=> ( ! [X4: A,N2: nat] :
( ( P @ N2 @ X4 )
=> ? [Y5: A] :
( ( P @ ( suc @ N2 ) @ Y5 )
& ( Q @ N2 @ X4 @ Y5 ) ) )
=> ? [F5: nat > A] :
! [N9: nat] :
( ( P @ N9 @ ( F5 @ N9 ) )
& ( Q @ N9 @ ( F5 @ N9 ) @ ( F5 @ ( suc @ N9 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_5348_root__def,axiom,
( root
= ( ^ [N3: nat,X: real] :
( if @ real
@ ( N3
= ( zero_zero @ nat ) )
@ ( zero_zero @ real )
@ ( the_inv_into @ real @ real @ ( top_top @ ( set @ real ) )
@ ^ [Y: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N3 ) )
@ X ) ) ) ) ).
% root_def
thf(fact_5349_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_5350_these__insert__Some,axiom,
! [A: $tType,X3: A,A4: set @ ( option @ A )] :
( ( these @ A @ ( insert @ ( option @ A ) @ ( some @ A @ X3 ) @ A4 ) )
= ( insert @ A @ X3 @ ( these @ A @ A4 ) ) ) ).
% these_insert_Some
thf(fact_5351_these__empty,axiom,
! [A: $tType] :
( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% these_empty
thf(fact_5352_these__image__Some__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( these @ A @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) )
= A4 ) ).
% these_image_Some_eq
thf(fact_5353_these__insert__None,axiom,
! [A: $tType,A4: set @ ( option @ A )] :
( ( these @ A @ ( insert @ ( option @ A ) @ ( none @ A ) @ A4 ) )
= ( these @ A @ A4 ) ) ).
% these_insert_None
thf(fact_5354_in__these__eq,axiom,
! [A: $tType,X3: A,A4: set @ ( option @ A )] :
( ( member @ A @ X3 @ ( these @ A @ A4 ) )
= ( member @ ( option @ A ) @ ( some @ A @ X3 ) @ A4 ) ) ).
% in_these_eq
thf(fact_5355_Option_Othese__def,axiom,
! [A: $tType] :
( ( these @ A )
= ( ^ [A9: set @ ( option @ A )] :
( image @ ( option @ A ) @ A @ ( the2 @ A )
@ ( collect @ ( option @ A )
@ ^ [X: option @ A] :
( ( member @ ( option @ A ) @ X @ A9 )
& ( X
!= ( none @ A ) ) ) ) ) ) ) ).
% Option.these_def
thf(fact_5356_these__not__empty__eq,axiom,
! [A: $tType,B6: set @ ( option @ A )] :
( ( ( these @ A @ B6 )
!= ( bot_bot @ ( set @ A ) ) )
= ( ( B6
!= ( bot_bot @ ( set @ ( option @ A ) ) ) )
& ( B6
!= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_not_empty_eq
thf(fact_5357_these__empty__eq,axiom,
! [A: $tType,B6: set @ ( option @ A )] :
( ( ( these @ A @ B6 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( B6
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( B6
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_empty_eq
thf(fact_5358_Some__image__these__eq,axiom,
! [A: $tType,A4: set @ ( option @ A )] :
( ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A4 ) )
= ( collect @ ( option @ A )
@ ^ [X: option @ A] :
( ( member @ ( option @ A ) @ X @ A4 )
& ( X
!= ( none @ A ) ) ) ) ) ).
% Some_image_these_eq
thf(fact_5359_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_5360_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_5361_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_5362_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_5363_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_5364_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_5365_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_5366_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_5367_char_Osize_I2_J,axiom,
! [X15: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_size @ char @ ( char2 @ X15 @ X2 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size(2)
thf(fact_5368_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_5369_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_5370_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_5371_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_5372_char_Osize__gen,axiom,
! [X15: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_char @ ( char2 @ X15 @ X2 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size_gen
thf(fact_5373_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_5374_DERIV__even__real__root,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N ) ) @ ( power_power @ real @ ( root @ N @ X3 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_5375_DERIV__real__root__generic,axiom,
! [N: nat,X3: real,D5: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X3
!= ( zero_zero @ real ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( D5
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X3 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ( D5
= ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X3 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( D5
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X3 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ D5 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_5376_DERIV__arctan__series,axiom,
! [X3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real
@ ^ [X10: real] :
( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X10 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) )
@ ( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( power_power @ real @ X3 @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_arctan_series
thf(fact_5377_DERIV__inverse_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X3: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ( F2 @ X3 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( inverse_inverse @ A @ ( F2 @ X ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X3 ) ) @ D5 ) @ ( inverse_inverse @ A @ ( F2 @ X3 ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% DERIV_inverse'
thf(fact_5378_DERIV__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X3: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ( G @ X3 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D5 @ ( G @ X3 ) ) @ ( times_times @ A @ ( F2 @ X3 ) @ E5 ) ) @ ( times_times @ A @ ( G @ X3 ) @ ( G @ X3 ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ).
% DERIV_divide
thf(fact_5379_DERIV__const,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [K: A,F4: filter @ A] :
( has_field_derivative @ A
@ ^ [X: A] : K
@ ( zero_zero @ A )
@ F4 ) ) ).
% DERIV_const
thf(fact_5380_has__real__derivative__neg__dec__left,axiom,
! [F2: real > real,L: real,X3: real,S2: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ S2 ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( minus_minus @ real @ X3 @ H5 ) @ S2 )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F2 @ X3 ) @ ( F2 @ ( minus_minus @ real @ X3 @ H5 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
thf(fact_5381_has__real__derivative__pos__inc__left,axiom,
! [F2: real > real,L: real,X3: real,S2: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ S2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( minus_minus @ real @ X3 @ H5 ) @ S2 )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X3 @ H5 ) ) @ ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
thf(fact_5382_has__real__derivative__pos__inc__right,axiom,
! [F2: real > real,L: real,X3: real,S2: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ S2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( plus_plus @ real @ X3 @ H5 ) @ S2 )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F2 @ X3 ) @ ( F2 @ ( plus_plus @ real @ X3 @ H5 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_5383_has__real__derivative__neg__dec__right,axiom,
! [F2: real > real,L: real,X3: real,S2: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ S2 ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( plus_plus @ real @ X3 @ H5 ) @ S2 )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F2 @ ( plus_plus @ real @ X3 @ H5 ) ) @ ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_5384_DERIV__isconst3,axiom,
! [A2: real,B2: real,X3: real,Y3: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( member @ real @ Y3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ! [X4: real] :
( ( member @ real @ X4 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( F2 @ X3 )
= ( F2 @ Y3 ) ) ) ) ) ) ).
% DERIV_isconst3
thf(fact_5385_DERIV__isconst__all,axiom,
! [F2: real > real,X3: real,Y3: real] :
( ! [X4: real] : ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( F2 @ X3 )
= ( F2 @ Y3 ) ) ) ).
% DERIV_isconst_all
thf(fact_5386_DERIV__neg__dec__right,axiom,
! [F2: real > real,L: real,X3: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F2 @ ( plus_plus @ real @ X3 @ H5 ) ) @ ( F2 @ X3 ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_5387_DERIV__pos__inc__right,axiom,
! [F2: real > real,L: real,X3: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F2 @ X3 ) @ ( F2 @ ( plus_plus @ real @ X3 @ H5 ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_5388_DERIV__pos__inc__left,axiom,
! [F2: real > real,L: real,X3: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X3 @ H5 ) ) @ ( F2 @ X3 ) ) ) ) ) ) ) ).
% DERIV_pos_inc_left
thf(fact_5389_DERIV__neg__dec__left,axiom,
! [F2: real > real,L: real,X3: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F2 @ X3 ) @ ( F2 @ ( minus_minus @ real @ X3 @ H5 ) ) ) ) ) ) ) ) ).
% DERIV_neg_dec_left
thf(fact_5390_DERIV__nonneg__imp__nondecreasing,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ord_less_eq @ real @ A2 @ X4 )
=> ( ( ord_less_eq @ real @ X4 @ B2 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y5 ) ) ) )
=> ( ord_less_eq @ real @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) ) ) ).
% DERIV_nonneg_imp_nondecreasing
thf(fact_5391_DERIV__nonpos__imp__nonincreasing,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ord_less_eq @ real @ A2 @ X4 )
=> ( ( ord_less_eq @ real @ X4 @ B2 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less_eq @ real @ Y5 @ ( zero_zero @ real ) ) ) ) )
=> ( ord_less_eq @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) ) ) ).
% DERIV_nonpos_imp_nonincreasing
thf(fact_5392_deriv__nonneg__imp__mono,axiom,
! [A2: real,B2: real,G: real > real,G4: real > real] :
( ! [X4: real] :
( ( member @ real @ X4 @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
=> ( has_field_derivative @ real @ G @ ( G4 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X4: real] :
( ( member @ real @ X4 @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( G4 @ X4 ) ) )
=> ( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ord_less_eq @ real @ ( G @ A2 ) @ ( G @ B2 ) ) ) ) ) ).
% deriv_nonneg_imp_mono
thf(fact_5393_DERIV__neg__imp__decreasing,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ord_less_eq @ real @ A2 @ X4 )
=> ( ( ord_less_eq @ real @ X4 @ B2 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y5 @ ( zero_zero @ real ) ) ) ) )
=> ( ord_less @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) ) ) ).
% DERIV_neg_imp_decreasing
thf(fact_5394_DERIV__pos__imp__increasing,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ord_less_eq @ real @ A2 @ X4 )
=> ( ( ord_less_eq @ real @ X4 @ B2 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) ) ) )
=> ( ord_less @ real @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) ) ) ).
% DERIV_pos_imp_increasing
thf(fact_5395_MVT2,axiom,
! [A2: real,B2: real,F2: real > real,F6: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ord_less_eq @ real @ A2 @ X4 )
=> ( ( ord_less_eq @ real @ X4 @ B2 )
=> ( has_field_derivative @ real @ F2 @ ( F6 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
& ( ord_less @ real @ Z3 @ B2 )
& ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( F6 @ Z3 ) ) ) ) ) ) ).
% MVT2
thf(fact_5396_DERIV__local__const,axiom,
! [F2: real > real,L: real,X3: real,D3: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
=> ( ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y4 ) ) @ D3 )
=> ( ( F2 @ X3 )
= ( F2 @ Y4 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_const
thf(fact_5397_DERIV__ln,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( inverse_inverse @ real @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln
thf(fact_5398_DERIV__power__Suc,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X3: A,S: set @ A,N: nat] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( power_power @ A @ ( F2 @ X ) @ ( suc @ N ) )
@ ( times_times @ A @ ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) ) @ ( times_times @ A @ D5 @ ( power_power @ A @ ( F2 @ X3 ) @ N ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ).
% DERIV_power_Suc
thf(fact_5399_DERIV__const__average,axiom,
! [A2: real,B2: real,V2: real > real,K: real] :
( ( A2 != B2 )
=> ( ! [X4: real] : ( has_field_derivative @ real @ V2 @ K @ ( topolo174197925503356063within @ real @ X4 @ ( 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_5400_DERIV__inverse,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X3: A,S: set @ A] :
( ( X3
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( inverse_inverse @ A ) @ ( uminus_uminus @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ).
% DERIV_inverse
thf(fact_5401_DERIV__power,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X3: A,S: set @ A,N: nat] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( power_power @ A @ ( F2 @ X ) @ N )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( times_times @ A @ D5 @ ( power_power @ A @ ( F2 @ X3 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ).
% DERIV_power
thf(fact_5402_DERIV__local__max,axiom,
! [F2: real > real,L: real,X3: real,D3: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
=> ( ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y4 ) ) @ D3 )
=> ( ord_less_eq @ real @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_max
thf(fact_5403_DERIV__local__min,axiom,
! [F2: real > real,L: real,X3: real,D3: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
=> ( ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y4 ) ) @ D3 )
=> ( ord_less_eq @ real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_min
thf(fact_5404_DERIV__ln__divide,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln_divide
thf(fact_5405_DERIV__pow,axiom,
! [N: nat,X3: real,S: set @ real] :
( has_field_derivative @ real
@ ^ [X: real] : ( power_power @ real @ X @ N )
@ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ X3 @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ S ) ) ).
% DERIV_pow
thf(fact_5406_termdiffs__strong__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X3: A] :
( ! [Y4: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Y4 @ N3 ) ) )
=> ( has_field_derivative @ A
@ ^ [X: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% termdiffs_strong_converges_everywhere
thf(fact_5407_at__within__Icc__at,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,X3: A,B2: A] :
( ( ord_less @ A @ A2 @ X3 )
=> ( ( ord_less @ A @ X3 @ B2 )
=> ( ( topolo174197925503356063within @ A @ X3 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% at_within_Icc_at
thf(fact_5408_DERIV__fun__pow,axiom,
! [G: real > real,M: real,X3: real,N: nat] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X: real] : ( power_power @ real @ ( G @ X ) @ N )
@ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( G @ X3 ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) @ M )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_fun_pow
thf(fact_5409_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_5410_DERIV__quotient,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D3: A,X3: A,S: set @ A,G: A > A,E2: A] :
( ( has_field_derivative @ A @ F2 @ D3 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( has_field_derivative @ A @ G @ E2 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ( G @ X3 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [Y: A] : ( divide_divide @ A @ ( F2 @ Y ) @ ( G @ Y ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D3 @ ( G @ X3 ) ) @ ( times_times @ A @ E2 @ ( F2 @ X3 ) ) ) @ ( power_power @ A @ ( G @ X3 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ).
% DERIV_quotient
thf(fact_5411_DERIV__inverse__fun,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D3: A,X3: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D3 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ( F2 @ X3 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( inverse_inverse @ A @ ( F2 @ X ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ D3 @ ( inverse_inverse @ A @ ( power_power @ A @ ( F2 @ X3 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% DERIV_inverse_fun
thf(fact_5412_termdiffs__sums__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K5: real,C2: nat > A,F2: A > A,F6: A,Z: A] :
( ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Z3 @ N3 ) )
@ ( F2 @ Z3 ) ) )
=> ( ( has_field_derivative @ A @ F2 @ F6 @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K5 )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) )
@ F6 ) ) ) ) ) ).
% termdiffs_sums_strong
thf(fact_5413_has__real__derivative__powr,axiom,
! [Z: real,R2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Z )
=> ( has_field_derivative @ real
@ ^ [Z6: real] : ( powr @ real @ Z6 @ R2 )
@ ( times_times @ real @ R2 @ ( powr @ real @ Z @ ( minus_minus @ real @ R2 @ ( one_one @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) ) ) ).
% has_real_derivative_powr
thf(fact_5414_DERIV__series_H,axiom,
! [F2: real > nat > real,F6: real > nat > real,X0: real,A2: real,B2: real,L5: nat > real] :
( ! [N2: nat] :
( has_field_derivative @ real
@ ^ [X: real] : ( F2 @ X @ N2 )
@ ( F6 @ X0 @ N2 )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) )
=> ( ! [X4: real] :
( ( member @ real @ X4 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( summable @ real @ ( F2 @ X4 ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( summable @ real @ ( F6 @ X0 ) )
=> ( ( summable @ real @ L5 )
=> ( ! [N2: nat,X4: real,Y4: real] :
( ( member @ real @ X4 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( member @ real @ Y4 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( F2 @ X4 @ N2 ) @ ( F2 @ Y4 @ N2 ) ) ) @ ( times_times @ real @ ( L5 @ N2 ) @ ( abs_abs @ real @ ( minus_minus @ real @ X4 @ Y4 ) ) ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X: real] : ( suminf @ real @ ( F2 @ X ) )
@ ( suminf @ real @ ( F6 @ X0 ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_series'
thf(fact_5415_termdiffs__strong_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K5: real,C2: nat > A,Z: A] :
( ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Z3 @ N3 ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K5 )
=> ( has_field_derivative @ A
@ ^ [Z6: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Z6 @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong'
thf(fact_5416_termdiffs__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K5: A,X3: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ K5 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( has_field_derivative @ A
@ ^ [X: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong
thf(fact_5417_termdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K5: A,X3: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ K5 @ N3 ) ) )
=> ( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ K5 @ N3 ) ) )
=> ( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N3 ) @ ( power_power @ A @ K5 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( has_field_derivative @ A
@ ^ [X: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% termdiffs
thf(fact_5418_DERIV__fun__powr,axiom,
! [G: real > real,M: real,X3: real,R2: real] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) )
=> ( has_field_derivative @ real
@ ^ [X: real] : ( powr @ real @ ( G @ X ) @ R2 )
@ ( times_times @ real @ ( times_times @ real @ R2 @ ( powr @ real @ ( G @ X3 ) @ ( minus_minus @ real @ R2 @ ( semiring_1_of_nat @ real @ ( one_one @ nat ) ) ) ) ) @ M )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_fun_powr
thf(fact_5419_DERIV__log,axiom,
! [X3: real,B2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( has_field_derivative @ real @ ( log @ B2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( times_times @ real @ ( ln_ln @ real @ B2 ) @ X3 ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_log
thf(fact_5420_DERIV__powr,axiom,
! [G: real > real,M: real,X3: real,F2: real > real,R2: real] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) )
=> ( ( has_field_derivative @ real @ F2 @ R2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X: real] : ( powr @ real @ ( G @ X ) @ ( F2 @ X ) )
@ ( times_times @ real @ ( powr @ real @ ( G @ X3 ) @ ( F2 @ X3 ) ) @ ( plus_plus @ real @ ( times_times @ real @ R2 @ ( ln_ln @ real @ ( G @ X3 ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ M @ ( F2 @ X3 ) ) @ ( G @ X3 ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_powr
thf(fact_5421_DERIV__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( tan @ A ) @ ( inverse_inverse @ A @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_tan
thf(fact_5422_DERIV__real__sqrt,axiom,
! [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( has_field_derivative @ real @ sqrt @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_real_sqrt
thf(fact_5423_DERIV__arctan,axiom,
! [X3: real] : ( has_field_derivative @ real @ arctan @ ( inverse_inverse @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ).
% DERIV_arctan
thf(fact_5424_arsinh__real__has__field__derivative,axiom,
! [X3: real,A4: set @ real] : ( has_field_derivative @ real @ ( arsinh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ A4 ) ) ).
% arsinh_real_has_field_derivative
thf(fact_5425_DERIV__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( sin @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( cot @ A ) @ ( uminus_uminus @ A @ ( inverse_inverse @ A @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_cot
thf(fact_5426_has__field__derivative__tanh,axiom,
! [A10: $tType] :
( ( ( real_Vector_banach @ A10 )
& ( real_V3459762299906320749_field @ A10 ) )
=> ! [G: A10 > A10,X3: A10,Db: A10,S: set @ A10] :
( ( ( cosh @ A10 @ ( G @ X3 ) )
!= ( zero_zero @ A10 ) )
=> ( ( has_field_derivative @ A10 @ G @ Db @ ( topolo174197925503356063within @ A10 @ X3 @ S ) )
=> ( has_field_derivative @ A10
@ ^ [X: A10] : ( tanh @ A10 @ ( G @ X ) )
@ ( times_times @ A10 @ ( minus_minus @ A10 @ ( one_one @ A10 ) @ ( power_power @ A10 @ ( tanh @ A10 @ ( G @ X3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ Db )
@ ( topolo174197925503356063within @ A10 @ X3 @ S ) ) ) ) ) ).
% has_field_derivative_tanh
thf(fact_5427_DERIV__real__sqrt__generic,axiom,
! [X3: real,D5: real] :
( ( X3
!= ( zero_zero @ real ) )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( D5
= ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) )
=> ( D5
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( has_field_derivative @ real @ sqrt @ D5 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_5428_arcosh__real__has__field__derivative,axiom,
! [X3: real,A4: set @ real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( has_field_derivative @ real @ ( arcosh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ A4 ) ) ) ).
% arcosh_real_has_field_derivative
thf(fact_5429_artanh__real__has__field__derivative,axiom,
! [X3: real,A4: set @ real] :
( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ ( artanh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ A4 ) ) ) ).
% artanh_real_has_field_derivative
thf(fact_5430_DERIV__power__series_H,axiom,
! [R3: real,F2: nat > real,X0: real] :
( ! [X4: real] :
( ( member @ real @ X4 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R3 ) @ R3 ) )
=> ( summable @ real
@ ^ [N3: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N3 ) @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) @ ( power_power @ real @ X4 @ N3 ) ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R3 ) @ R3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ( has_field_derivative @ real
@ ^ [X: real] :
( suminf @ real
@ ^ [N3: nat] : ( times_times @ real @ ( F2 @ N3 ) @ ( power_power @ real @ X @ ( 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_5431_DERIV__real__root,axiom,
! [N: nat,X3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X3 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_real_root
thf(fact_5432_DERIV__arccos,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arccos @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arccos
thf(fact_5433_DERIV__arcsin,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arcsin @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arcsin
thf(fact_5434_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F2: real > real,X3: real,N: nat] :
( ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
& ! [M4: nat,X4: real] : ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X3 ) )
& ( ( F2 @ X3 )
= ( 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 @ X3 @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X3 @ N ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_5435_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F2: real > real,X3: real,N: nat] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M4: nat,X4: real] : ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X3 ) )
& ( ( F2 @ X3 )
= ( 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 @ X3 @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X3 @ N ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_5436_DERIV__odd__real__root,axiom,
! [N: nat,X3: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( X3
!= ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X3 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_5437_Maclaurin,axiom,
! [H: real,N: nat,Diff: nat > real > real,F2: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H )
=> ( ( 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 @ H ) )
=> ( 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 @ H )
& ( ( F2 @ H )
= ( 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 @ H @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_5438_Maclaurin2,axiom,
! [H: real,Diff: nat > real > real,F2: real > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H )
=> ( ( ( 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 @ H ) )
=> ( 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 @ H )
& ( ( F2 @ H )
= ( 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 @ H @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_5439_Maclaurin__minus,axiom,
! [H: real,N: nat,Diff: nat > real > real,F2: real > real] :
( ( ord_less @ real @ H @ ( 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 @ H @ 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 @ H @ T4 )
& ( ord_less @ real @ T4 @ ( zero_zero @ real ) )
& ( ( F2 @ H )
= ( 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 @ H @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_5440_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F2: real > real,N: nat,X3: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X3
!= ( zero_zero @ real ) )
=> ( ! [M4: nat,X4: real] : ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( 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 @ X3 ) )
& ( ( F2 @ X3 )
= ( 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 @ X3 @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X3 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_5441_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F2: real > real,N: nat,X3: 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 @ X3 ) ) )
=> ( 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 @ X3 ) )
& ( ( F2 @ X3 )
= ( 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 @ X3 @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X3 @ N ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_5442_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_5443_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_5444_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real,X3: 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 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ( ( X3 != C2 )
=> ? [T4: real] :
( ( ( ord_less @ real @ X3 @ C2 )
=> ( ( ord_less @ real @ X3 @ T4 )
& ( ord_less @ real @ T4 @ C2 ) ) )
& ( ~ ( ord_less @ real @ X3 @ C2 )
=> ( ( ord_less @ real @ C2 @ T4 )
& ( ord_less @ real @ T4 @ X3 ) ) )
& ( ( F2 @ X3 )
= ( 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 @ X3 @ 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 @ X3 @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_5445_Maclaurin__lemma2,axiom,
! [N: nat,H: real,Diff: nat > real > real,K: nat,B6: real] :
( ! [M4: nat,T4: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ H ) )
=> ( 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 @ H ) )
=> ( has_field_derivative @ real
@ ^ [U2: real] :
( minus_minus @ real @ ( Diff @ M2 @ U2 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ M2 @ P5 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ real @ U2 @ P5 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ M2 ) ) )
@ ( times_times @ real @ B6 @ ( divide_divide @ real @ ( power_power @ real @ U2 @ ( 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
@ ^ [P5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ ( suc @ M2 ) @ P5 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ real @ T8 @ P5 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ M2 ) ) ) )
@ ( times_times @ real @ B6 @ ( 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_5446_has__derivative__arcsin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X3: A,G4: A > real,S: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X3 ) )
=> ( ( ord_less @ real @ ( G @ X3 ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( arcsin @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G4 @ X ) @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ).
% has_derivative_arcsin
thf(fact_5447_has__derivative__arccos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X3: A,G4: A > real,S: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X3 ) )
=> ( ( ord_less @ real @ ( G @ X3 ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( arccos @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G4 @ X ) @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ).
% has_derivative_arccos
thf(fact_5448_has__derivative__tan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X3: A,G4: A > real,S: set @ A] :
( ( ( cos @ real @ ( G @ X3 ) )
!= ( zero_zero @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( tan @ real @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G4 @ X ) @ ( inverse_inverse @ real @ ( power_power @ real @ ( cos @ real @ ( G @ X3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% has_derivative_tan
thf(fact_5449_has__derivative__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [C2: B,F4: filter @ A] :
( has_derivative @ A @ B
@ ^ [X: A] : C2
@ ^ [X: A] : ( zero_zero @ B )
@ F4 ) ) ).
% has_derivative_const
thf(fact_5450_has__derivative__zero__unique,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F4: A > B,X3: A] :
( ( has_derivative @ A @ B
@ ^ [X: A] : ( zero_zero @ B )
@ F4
@ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
=> ( F4
= ( ^ [H2: A] : ( zero_zero @ B ) ) ) ) ) ).
% has_derivative_zero_unique
thf(fact_5451_has__derivative__divide_H,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,F6: C > A,X3: C,S2: set @ C,G: C > A,G4: C > A] :
( ( has_derivative @ C @ A @ F2 @ F6 @ ( topolo174197925503356063within @ C @ X3 @ S2 ) )
=> ( ( has_derivative @ C @ A @ G @ G4 @ ( topolo174197925503356063within @ C @ X3 @ S2 ) )
=> ( ( ( G @ X3 )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X: C] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ^ [H2: C] : ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ ( F6 @ H2 ) @ ( G @ X3 ) ) @ ( times_times @ A @ ( F2 @ X3 ) @ ( G4 @ H2 ) ) ) @ ( times_times @ A @ ( G @ X3 ) @ ( G @ X3 ) ) )
@ ( topolo174197925503356063within @ C @ X3 @ S2 ) ) ) ) ) ) ).
% has_derivative_divide'
thf(fact_5452_has__derivative__inverse,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: C > A,X3: C,F6: C > A,S2: set @ C] :
( ( ( F2 @ X3 )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F2 @ F6 @ ( topolo174197925503356063within @ C @ X3 @ S2 ) )
=> ( has_derivative @ C @ A
@ ^ [X: C] : ( inverse_inverse @ A @ ( F2 @ X ) )
@ ^ [H2: C] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X3 ) ) @ ( F6 @ H2 ) ) @ ( inverse_inverse @ A @ ( F2 @ X3 ) ) ) )
@ ( topolo174197925503356063within @ C @ X3 @ S2 ) ) ) ) ) ).
% has_derivative_inverse
thf(fact_5453_has__derivative__inverse_H,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X3: A,S2: set @ A] :
( ( X3
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A @ ( inverse_inverse @ A )
@ ^ [H2: A] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ X3 ) @ H2 ) @ ( inverse_inverse @ A @ X3 ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S2 ) ) ) ) ).
% has_derivative_inverse'
thf(fact_5454_has__derivative__power,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,F6: A > B,X3: A,S2: set @ A,N: nat] :
( ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X3 @ S2 ) )
=> ( has_derivative @ A @ B
@ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N )
@ ^ [Y: A] : ( times_times @ B @ ( times_times @ B @ ( semiring_1_of_nat @ B @ N ) @ ( F6 @ Y ) ) @ ( power_power @ B @ ( F2 @ X3 ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S2 ) ) ) ) ).
% has_derivative_power
thf(fact_5455_has__derivative__ln,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X3: A,G4: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( ln_ln @ real @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G4 @ X ) @ ( inverse_inverse @ real @ ( G @ X3 ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% has_derivative_ln
thf(fact_5456_has__derivative__divide,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: C > A,F6: C > A,X3: C,S2: set @ C,G: C > A,G4: C > A] :
( ( has_derivative @ C @ A @ F2 @ F6 @ ( topolo174197925503356063within @ C @ X3 @ S2 ) )
=> ( ( has_derivative @ C @ A @ G @ G4 @ ( topolo174197925503356063within @ C @ X3 @ S2 ) )
=> ( ( ( G @ X3 )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X: C] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ^ [H2: C] : ( plus_plus @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( F2 @ X3 ) ) @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( G @ X3 ) ) @ ( G4 @ H2 ) ) @ ( inverse_inverse @ A @ ( G @ X3 ) ) ) ) @ ( divide_divide @ A @ ( F6 @ H2 ) @ ( G @ X3 ) ) )
@ ( topolo174197925503356063within @ C @ X3 @ S2 ) ) ) ) ) ) ).
% has_derivative_divide
thf(fact_5457_has__derivative__powr,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G4: A > real,X3: A,X8: set @ A,F2: A > real,F6: A > real] :
( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X3 @ X8 ) )
=> ( ( has_derivative @ A @ real @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X3 @ X8 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) )
=> ( ( member @ A @ X3 @ X8 )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( powr @ real @ ( G @ X ) @ ( F2 @ X ) )
@ ^ [H2: A] : ( times_times @ real @ ( powr @ real @ ( G @ X3 ) @ ( F2 @ X3 ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( F6 @ H2 ) @ ( ln_ln @ real @ ( G @ X3 ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ ( G4 @ H2 ) @ ( F2 @ X3 ) ) @ ( G @ X3 ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ X8 ) ) ) ) ) ) ) ).
% has_derivative_powr
thf(fact_5458_has__derivative__real__sqrt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X3: A,G4: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( sqrt @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G4 @ X ) @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ ( G @ X3 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% has_derivative_real_sqrt
thf(fact_5459_has__derivative__arctan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G4: A > real,X3: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( arctan @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G4 @ X ) @ ( inverse_inverse @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ).
% has_derivative_arctan
thf(fact_5460_has__derivative__floor,axiom,
! [Aa: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( archim2362893244070406136eiling @ Aa )
& ( topolo2564578578187576103pology @ Aa ) )
=> ! [G: A > real,X3: A,F2: real > Aa,G4: A > real,S: set @ A] :
( ( topolo3448309680560233919inuous @ real @ Aa @ ( topolo174197925503356063within @ real @ ( G @ X3 ) @ ( top_top @ ( set @ real ) ) ) @ F2 )
=> ( ~ ( member @ Aa @ ( F2 @ ( G @ X3 ) ) @ ( ring_1_Ints @ Aa ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ Aa @ ( F2 @ ( G @ X ) ) ) )
@ ^ [X: A] : ( times_times @ real @ ( G4 @ X ) @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ).
% has_derivative_floor
thf(fact_5461_termdiffs__aux,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K5: A,X3: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N3 ) @ ( power_power @ A @ K5 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( filterlim @ A @ A
@ ^ [H2: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ X3 @ H2 ) @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( power_power @ A @ X3 @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_aux
thf(fact_5462_isCont__powser,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K5: A,X3: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ K5 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ) ) ).
% isCont_powser
thf(fact_5463_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
@ ^ [X: B] : ( times_times @ A @ ( F2 @ X ) @ 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_5464_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
@ ^ [X: B] : ( times_times @ A @ C2 @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ L ) )
@ F4 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% tendsto_mult_left_iff
thf(fact_5465_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
@ ^ [X: A] : ( power_power @ real @ ( F2 @ X ) @ 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_5466_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 )
& ! [X4: A] :
( ( ( X4 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X4 @ A2 ) ) @ D2 ) )
=> ( ( F2 @ X4 )
!= ( F2 @ A2 ) ) ) )
=> ( filterlim @ A @ C
@ ^ [X: A] : ( G @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ C @ L )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% isCont_LIM_compose2
thf(fact_5467_isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X3: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
= ( filterlim @ A @ B
@ ^ [H2: A] : ( F2 @ ( plus_plus @ A @ X3 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ X3 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% isCont_iff
thf(fact_5468_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
@ ^ [H2: A] : ( F2 @ ( plus_plus @ A @ A2 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_isCont_iff
thf(fact_5469_LIM__offset__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L5: B,A2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H2: A] : ( F2 @ ( plus_plus @ A @ A2 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero
thf(fact_5470_LIM__offset__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A2: A,L5: B] :
( ( filterlim @ A @ B
@ ^ [H2: A] : ( F2 @ ( plus_plus @ A @ A2 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero_cancel
thf(fact_5471_LIM__not__zero,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topolo8386298272705272623_space @ A )
& ( zero @ Aa )
& ( topological_t2_space @ Aa ) )
=> ! [K: Aa,A2: A] :
( ( K
!= ( zero_zero @ Aa ) )
=> ~ ( filterlim @ A @ Aa
@ ^ [X: A] : K
@ ( topolo7230453075368039082e_nhds @ Aa @ ( zero_zero @ Aa ) )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_not_zero
thf(fact_5472_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 ) ) ) )
=> ( ! [X4: A] :
( ( X4 != A2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( G @ X4 ) ) )
=> ( ! [X4: A] :
( ( X4 != A2 )
=> ( ord_less_eq @ real @ ( G @ X4 ) @ ( F2 @ X4 ) ) )
=> ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% real_LIM_sandwich_zero
thf(fact_5473_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
@ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% tendsto_null_power
thf(fact_5474_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
@ ^ [X: B] : ( divide_divide @ A @ ( F2 @ X ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_divide_zero
thf(fact_5475_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
@ ^ [X: B] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( divide_divide @ A @ A2 @ B2 ) )
@ F4 ) ) ) ) ) ).
% tendsto_divide
thf(fact_5476_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
@ ^ [X: B] : ( arcosh @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A2 ) )
@ F4 ) ) ) ).
% tendsto_arcosh
thf(fact_5477_tendsto__rabs__zero__cancel,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X: A] : ( abs_abs @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ).
% tendsto_rabs_zero_cancel
thf(fact_5478_tendsto__rabs__zero__iff,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X: A] : ( abs_abs @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
= ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ).
% tendsto_rabs_zero_iff
thf(fact_5479_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
@ ^ [X: A] : ( abs_abs @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ).
% tendsto_rabs_zero
thf(fact_5480_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
@ ^ [X: A] : ( ring_1_of_int @ C @ ( archim6421214686448440834_floor @ B @ ( F2 @ X ) ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( ring_1_of_int @ C @ ( archim6421214686448440834_floor @ B @ L ) ) )
@ F4 ) ) ) ) ).
% tendsto_of_int_floor
thf(fact_5481_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
@ ^ [X: A] : ( ring_1_of_int @ C @ ( archimedean_ceiling @ B @ ( F2 @ X ) ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( ring_1_of_int @ C @ ( archimedean_ceiling @ B @ L ) ) )
@ F4 ) ) ) ) ).
% tendsto_of_int_ceiling
thf(fact_5482_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
@ ^ [X: A] : ( root @ N @ ( F2 @ X ) ) ) ) ) ).
% continuous_real_root
thf(fact_5483_tendsto__real__root,axiom,
! [A: $tType,F2: A > real,X3: real,F4: filter @ A,N: nat] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ X3 ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( root @ N @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( root @ N @ X3 ) )
@ F4 ) ) ).
% tendsto_real_root
thf(fact_5484_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
@ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( power_power @ B @ A2 @ N ) )
@ F4 ) ) ) ).
% tendsto_power
thf(fact_5485_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
@ ^ [X: C] : ( power_power @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_power'
thf(fact_5486_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
@ ^ [X: C] : ( power_power @ B @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( power_power @ B @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_power_strong
thf(fact_5487_tendsto__real__sqrt,axiom,
! [A: $tType,F2: A > real,X3: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ X3 ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( sqrt @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( sqrt @ X3 ) )
@ F4 ) ) ).
% tendsto_real_sqrt
thf(fact_5488_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
@ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N ) ) ) ) ).
% continuous_power
thf(fact_5489_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
@ ^ [X: A] : ( sqrt @ ( F2 @ X ) ) ) ) ) ).
% continuous_real_sqrt
thf(fact_5490_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
@ ^ [X: A] : ( cot @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cot @ A @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_cot
thf(fact_5491_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
@ ^ [X: C] : ( tanh @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tanh @ A @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_tanh
thf(fact_5492_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
@ ^ [X: B] : ( sgn_sgn @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sgn_sgn @ A @ L ) )
@ F4 ) ) ) ) ).
% tendsto_sgn
thf(fact_5493_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
@ ^ [X: A] : ( tan @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tan @ A @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_tan
thf(fact_5494_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
@ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_powr
thf(fact_5495_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
@ ^ [X: A] : ( ln_ln @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( ln_ln @ real @ A2 ) )
@ F4 ) ) ) ).
% tendsto_ln
thf(fact_5496_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
@ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% tendsto_norm_zero
thf(fact_5497_tendsto__norm__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
= ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 ) ) ) ).
% tendsto_norm_zero_iff
thf(fact_5498_tendsto__norm__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 ) ) ) ).
% tendsto_norm_zero_cancel
thf(fact_5499_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
@ ^ [X: B] : ( inverse_inverse @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( inverse_inverse @ A @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_inverse
thf(fact_5500_tendsto__add__zero,axiom,
! [B: $tType,D: $tType] :
( ( topolo6943815403480290642id_add @ B )
=> ! [F2: D > B,F4: filter @ D,G: D > B] :
( ( filterlim @ D @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( filterlim @ D @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( filterlim @ D @ B
@ ^ [X: D] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% tendsto_add_zero
thf(fact_5501_tendsto__mult__right__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D > A,F4: filter @ D,C2: A] :
( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D @ A
@ ^ [X: D] : ( times_times @ A @ C2 @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_mult_right_zero
thf(fact_5502_tendsto__mult__left__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D > A,F4: filter @ D,C2: A] :
( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D @ A
@ ^ [X: D] : ( times_times @ A @ ( F2 @ X ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_mult_left_zero
thf(fact_5503_tendsto__mult__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D > A,F4: filter @ D,G: D > A] :
( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( ( filterlim @ D @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D @ A
@ ^ [X: D] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_mult_zero
thf(fact_5504_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
@ ^ [X: B] : ( minus_minus @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( 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_5505_LIM__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 ) ) ) ).
% LIM_zero_cancel
thf(fact_5506_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
@ ^ [X: B] : ( minus_minus @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 ) ) ) ) ).
% Lim_transform2
thf(fact_5507_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
@ ^ [X: B] : ( minus_minus @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 ) ) ) ) ).
% Lim_transform
thf(fact_5508_LIM__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 )
= ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 ) ) ) ).
% LIM_zero_iff
thf(fact_5509_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
@ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ).
% LIM_zero
thf(fact_5510_tendsto__null__sum,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I6: set @ B,F2: A > B > C,F4: filter @ A] :
( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( filterlim @ A @ C
@ ^ [X: A] : ( F2 @ X @ I3 )
@ ( 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_5511_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
@ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( log @ A2 @ B2 ) )
@ F4 ) ) ) ) ) ) ).
% tendsto_log
thf(fact_5512_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
@ ^ [X: A] : ( artanh @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( artanh @ real @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_artanh
thf(fact_5513_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 ) ) ) )
=> ( ! [X4: A] :
( ( X4 != A2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( minus_minus @ C @ ( G @ X4 ) @ M ) ) @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X4 ) @ L ) ) ) )
=> ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ M ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_imp_LIM
thf(fact_5514_IVT,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,A2: A,Y3: B,B2: A] :
( ( ord_less_eq @ B @ ( F2 @ A2 ) @ Y3 )
=> ( ( ord_less_eq @ B @ Y3 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ! [X4: A] :
( ( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less_eq @ A @ X4 @ B2 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X4 @ ( top_top @ ( set @ A ) ) ) @ F2 ) )
=> ? [X4: A] :
( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less_eq @ A @ X4 @ B2 )
& ( ( F2 @ X4 )
= Y3 ) ) ) ) ) ) ) ).
% IVT
thf(fact_5515_IVT2,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,B2: A,Y3: B,A2: A] :
( ( ord_less_eq @ B @ ( F2 @ B2 ) @ Y3 )
=> ( ( ord_less_eq @ B @ Y3 @ ( F2 @ A2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ! [X4: A] :
( ( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less_eq @ A @ X4 @ B2 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X4 @ ( top_top @ ( set @ A ) ) ) @ F2 ) )
=> ? [X4: A] :
( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less_eq @ A @ X4 @ B2 )
& ( ( F2 @ X4 )
= Y3 ) ) ) ) ) ) ) ).
% IVT2
thf(fact_5516_LIM__D,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,L5: B,A2: A,R2: real] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [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 ) @ L5 ) ) @ R2 ) ) ) ) ) ) ).
% LIM_D
thf(fact_5517_LIM__I,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F2: A > B,L5: B] :
( ! [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [S7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S7 )
& ! [X4: A] :
( ( ( X4 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X4 @ A2 ) ) @ S7 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X4 ) @ L5 ) ) @ R ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_I
thf(fact_5518_LIM__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,L5: B,A2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S6 )
& ! [X: A] :
( ( ( X != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ A2 ) ) @ S6 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X ) @ L5 ) ) @ R5 ) ) ) ) ) ) ) ).
% LIM_eq
thf(fact_5519_LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [R3: real,A2: A,F2: A > B,G: A > B,L: B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ( ! [X4: A] :
( ( X4 != A2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X4 @ A2 ) ) @ R3 )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( ( 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_5520_DERIV__LIM__iff,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > A,A2: A,D5: A] :
( ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ A2 @ H2 ) ) @ ( F2 @ A2 ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [X: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ X ) @ ( F2 @ A2 ) ) @ ( minus_minus @ A @ X @ A2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ D5 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_LIM_iff
thf(fact_5521_isCont__Lb__Ub,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [L6: real,M8: real] :
( ! [X6: real] :
( ( ( ord_less_eq @ real @ A2 @ X6 )
& ( ord_less_eq @ real @ X6 @ B2 ) )
=> ( ( ord_less_eq @ real @ L6 @ ( F2 @ X6 ) )
& ( ord_less_eq @ real @ ( F2 @ X6 ) @ M8 ) ) )
& ! [Y5: real] :
( ( ( ord_less_eq @ real @ L6 @ Y5 )
& ( ord_less_eq @ real @ Y5 @ M8 ) )
=> ? [X4: real] :
( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 )
& ( ( F2 @ X4 )
= Y5 ) ) ) ) ) ) ).
% isCont_Lb_Ub
thf(fact_5522_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_5523_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_5524_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_5525_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 )
& ! [X4: A] :
( ( ( X4 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X4 @ A2 ) ) @ D2 ) )
=> ( ( F2 @ X4 )
!= B2 ) ) )
=> ( filterlim @ A @ C
@ ^ [X: A] : ( G @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ C @ C2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_compose2
thf(fact_5526_isCont__real__sqrt,axiom,
! [X3: real] : ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ sqrt ) ).
% isCont_real_sqrt
thf(fact_5527_isCont__real__root,axiom,
! [X3: real,N: nat] : ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ ( root @ N ) ) ).
% isCont_real_root
thf(fact_5528_continuous__at__within__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,S: set @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ G )
=> ( ( ( G @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% continuous_at_within_divide
thf(fact_5529_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 ) ) )
@ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N ) ) ) ) ).
% isCont_power
thf(fact_5530_continuous__at__within__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A2: A,S: set @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X: A] : ( inverse_inverse @ B @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_at_within_inverse
thf(fact_5531_continuous__at__within__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,S: set @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X: A] : ( sgn_sgn @ B @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_at_within_sgn
thf(fact_5532_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 )
@ ^ [X: C] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% continuous_at_within_powr
thf(fact_5533_continuous__within__ln,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X3: A,S: set @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X3 @ S ) @ F2 )
=> ( ( ( F2 @ X3 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X3 @ S )
@ ^ [X: A] : ( ln_ln @ real @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_within_ln
thf(fact_5534_DERIV__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X3: A] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X3 @ H2 ) ) @ ( F2 @ X3 ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_def
thf(fact_5535_DERIV__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X3: A] :
( ( has_field_derivative @ A @ F2 @ D5 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X3 @ H2 ) ) @ ( F2 @ X3 ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_D
thf(fact_5536_lim__exp__minus__1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( filterlim @ A @ A
@ ^ [Z6: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z6 ) @ ( one_one @ A ) ) @ Z6 )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% lim_exp_minus_1
thf(fact_5537_lemma__termdiff4,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [K: real,F2: A > B,K5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ! [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 @ K5 @ ( 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_5538_isCont__eq__Lb,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X4 @ ( 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 ) ) )
& ? [X4: real] :
( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 )
& ( ( F2 @ X4 )
= M8 ) ) ) ) ) ) ).
% isCont_eq_Lb
thf(fact_5539_isCont__eq__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X4 @ ( 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 ) )
& ? [X4: real] :
( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 )
& ( ( F2 @ X4 )
= M8 ) ) ) ) ) ) ).
% isCont_eq_Ub
thf(fact_5540_isCont__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X4 @ ( 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_5541_isCont__inverse__function2,axiom,
! [A2: real,X3: real,B2: real,G: real > real,F2: real > real] :
( ( ord_less @ real @ A2 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A2 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B2 )
=> ( ( G @ ( F2 @ Z3 ) )
= Z3 ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A2 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F2 @ X3 ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ) ).
% isCont_inverse_function2
thf(fact_5542_field__has__derivative__at,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D5: A,X3: A] :
( ( has_derivative @ A @ A @ F2 @ ( times_times @ A @ D5 ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X3 @ H2 ) ) @ ( F2 @ X3 ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% field_has_derivative_at
thf(fact_5543_isCont__ln,axiom,
! [X3: real] :
( ( X3
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ ( ln_ln @ real ) ) ) ).
% isCont_ln
thf(fact_5544_isCont__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( ( G @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% isCont_divide
thf(fact_5545_isCont__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( sgn_sgn @ B @ ( F2 @ X ) ) ) ) ) ) ).
% isCont_sgn
thf(fact_5546_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
@ ^ [X: A] : ( F2 @ ( plus_plus @ A @ X @ A2 ) )
@ F4
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_at_to_0
thf(fact_5547_continuous__within__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,S: set @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ S ) @ F2 )
=> ( ( ( cos @ A @ ( F2 @ X3 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ S )
@ ^ [X: A] : ( tan @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_within_tan
thf(fact_5548_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 ) ) )
@ ^ [X: C] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% isCont_powr
thf(fact_5549_isCont__ln_H,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X3: A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( F2 @ X3 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( ln_ln @ real @ ( F2 @ X ) ) ) ) ) ) ).
% isCont_ln'
thf(fact_5550_continuous__within__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A,S: set @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ S ) @ F2 )
=> ( ( ( sin @ A @ ( F2 @ X3 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ S )
@ ^ [X: A] : ( cot @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_within_cot
thf(fact_5551_continuous__at__within__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: C,A4: set @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X3 @ A4 ) @ F2 )
=> ( ( ( cosh @ A @ ( F2 @ X3 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X3 @ A4 )
@ ^ [X: C] : ( tanh @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_at_within_tanh
thf(fact_5552_isCont__has__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X4 @ ( 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 ) )
& ! [N7: A] :
( ( ord_less @ A @ N7 @ M8 )
=> ? [X4: real] :
( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 )
& ( ord_less @ A @ N7 @ ( F2 @ X4 ) ) ) ) ) ) ) ) ).
% isCont_has_Ub
thf(fact_5553_isCont__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( cos @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ ( tan @ A ) ) ) ) ).
% isCont_tan
thf(fact_5554_isCont__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( sin @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ ( cot @ A ) ) ) ) ).
% isCont_cot
thf(fact_5555_isCont__tanh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ( cosh @ A @ X3 )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ ( tanh @ A ) ) ) ) ).
% isCont_tanh
thf(fact_5556_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 )
=> ( ! [X4: A] :
( ( X4
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ S )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( A2 @ N3 ) @ ( power_power @ A @ X4 @ N3 ) )
@ ( F2 @ X4 ) ) ) )
=> ( 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_5557_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 )
=> ( ! [X4: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ S )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( A2 @ N3 ) @ ( power_power @ A @ X4 @ N3 ) )
@ ( F2 @ X4 ) ) )
=> ( 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_5558_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
@ ^ [H2: A] : ( suminf @ B @ ( G @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% lemma_termdiff5
thf(fact_5559_isCont__tan_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( cos @ A @ ( F2 @ A2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( tan @ A @ ( F2 @ X ) ) ) ) ) ) ).
% isCont_tan'
thf(fact_5560_continuous__at__within__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A,S: set @ A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S ) @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ A2 ) )
=> ( ( ( F2 @ A2 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ A2 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ).
% continuous_at_within_log
thf(fact_5561_isCont__arcosh,axiom,
! [X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ ( arcosh @ real ) ) ) ).
% isCont_arcosh
thf(fact_5562_LIM__cos__div__sin,axiom,
( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( cos @ real @ X ) @ ( sin @ real @ X ) )
@ ( 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_5563_isCont__cot_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( sin @ A @ ( F2 @ A2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( cot @ A @ ( F2 @ X ) ) ) ) ) ) ).
% isCont_cot'
thf(fact_5564_DERIV__inverse__function,axiom,
! [F2: real > real,D5: real,G: real > real,X3: real,A2: real,B2: real] :
( ( has_field_derivative @ real @ F2 @ D5 @ ( topolo174197925503356063within @ real @ ( G @ X3 ) @ ( top_top @ ( set @ real ) ) ) )
=> ( ( D5
!= ( zero_zero @ real ) )
=> ( ( ord_less @ real @ A2 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ( ! [Y4: real] :
( ( ord_less @ real @ A2 @ Y4 )
=> ( ( ord_less @ real @ Y4 @ B2 )
=> ( ( F2 @ ( G @ Y4 ) )
= Y4 ) ) )
=> ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ G )
=> ( has_field_derivative @ real @ G @ ( inverse_inverse @ real @ D5 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_inverse_function
thf(fact_5565_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_5566_isCont__arccos,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ arccos ) ) ) ).
% isCont_arccos
thf(fact_5567_isCont__arcsin,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ arcsin ) ) ) ).
% isCont_arcsin
thf(fact_5568_isCont__powser__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X3: A] :
( ! [Y4: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Y4 @ N3 ) ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ) ).
% isCont_powser_converges_everywhere
thf(fact_5569_LIM__less__bound,axiom,
! [B2: real,X3: real,F2: real > real] :
( ( ord_less @ real @ B2 @ X3 )
=> ( ! [X4: real] :
( ( member @ real @ X4 @ ( set_or5935395276787703475ssThan @ real @ B2 @ X3 ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X4 ) ) )
=> ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) ) ) ) ) ).
% LIM_less_bound
thf(fact_5570_isCont__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ A2 ) )
=> ( ( ( F2 @ A2 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ A2 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ).
% isCont_log
thf(fact_5571_isCont__artanh,axiom,
! [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ ( artanh @ real ) ) ) ) ).
% isCont_artanh
thf(fact_5572_isCont__inverse__function,axiom,
! [D3: real,X3: real,G: real > real,F2: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z3 @ X3 ) ) @ D3 )
=> ( ( G @ ( F2 @ Z3 ) )
= Z3 ) )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z3 @ X3 ) ) @ D3 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F2 @ X3 ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ).
% isCont_inverse_function
thf(fact_5573_GMVT_H,axiom,
! [A2: real,B2: real,F2: real > real,G: real > real,G4: real > real,F6: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A2 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A2 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ G ) ) )
=> ( ! [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
=> ( ( ord_less @ real @ Z3 @ B2 )
=> ( has_field_derivative @ real @ G @ ( G4 @ Z3 ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ! [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
=> ( ( ord_less @ real @ Z3 @ B2 )
=> ( has_field_derivative @ real @ F2 @ ( F6 @ Z3 ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [C3: real] :
( ( ord_less @ real @ A2 @ C3 )
& ( ord_less @ real @ C3 @ B2 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) @ ( G4 @ C3 ) )
= ( times_times @ real @ ( minus_minus @ real @ ( G @ B2 ) @ ( G @ A2 ) ) @ ( F6 @ C3 ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_5574_floor__has__real__derivative,axiom,
! [A: $tType] :
( ( ( archim2362893244070406136eiling @ A )
& ( topolo2564578578187576103pology @ A ) )
=> ! [X3: real,F2: real > A] :
( ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 )
=> ( ~ ( member @ A @ ( F2 @ X3 ) @ ( ring_1_Ints @ A ) )
=> ( has_field_derivative @ real
@ ^ [X: real] : ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ A @ ( F2 @ X ) ) )
@ ( zero_zero @ real )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% floor_has_real_derivative
thf(fact_5575_isCont__powser_H,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ Aa )
& ( real_V3459762299906320749_field @ Aa ) )
=> ! [A2: A,F2: A > Aa,C2: nat > Aa,K5: Aa] :
( ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( summable @ Aa
@ ^ [N3: nat] : ( times_times @ Aa @ ( C2 @ N3 ) @ ( power_power @ Aa @ K5 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ Aa @ ( F2 @ A2 ) ) @ ( real_V7770717601297561774m_norm @ Aa @ K5 ) )
=> ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] :
( suminf @ Aa
@ ^ [N3: nat] : ( times_times @ Aa @ ( C2 @ N3 ) @ ( power_power @ Aa @ ( F2 @ X ) @ N3 ) ) ) ) ) ) ) ) ).
% isCont_powser'
thf(fact_5576_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_5577_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_5578_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_5579_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_5580_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_5581_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_5582_approx__from__below__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [Y3: A,X3: A] :
( ( ord_less @ A @ Y3 @ X3 )
=> ? [U4: nat > A] :
( ! [N9: nat] : ( ord_less @ A @ ( U4 @ N9 ) @ X3 )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_below_dense_linorder
thf(fact_5583_approx__from__above__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ? [U4: nat > A] :
( ! [N9: nat] : ( ord_less @ A @ X3 @ ( U4 @ N9 ) )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_above_dense_linorder
thf(fact_5584_lim__mono,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [N4: nat,X8: nat > A,Y7: nat > A,X3: A,Y3: A] :
( ! [N2: nat] :
( ( ord_less_eq @ nat @ N4 @ N2 )
=> ( ord_less_eq @ A @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y7 @ ( topolo7230453075368039082e_nhds @ A @ Y3 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ).
% lim_mono
thf(fact_5585_LIMSEQ__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,X3: A,Y7: nat > A,Y3: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y7 @ ( topolo7230453075368039082e_nhds @ A @ Y3 ) @ ( at_top @ nat ) )
=> ( ? [N7: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N7 @ N2 )
=> ( ord_less_eq @ A @ ( X8 @ N2 ) @ ( Y7 @ N2 ) ) )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ).
% LIMSEQ_le
thf(fact_5586_Lim__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: nat > A,L: A,M7: nat,C5: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ M7 @ N2 )
=> ( ord_less_eq @ A @ ( F2 @ N2 ) @ C5 ) )
=> ( ord_less_eq @ A @ L @ C5 ) ) ) ) ).
% Lim_bounded
thf(fact_5587_Lim__bounded2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: nat > A,L: A,N4: nat,C5: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N4 @ N2 )
=> ( ord_less_eq @ A @ C5 @ ( F2 @ N2 ) ) )
=> ( ord_less_eq @ A @ C5 @ L ) ) ) ) ).
% Lim_bounded2
thf(fact_5588_LIMSEQ__le__const,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,X3: A,A2: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) )
=> ( ? [N7: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N7 @ N2 )
=> ( ord_less_eq @ A @ A2 @ ( X8 @ N2 ) ) )
=> ( ord_less_eq @ A @ A2 @ X3 ) ) ) ) ).
% LIMSEQ_le_const
thf(fact_5589_LIMSEQ__le__const2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,X3: A,A2: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) )
=> ( ? [N7: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N7 @ N2 )
=> ( ord_less_eq @ A @ ( X8 @ N2 ) @ A2 ) )
=> ( ord_less_eq @ A @ X3 @ A2 ) ) ) ) ).
% LIMSEQ_le_const2
thf(fact_5590_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_5591_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_5592_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_5593_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_5594_mult__nat__right__at__top,axiom,
! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( filterlim @ nat @ nat
@ ^ [X: nat] : ( times_times @ nat @ X @ C2 )
@ ( at_top @ nat )
@ ( at_top @ nat ) ) ) ).
% mult_nat_right_at_top
thf(fact_5595_monoseq__convergent,axiom,
! [X8: nat > real,B6: real] :
( ( topological_monoseq @ real @ X8 )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( X8 @ I3 ) ) @ B6 )
=> ~ ! [L6: real] :
~ ( filterlim @ nat @ real @ X8 @ ( topolo7230453075368039082e_nhds @ real @ L6 ) @ ( at_top @ nat ) ) ) ) ).
% monoseq_convergent
thf(fact_5596_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_5597_monoseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: nat > A,X3: A] :
( ( topological_monoseq @ A @ A2 )
=> ( ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) )
=> ( ( ! [N9: nat] : ( ord_less_eq @ A @ ( A2 @ N9 ) @ X3 )
& ! [M2: nat,N9: nat] :
( ( ord_less_eq @ nat @ M2 @ N9 )
=> ( ord_less_eq @ A @ ( A2 @ M2 ) @ ( A2 @ N9 ) ) ) )
| ( ! [N9: nat] : ( ord_less_eq @ A @ X3 @ ( A2 @ N9 ) )
& ! [M2: nat,N9: nat] :
( ( ord_less_eq @ nat @ M2 @ N9 )
=> ( ord_less_eq @ A @ ( A2 @ N9 ) @ ( A2 @ M2 ) ) ) ) ) ) ) ) ).
% monoseq_le
thf(fact_5598_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_5599_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_5600_LIMSEQ__linear,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X8: nat > A,X3: A,L: nat] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( X8 @ ( times_times @ nat @ N3 @ L ) )
@ ( topolo7230453075368039082e_nhds @ A @ X3 )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_linear
thf(fact_5601_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 ) )
=> ? [L4: real] :
( ! [N9: nat] : ( ord_less_eq @ real @ ( F2 @ N9 ) @ L4 )
& ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L4 ) @ ( at_top @ nat ) )
& ! [N9: nat] : ( ord_less_eq @ real @ L4 @ ( G @ N9 ) )
& ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ L4 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_5602_LIMSEQ__inverse__zero,axiom,
! [X8: nat > real] :
( ! [R: real] :
? [N7: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N7 @ N2 )
=> ( ord_less @ real @ R @ ( X8 @ N2 ) ) )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( inverse_inverse @ real @ ( X8 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_zero
thf(fact_5603_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_5604_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_5605_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_5606_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N3: nat] : ( plus_plus @ real @ R2 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_5607_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_5608_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_5609_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_5610_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_5611_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_5612_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_5613_LIMSEQ__realpow__zero,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ X3 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_realpow_zero
thf(fact_5614_LIMSEQ__divide__realpow__zero,axiom,
! [X3: real,A2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( divide_divide @ real @ A2 @ ( power_power @ real @ X3 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_5615_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_5616_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_5617_LIMSEQ__inverse__realpow__zero,axiom,
! [X3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X3 )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( inverse_inverse @ real @ ( power_power @ real @ X3 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_5618_sums__def_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F3: nat > A,S6: A] :
( filterlim @ nat @ A
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S6 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def'
thf(fact_5619_root__test__convergence,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,X3: real] :
( ( filterlim @ nat @ real
@ ^ [N3: nat] : ( root @ N3 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ X3 )
@ ( at_top @ nat ) )
=> ( ( ord_less @ real @ X3 @ ( one_one @ real ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% root_test_convergence
thf(fact_5620_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N3: nat] : ( plus_plus @ real @ R2 @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_5621_LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,L5: A,R2: real] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ No @ N9 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N9 ) @ L5 ) ) @ R2 ) ) ) ) ) ).
% LIMSEQ_D
thf(fact_5622_LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,L5: 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 @ ( X8 @ N2 ) @ L5 ) ) @ R ) ) )
=> ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_I
thf(fact_5623_LIMSEQ__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No3 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N3 ) @ L5 ) ) @ R5 ) ) ) ) ) ) ).
% LIMSEQ_iff
thf(fact_5624_LIMSEQ__power__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X3 ) @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_power_zero
thf(fact_5625_tendsto__exp__limit__sequentially,axiom,
! [X3: real] :
( filterlim @ nat @ real
@ ^ [N3: nat] : ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X3 @ ( semiring_1_of_nat @ real @ N3 ) ) ) @ N3 )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X3 ) )
@ ( at_top @ nat ) ) ).
% tendsto_exp_limit_sequentially
thf(fact_5626_tendsto__power__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F2: B > nat,F4: filter @ B,X3: A] :
( ( filterlim @ B @ nat @ F2 @ ( at_top @ nat ) @ F4 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( one_one @ real ) )
=> ( filterlim @ B @ A
@ ^ [Y: B] : ( power_power @ A @ X3 @ ( F2 @ Y ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_power_zero
thf(fact_5627_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N3: nat] : ( times_times @ real @ R2 @ ( plus_plus @ real @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_5628_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_5629_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_5630_field__derivative__lim__unique,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Df: A,Z: A,S: nat > A,A2: A] :
( ( has_field_derivative @ A @ F2 @ Df @ ( topolo174197925503356063within @ A @ Z @ ( 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 @ Z @ ( S @ N3 ) ) ) @ ( F2 @ Z ) ) @ ( S @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) )
=> ( Df = A2 ) ) ) ) ) ) ).
% field_derivative_lim_unique
thf(fact_5631_powser__times__n__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( power_power @ A @ X3 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% powser_times_n_limit_0
thf(fact_5632_lim__n__over__pown,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X3: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( power_power @ A @ X3 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% lim_n_over_pown
thf(fact_5633_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_5634_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_5635_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_5636_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_5637_zeroseq__arctan__series,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% zeroseq_arctan_series
thf(fact_5638_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_5639_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_5640_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 ) )
=> ? [L4: real] :
( ! [N9: nat] :
( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) ) )
@ L4 )
& ( filterlim @ nat @ real
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L4 )
@ ( at_top @ nat ) )
& ! [N9: nat] :
( ord_less_eq @ real @ L4
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( 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 @ L4 )
@ ( at_top @ nat ) ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_5641_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_5642_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_5643_has__derivative__at2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F6: A > B,X3: A] :
( ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F6 )
& ( filterlim @ A @ B
@ ^ [Y: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X3 ) ) ) @ ( minus_minus @ B @ ( F2 @ Y ) @ ( plus_plus @ B @ ( F2 @ X3 ) @ ( F6 @ ( minus_minus @ A @ Y @ X3 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at2
thf(fact_5644_has__derivative__at,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,D5: A > B,X3: A] :
( ( has_derivative @ A @ B @ F2 @ D5 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ D5 )
& ( filterlim @ A @ real
@ ^ [H2: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ ( plus_plus @ A @ X3 @ H2 ) ) @ ( F2 @ X3 ) ) @ ( D5 @ H2 ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at
thf(fact_5645_has__derivative__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F6: A > B,X3: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F6 )
& ( filterlim @ A @ B
@ ^ [Y: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X3 ) ) ) @ ( minus_minus @ B @ ( F2 @ Y ) @ ( plus_plus @ B @ ( F2 @ X3 ) @ ( F6 @ ( minus_minus @ A @ Y @ X3 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% has_derivative_within
thf(fact_5646_bounded__linear__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X: A] : ( zero_zero @ B ) ) ) ).
% bounded_linear_zero
thf(fact_5647_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_5648_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
@ ^ [X: C] : ( F2 @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% bounded_linear.tendsto_zero
thf(fact_5649_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_5650_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_5651_bounded__linear__intro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,K5: real] :
( ! [X4: A,Y4: A] :
( ( F2 @ ( plus_plus @ A @ X4 @ Y4 ) )
= ( plus_plus @ B @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ! [R: real,X4: A] :
( ( F2 @ ( real_V8093663219630862766scaleR @ A @ R @ X4 ) )
= ( real_V8093663219630862766scaleR @ B @ R @ ( F2 @ X4 ) ) )
=> ( ! [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X4 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ K5 ) )
=> ( real_V3181309239436604168linear @ A @ B @ F2 ) ) ) ) ) ).
% bounded_linear_intro
thf(fact_5652_has__derivative__iff__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F6: A > B,X3: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F6 )
& ( filterlim @ A @ real
@ ^ [Y: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y ) @ ( F2 @ X3 ) ) @ ( F6 @ ( minus_minus @ A @ Y @ X3 ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% has_derivative_iff_norm
thf(fact_5653_has__derivativeI,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F6: A > B,X3: A,F2: A > B,S: set @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F6 )
=> ( ( filterlim @ A @ B
@ ^ [Y: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X3 ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y ) @ ( F2 @ X3 ) ) @ ( F6 @ ( minus_minus @ A @ Y @ X3 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% has_derivativeI
thf(fact_5654_has__derivative__at__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F6: A > B,X3: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F6 )
& ( filterlim @ A @ B
@ ^ [Y: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X3 ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y ) @ ( F2 @ X3 ) ) @ ( F6 @ ( minus_minus @ A @ Y @ X3 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% has_derivative_at_within
thf(fact_5655_has__derivative__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F6: A > B,X3: A] :
( ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F6 )
& ? [E3: A > B] :
( ! [H2: A] :
( ( F2 @ ( plus_plus @ A @ X3 @ H2 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X3 ) @ ( F6 @ H2 ) ) @ ( E3 @ H2 ) ) )
& ( filterlim @ A @ real
@ ^ [H2: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E3 @ H2 ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% has_derivative_iff_Ex
thf(fact_5656_has__derivative__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( has_derivative @ A @ B )
= ( ^ [F3: A > B,F7: A > B,F8: filter @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( filterlim @ A @ B
@ ^ [Y: A] :
( real_V8093663219630862766scaleR @ B
@ ( inverse_inverse @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( minus_minus @ A @ Y
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X: A] : X ) ) ) )
@ ( minus_minus @ B
@ ( minus_minus @ B @ ( F3 @ Y )
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X: A] : X ) ) )
@ ( F7
@ ( minus_minus @ A @ Y
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X: A] : X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F8 ) ) ) ) ) ).
% has_derivative_def
thf(fact_5657_has__derivative__at__within__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [X3: A,S2: set @ A,F2: A > B,F6: A > B] :
( ( member @ A @ X3 @ S2 )
=> ( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X3 @ S2 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F6 )
& ? [E3: A > B] :
( ! [H2: A] :
( ( member @ A @ ( plus_plus @ A @ X3 @ H2 ) @ S2 )
=> ( ( F2 @ ( plus_plus @ A @ X3 @ H2 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X3 ) @ ( F6 @ H2 ) ) @ ( E3 @ H2 ) ) ) )
& ( filterlim @ A @ real
@ ^ [H2: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E3 @ H2 ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% has_derivative_at_within_iff_Ex
thf(fact_5658_has__derivativeI__sandwich,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [E2: real,F6: A > B,S: set @ A,X3: A,F2: A > B,H6: A > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( ( real_V3181309239436604168linear @ A @ B @ F6 )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ S )
=> ( ( Y4 != X3 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y4 @ X3 ) @ E2 )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) @ ( F6 @ ( minus_minus @ A @ Y4 @ X3 ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y4 @ X3 ) ) ) @ ( H6 @ Y4 ) ) ) ) )
=> ( ( filterlim @ A @ real @ H6 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ) ).
% has_derivativeI_sandwich
thf(fact_5659_dist__self,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X3: A] :
( ( real_V557655796197034286t_dist @ A @ X3 @ X3 )
= ( zero_zero @ real ) ) ) ).
% dist_self
thf(fact_5660_dist__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X3: A,Y3: A] :
( ( ( real_V557655796197034286t_dist @ A @ X3 @ Y3 )
= ( zero_zero @ real ) )
= ( X3 = Y3 ) ) ) ).
% dist_eq_0_iff
thf(fact_5661_dist__0__norm,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X3: A] :
( ( real_V557655796197034286t_dist @ A @ ( zero_zero @ A ) @ X3 )
= ( real_V7770717601297561774m_norm @ A @ X3 ) ) ) ).
% dist_0_norm
thf(fact_5662_zero__less__dist__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X3 @ Y3 ) )
= ( X3 != Y3 ) ) ) ).
% zero_less_dist_iff
thf(fact_5663_dist__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Y3 ) @ ( zero_zero @ real ) )
= ( X3 = Y3 ) ) ) ).
% dist_le_zero_iff
thf(fact_5664_dist__triangle__less__add,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X15: A,Y3: A,E1: real,X2: A,E22: real] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ Y3 ) @ E1 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y3 ) @ E22 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X2 ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% dist_triangle_less_add
thf(fact_5665_dist__triangle__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X3: A,Z: A,Y3: A,E2: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y3 @ Z ) ) @ E2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Y3 ) @ E2 ) ) ) ).
% dist_triangle_lt
thf(fact_5666_dist__pos__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X3: A,Y3: A] :
( ( X3 != Y3 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X3 @ Y3 ) ) ) ) ).
% dist_pos_lt
thf(fact_5667_dist__not__less__zero,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X3: A,Y3: A] :
~ ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Y3 ) @ ( zero_zero @ real ) ) ) ).
% dist_not_less_zero
thf(fact_5668_open__dist,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S8: set @ A] :
! [X: A] :
( ( member @ A @ X @ S8 )
=> ? [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
& ! [Y: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X ) @ E3 )
=> ( member @ A @ Y @ S8 ) ) ) ) ) ) ) ).
% open_dist
thf(fact_5669_dist__commute__lessI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y3: A,X3: A,E2: real] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y3 @ X3 ) @ E2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Y3 ) @ E2 ) ) ) ).
% dist_commute_lessI
thf(fact_5670_open__ball,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X3: A,D3: real] :
( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [Y: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Y ) @ D3 ) ) ) ) ).
% open_ball
thf(fact_5671_norm__conv__dist,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A )
= ( ^ [X: A] : ( real_V557655796197034286t_dist @ A @ X @ ( zero_zero @ A ) ) ) ) ) ).
% norm_conv_dist
thf(fact_5672_zero__le__dist,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X3 @ Y3 ) ) ) ).
% zero_le_dist
thf(fact_5673_dist__triangle,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X3: A,Z: A,Y3: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Z ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Y3 ) @ ( real_V557655796197034286t_dist @ A @ Y3 @ Z ) ) ) ) ).
% dist_triangle
thf(fact_5674_dist__triangle2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X3: A,Y3: A,Z: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Y3 ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y3 @ Z ) ) ) ) ).
% dist_triangle2
thf(fact_5675_dist__triangle3,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X3: A,Y3: A,A2: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Y3 ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ A2 @ X3 ) @ ( real_V557655796197034286t_dist @ A @ A2 @ Y3 ) ) ) ) ).
% dist_triangle3
thf(fact_5676_dist__triangle__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X3: A,Z: A,Y3: A,E2: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y3 @ Z ) ) @ E2 )
=> ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Y3 ) @ E2 ) ) ) ).
% dist_triangle_le
thf(fact_5677_Inf__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A4: set @ A,X3: A] :
( ( topolo1002775350975398744n_open @ A @ A4 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less @ A @ X3 @ X4 ) )
=> ~ ( member @ A @ ( complete_Inf_Inf @ A @ A4 ) @ A4 ) ) ) ) ).
% Inf_notin_open
thf(fact_5678_Sup__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A4: set @ A,X3: A] :
( ( topolo1002775350975398744n_open @ A @ A4 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less @ A @ X4 @ X3 ) )
=> ~ ( member @ A @ ( complete_Sup_Sup @ A @ A4 ) @ A4 ) ) ) ) ).
% Sup_notin_open
thf(fact_5679_open__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S2: set @ A,X3: A,Y3: A] :
( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( member @ A @ X3 @ S2 )
=> ( ( ord_less @ A @ X3 @ Y3 )
=> ? [B5: A] :
( ( ord_less @ A @ X3 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ X3 @ B5 ) @ S2 ) ) ) ) ) ) ).
% open_right
thf(fact_5680_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_5681_filterlim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [G: A > B,G5: filter @ B,X3: A,S2: set @ A,F4: filter @ B,D3: real,F2: A > B] :
( ( filterlim @ A @ B @ G @ G5 @ ( topolo174197925503356063within @ A @ X3 @ S2 ) )
=> ( ( ord_less_eq @ ( filter @ B ) @ G5 @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
=> ( ! [X16: A] :
( ( member @ A @ X16 @ S2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X16 @ X3 ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ X3 ) @ D3 )
=> ( ( F2 @ X16 )
= ( G @ X16 ) ) ) ) )
=> ( filterlim @ A @ B @ F2 @ F4 @ ( topolo174197925503356063within @ A @ X3 @ S2 ) ) ) ) ) ) ) ).
% filterlim_transform_within
thf(fact_5682_has__field__derivative__transform__within,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F6: A,A2: A,S2: set @ A,D3: real,G: A > A] :
( ( has_field_derivative @ A @ F2 @ F6 @ ( topolo174197925503356063within @ A @ A2 @ S2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
=> ( ( member @ A @ A2 @ S2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ A2 ) @ D3 )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( has_field_derivative @ A @ G @ F6 @ ( topolo174197925503356063within @ A @ A2 @ S2 ) ) ) ) ) ) ) ).
% has_field_derivative_transform_within
thf(fact_5683_has__derivative__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F6: A > B,X3: A,S: set @ A,D3: real,G: A > B] :
( ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
=> ( ( member @ A @ X3 @ S )
=> ( ! [X16: A] :
( ( member @ A @ X16 @ S )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ X3 ) @ D3 )
=> ( ( F2 @ X16 )
= ( G @ X16 ) ) ) )
=> ( has_derivative @ A @ B @ G @ F6 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ) ).
% has_derivative_transform_within
thf(fact_5684_metric__CauchyI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: 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 @ ( X8 @ M4 ) @ ( X8 @ N2 ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ).
% metric_CauchyI
thf(fact_5685_metric__CauchyD,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,E2: real] :
( ( topolo3814608138187158403Cauchy @ A @ X8 )
=> ( ( 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 @ ( X8 @ M2 ) @ ( X8 @ N9 ) ) @ E2 ) ) ) ) ) ) ).
% metric_CauchyD
thf(fact_5686_Cauchy__altdef2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [S6: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [N6: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( S6 @ N3 ) @ ( S6 @ N6 ) ) @ E3 ) ) ) ) ) ) ).
% Cauchy_altdef2
thf(fact_5687_Cauchy__def,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X9: 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 @ ( X9 @ M3 ) @ ( X9 @ N3 ) ) @ E3 ) ) ) ) ) ) ) ).
% Cauchy_def
thf(fact_5688_lim__explicit,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,F0: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ F0 ) @ ( at_top @ nat ) )
= ( ! [S8: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S8 )
=> ( ( member @ A @ F0 @ S8 )
=> ? [N6: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( member @ A @ ( F2 @ N3 ) @ S8 ) ) ) ) ) ) ) ).
% lim_explicit
thf(fact_5689_dist__of__int,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [M: int,N: int] :
( ( real_V557655796197034286t_dist @ A @ ( ring_1_of_int @ A @ M ) @ ( ring_1_of_int @ A @ N ) )
= ( ring_1_of_int @ real @ ( abs_abs @ int @ ( minus_minus @ int @ M @ N ) ) ) ) ) ).
% dist_of_int
thf(fact_5690_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
@ ^ [X: A] : X ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% continuous_divide
thf(fact_5691_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
@ ^ [X: A] : X ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X: A] : ( inverse_inverse @ B @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_inverse
thf(fact_5692_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
@ ^ [X: A] : X ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X: A] : ( sgn_sgn @ B @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_sgn
thf(fact_5693_dist__triangle__half__l,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X15: A,Y3: A,E2: real,X2: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ Y3 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y3 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X2 ) @ E2 ) ) ) ) ).
% dist_triangle_half_l
thf(fact_5694_dist__triangle__half__r,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y3: A,X15: A,E2: real,X2: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y3 @ X15 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y3 @ X2 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X2 ) @ E2 ) ) ) ) ).
% dist_triangle_half_r
thf(fact_5695_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 ) ) ) )
=> ( ! [X4: C] :
( ( X4 != A2 )
=> ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( G @ X4 ) @ M ) @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X4 ) @ L ) ) )
=> ( filterlim @ C @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ M ) @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) ) ) ) ) ).
% metric_LIM_imp_LIM
thf(fact_5696_Lim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L: B,X3: A,S2: set @ A,D3: real,G: A > B] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ X3 @ S2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
=> ( ! [X16: A] :
( ( member @ A @ X16 @ S2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X16 @ X3 ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ X3 ) @ D3 )
=> ( ( F2 @ X16 )
= ( G @ X16 ) ) ) ) )
=> ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ X3 @ S2 ) ) ) ) ) ) ).
% Lim_transform_within
thf(fact_5697_dist__triangle__third,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X15: A,X2: A,E2: real,X32: A,X42: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X2 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ 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_5698_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
@ ^ [X: A] : X ) )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% continuous_powr
thf(fact_5699_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
@ ^ [X: A] : X ) )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X: A] : ( ln_ln @ real @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_ln
thf(fact_5700_CauchyI_H,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: 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 @ ( X8 @ M4 ) @ ( X8 @ N2 ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ).
% CauchyI'
thf(fact_5701_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_5702_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_5703_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
@ ^ [X: A] : ( real_V557655796197034286t_dist @ B @ ( F2 @ X ) @ L )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% tendsto_dist_iff
thf(fact_5704_LIM__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F2: A > B,L5: B,A2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S6 )
& ! [X: A] :
( ( ( X != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ A2 ) @ S6 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X ) @ L5 ) @ R5 ) ) ) ) ) ) ) ).
% LIM_def
thf(fact_5705_metric__LIM__D,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F2: A > B,L5: B,A2: A,R2: real] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [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 ) @ L5 ) @ R2 ) ) ) ) ) ) ).
% metric_LIM_D
thf(fact_5706_metric__LIM__I,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [A2: A,F2: A > B,L5: B] :
( ! [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [S7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S7 )
& ! [X4: A] :
( ( ( X4 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ A2 ) @ S7 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X4 ) @ L5 ) @ R ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% metric_LIM_I
thf(fact_5707_metric__LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [G: A > B,L: B,A2: A,R3: 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 ) @ R3 )
=> ( ! [X4: A] :
( ( X4 != A2 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ A2 ) @ R3 )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_equal2
thf(fact_5708_metric__LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L5: A,R2: real] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ No @ N9 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N9 ) @ L5 ) @ R2 ) ) ) ) ) ).
% metric_LIMSEQ_D
thf(fact_5709_metric__LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L5: 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 @ ( X8 @ N2 ) @ L5 ) @ R ) ) )
=> ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) ) ) ).
% metric_LIMSEQ_I
thf(fact_5710_lim__sequentially,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No3 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N3 ) @ L5 ) @ R5 ) ) ) ) ) ) ).
% lim_sequentially
thf(fact_5711_metric__Cauchy__iff2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X9: 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 @ ( X9 @ M3 ) @ ( X9 @ N3 ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ) ).
% metric_Cauchy_iff2
thf(fact_5712_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 )
& ! [X4: A] :
( ( ( X4 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ A2 ) @ D2 ) )
=> ( ( F2 @ X4 )
!= B2 ) ) )
=> ( filterlim @ A @ C
@ ^ [X: A] : ( G @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ C @ C2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_compose2
thf(fact_5713_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
@ ^ [X: A] : X ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F4
@ ^ [X: A] : ( tan @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_tan
thf(fact_5714_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
@ ^ [X: A] : X ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F4
@ ^ [X: A] : ( cot @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_cot
thf(fact_5715_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
@ ^ [X: C] : X ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X: C] : ( tanh @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_tanh
thf(fact_5716_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
@ ^ [X: A] : X ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X: A] : ( arcosh @ real @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_arcosh
thf(fact_5717_metric__isCont__LIM__compose2,axiom,
! [D: $tType,C: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ D ) )
=> ! [A2: A,F2: A > C,G: C > D,L: D] :
( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( filterlim @ C @ D @ G @ ( topolo7230453075368039082e_nhds @ D @ L ) @ ( topolo174197925503356063within @ C @ ( F2 @ A2 ) @ ( top_top @ ( set @ C ) ) ) )
=> ( ? [D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
& ! [X4: A] :
( ( ( X4 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ A2 ) @ D2 ) )
=> ( ( F2 @ X4 )
!= ( F2 @ A2 ) ) ) )
=> ( filterlim @ A @ D
@ ^ [X: A] : ( G @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ D @ L )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_isCont_LIM_compose2
thf(fact_5718_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
@ ^ [X: A] : X ) ) )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X: A] : X ) )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( G
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X: A] : X ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ).
% continuous_log
thf(fact_5719_LIMSEQ__iff__nz,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ No3 )
& ! [N3: nat] :
( ( ord_less_eq @ nat @ No3 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N3 ) @ L5 ) @ R5 ) ) ) ) ) ) ) ).
% LIMSEQ_iff_nz
thf(fact_5720_totally__bounded__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo6688025880775521714ounded @ A )
= ( ^ [S8: set @ A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [K3: set @ A] :
( ( finite_finite @ A @ K3 )
& ( ord_less_eq @ ( set @ A ) @ S8
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image @ A @ ( set @ A )
@ ^ [X: A] :
( collect @ A
@ ^ [Y: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E3 ) )
@ K3 ) ) ) ) ) ) ) ) ).
% totally_bounded_metric
thf(fact_5721_tendsto__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A2: A,S2: set @ A,F2: A > D,L5: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 )
=> ( ( member @ A @ A2 @ S2 )
=> ( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( filterlim @ A @ D @ F2 @ ( topolo7230453075368039082e_nhds @ D @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ S2 ) )
= ( filterlim @ A @ D
@ ^ [H2: A] : ( F2 @ ( plus_plus @ A @ A2 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ D @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% tendsto_offset_zero_iff
thf(fact_5722_tendsto__exp__limit__at__right,axiom,
! [X3: real] :
( filterlim @ real @ real
@ ^ [Y: real] : ( powr @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ X3 @ Y ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ Y ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X3 ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ).
% tendsto_exp_limit_at_right
thf(fact_5723_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_5724_cInf__greaterThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A )
& ( no_top @ A ) )
=> ! [X3: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_greaterThan @ A @ X3 ) )
= X3 ) ) ).
% cInf_greaterThan
thf(fact_5725_greaterThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X3 ) @ ( set_ord_greaterThan @ A @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% greaterThan_subset_iff
thf(fact_5726_Sup__greaterThanAtLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A] :
( ( ord_less @ A @ X3 @ ( top_top @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( set_ord_greaterThan @ A @ X3 ) )
= ( top_top @ A ) ) ) ) ).
% Sup_greaterThanAtLeast
thf(fact_5727_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less @ A @ L2 ) ) ) ) ) ).
% greaterThan_def
thf(fact_5728_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_5729_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
@ ^ [X: real] : ( F2 @ ( plus_plus @ real @ X @ A2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% filterlim_at_right_to_0
thf(fact_5730_filterlim__times__pos,axiom,
! [A: $tType,B: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: B > A,P6: A,F13: filter @ B,C2: A,L: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ P6 @ ( set_ord_greaterThan @ A @ P6 ) ) @ F13 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( L
= ( times_times @ A @ C2 @ P6 ) )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( times_times @ A @ C2 @ ( F2 @ X ) )
@ ( topolo174197925503356063within @ A @ L @ ( set_ord_greaterThan @ A @ L ) )
@ F13 ) ) ) ) ) ).
% filterlim_times_pos
thf(fact_5731_power__minus_H,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: A,N: nat] :
( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X3 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ X3 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ X3 @ N ) ) ) ) ) ).
% power_minus'
thf(fact_5732_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_5733_isCont__If__ge,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,G: A > B,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) @ G )
=> ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( G @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( if @ B @ ( ord_less_eq @ A @ X @ A2 ) @ ( G @ X ) @ ( F2 @ X ) ) ) ) ) ) ).
% isCont_If_ge
thf(fact_5734_LIM__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A2: A,F2: A > D,L5: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 )
=> ( ( filterlim @ A @ D @ F2 @ ( topolo7230453075368039082e_nhds @ D @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ D
@ ^ [H2: A] : ( F2 @ ( plus_plus @ A @ A2 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ D @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_offset_zero_iff
thf(fact_5735_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X3: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X3 @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self1_no_field
thf(fact_5736_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X3: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X3 @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
thf(fact_5737_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_5738_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_5739_exp__at__bot,axiom,
filterlim @ real @ real @ ( exp @ real ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_bot @ real ) ).
% exp_at_bot
thf(fact_5740_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_5741_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
@ ^ [X: A] : ( times_times @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_bot
thf(fact_5742_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_5743_DERIV__pos__imp__increasing__at__bot,axiom,
! [B2: real,F2: real > real,Flim: real] :
( ! [X4: real] :
( ( ord_less_eq @ real @ X4 @ B2 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) ) )
=> ( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_bot @ real ) )
=> ( ord_less @ real @ Flim @ ( F2 @ B2 ) ) ) ) ).
% DERIV_pos_imp_increasing_at_bot
thf(fact_5744_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
@ ^ [X: real] : ( power_power @ real @ ( F2 @ X ) @ N )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_5745_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_5746_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
@ ^ [X: real] : ( power_power @ real @ ( F2 @ X ) @ N )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_5747_lim__zero__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L: A] :
( ( filterlim @ A @ A
@ ^ [X: A] : ( F2 @ ( divide_divide @ A @ ( one_one @ A ) @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) ) ) ) ).
% lim_zero_infinity
thf(fact_5748_rat__inverse__code,axiom,
! [P6: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P6 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,B4: 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 ) @ B4 ) @ ( abs_abs @ int @ A5 ) ) )
@ ( quotient_of @ P6 ) ) ) ).
% rat_inverse_code
thf(fact_5749_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_5750_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_5751_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_5752_sqrt__at__top,axiom,
filterlim @ real @ real @ sqrt @ ( at_top @ real ) @ ( at_top @ real ) ).
% sqrt_at_top
thf(fact_5753_quotient__of__denom__pos,axiom,
! [R2: rat,P6: int,Q3: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair @ int @ int @ P6 @ Q3 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q3 ) ) ).
% quotient_of_denom_pos
thf(fact_5754_filterlim__real__sequentially,axiom,
filterlim @ nat @ real @ ( semiring_1_of_nat @ real ) @ ( at_top @ real ) @ ( at_top @ nat ) ).
% filterlim_real_sequentially
thf(fact_5755_quotient__of__denom__pos_H,axiom,
! [R2: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ ( quotient_of @ R2 ) ) ) ).
% quotient_of_denom_pos'
thf(fact_5756_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_5757_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_5758_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
@ ^ [X: A] : ( power_power @ real @ ( F2 @ X ) @ N )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_pow_at_top
thf(fact_5759_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
@ ^ [X: A] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% real_tendsto_divide_at_top
thf(fact_5760_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
@ ^ [X: A] : ( inverse_inverse @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ).
% tendsto_inverse_0_at_top
thf(fact_5761_rat__less__eq__code,axiom,
( ( ord_less_eq @ rat )
= ( ^ [P5: rat,Q5: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A5: int,C4: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B4: int,D4: int] : ( ord_less_eq @ int @ ( times_times @ int @ A5 @ D4 ) @ ( times_times @ int @ C4 @ B4 ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P5 ) ) ) ) ).
% rat_less_eq_code
thf(fact_5762_rat__less__code,axiom,
( ( ord_less @ rat )
= ( ^ [P5: rat,Q5: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A5: int,C4: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B4: int,D4: int] : ( ord_less @ int @ ( times_times @ int @ A5 @ D4 ) @ ( times_times @ int @ C4 @ B4 ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P5 ) ) ) ) ).
% rat_less_code
thf(fact_5763_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
@ ^ [X: A] : ( semiring_1_of_nat @ real @ ( F2 @ X ) )
@ ( at_top @ real )
@ F4 ) ) ).
% filterlim_sequentially_iff_filterlim_real
thf(fact_5764_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_5765_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
@ ^ [X: A] : ( times_times @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_top
thf(fact_5766_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
@ ^ [X: A] : ( times_times @ real @ ( G @ X ) @ ( F2 @ X ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_at_top_mult_tendsto_pos
thf(fact_5767_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
@ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ S )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% tendsto_neg_powr
thf(fact_5768_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
@ ^ [X: B] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_infinity @ A )
@ F4 ) ) ) ) ) ).
% tendsto_mult_filterlim_at_infinity
thf(fact_5769_ln__x__over__x__tendsto__0,axiom,
( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( ln_ln @ real @ X ) @ X )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ real ) ) ).
% ln_x_over_x_tendsto_0
thf(fact_5770_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
@ ^ [X: C] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_divide_0
thf(fact_5771_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
@ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N )
@ ( at_infinity @ B )
@ F4 ) ) ) ) ).
% filterlim_power_at_infinity
thf(fact_5772_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
@ ^ [X: real] : ( F2 @ ( inverse_inverse @ real @ X ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% filterlim_at_top_to_right
thf(fact_5773_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
@ ^ [X: real] : ( F2 @ ( inverse_inverse @ real @ X ) )
@ F4
@ ( at_top @ real ) ) ) ).
% filterlim_at_right_to_top
thf(fact_5774_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_5775_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_5776_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_5777_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
@ ^ [X: A] : ( times_times @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_neg_mult_at_bot
thf(fact_5778_tendsto__power__div__exp__0,axiom,
! [K: nat] :
( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( power_power @ real @ X @ K ) @ ( exp @ real @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ real ) ) ).
% tendsto_power_div_exp_0
thf(fact_5779_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_5780_filterlim__inverse__at__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [G: A > B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X: A] : ( inverse_inverse @ B @ ( G @ X ) )
@ ( topolo174197925503356063within @ B @ ( zero_zero @ B ) @ ( top_top @ ( set @ B ) ) )
@ F4 )
= ( filterlim @ A @ B @ G @ ( at_infinity @ B ) @ F4 ) ) ) ).
% filterlim_inverse_at_iff
thf(fact_5781_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
@ ^ [X: A] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_infinity @ A )
@ F4 ) ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_5782_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_5783_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_5784_DERIV__neg__imp__decreasing__at__top,axiom,
! [B2: real,F2: real > real,Flim: real] :
( ! [X4: real] :
( ( ord_less_eq @ real @ B2 @ X4 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y5 @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_top @ real ) )
=> ( ord_less @ real @ Flim @ ( F2 @ B2 ) ) ) ) ).
% DERIV_neg_imp_decreasing_at_top
thf(fact_5785_filterlim__realpow__sequentially__gt1,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X3: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X3 ) @ ( at_infinity @ A ) @ ( at_top @ nat ) ) ) ) ).
% filterlim_realpow_sequentially_gt1
thf(fact_5786_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_5787_polyfun__extremal,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [C2: nat > A,K: nat,N: nat,B6: real] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( eventually @ A
@ ^ [Z6: A] :
( ord_less_eq @ real @ B6
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z6 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
@ ( at_infinity @ A ) ) ) ) ) ) ).
% polyfun_extremal
thf(fact_5788_lhopital__left__at__top,axiom,
! [G: real > real,X3: real,G4: real > real,F2: real > real,F6: real > real,Y3: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_lessThan @ real @ X3 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G4 @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_lessThan @ real @ X3 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_lessThan @ real @ X3 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_lessThan @ real @ X3 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F6 @ X ) @ ( G4 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y3 )
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_lessThan @ real @ X3 ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y3 )
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_lessThan @ real @ X3 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top
thf(fact_5789_lhopital__right__at__top,axiom,
! [G: real > real,X3: real,G4: real > real,F2: real > real,F6: real > real,Y3: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_greaterThan @ real @ X3 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G4 @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_greaterThan @ real @ X3 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_greaterThan @ real @ X3 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_greaterThan @ real @ X3 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F6 @ X ) @ ( G4 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y3 )
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_greaterThan @ real @ X3 ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y3 )
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_greaterThan @ real @ X3 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top
thf(fact_5790_obtain__pos__sum,axiom,
! [R2: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 )
=> ~ ! [S3: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ S3 )
=> ! [T4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ T4 )
=> ( R2
!= ( plus_plus @ rat @ S3 @ T4 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_5791_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_5792_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_5793_eventually__nhds__top,axiom,
! [A: $tType] :
( ( ( order_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: A,P: A > $o] :
( ( ord_less @ A @ B2 @ ( top_top @ A ) )
=> ( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ ( top_top @ A ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ B4 @ ( top_top @ A ) )
& ! [Z6: A] :
( ( ord_less @ A @ B4 @ Z6 )
=> ( P @ Z6 ) ) ) ) ) ) ) ).
% eventually_nhds_top
thf(fact_5794_eventually__at__left__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P: A > $o,X3: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X3 @ ( set_ord_lessThan @ A @ X3 ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ B4 @ X3 )
& ! [Y: A] :
( ( ord_less @ A @ B4 @ Y )
=> ( ( ord_less @ A @ Y @ X3 )
=> ( P @ Y ) ) ) ) ) ) ) ).
% eventually_at_left_field
thf(fact_5795_eventually__at__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [Y3: A,X3: A,P: A > $o] :
( ( ord_less @ A @ Y3 @ X3 )
=> ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X3 @ ( set_ord_lessThan @ A @ X3 ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ B4 @ X3 )
& ! [Y: A] :
( ( ord_less @ A @ B4 @ Y )
=> ( ( ord_less @ A @ Y @ X3 )
=> ( P @ Y ) ) ) ) ) ) ) ) ).
% eventually_at_left
thf(fact_5796_eventually__at__right__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P: A > $o,X3: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X3 @ ( set_ord_greaterThan @ A @ X3 ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ X3 @ B4 )
& ! [Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less @ A @ Y @ B4 )
=> ( P @ Y ) ) ) ) ) ) ) ).
% eventually_at_right_field
thf(fact_5797_eventually__at__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X3: A,Y3: A,P: A > $o] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X3 @ ( set_ord_greaterThan @ A @ X3 ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ X3 @ B4 )
& ! [Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less @ A @ Y @ B4 )
=> ( P @ Y ) ) ) ) ) ) ) ) ).
% eventually_at_right
thf(fact_5798_eventually__at__infinity,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_infinity @ A ) )
= ( ? [B4: real] :
! [X: A] :
( ( ord_less_eq @ real @ B4 @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( P @ X ) ) ) ) ) ).
% eventually_at_infinity
thf(fact_5799_tendsto__sandwich,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,G: B > A,Net: filter @ B,H: 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 ) @ ( H @ N3 ) )
@ Net )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net )
=> ( ( filterlim @ B @ A @ H @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net )
=> ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net ) ) ) ) ) ) ).
% tendsto_sandwich
thf(fact_5800_tendsto__le,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F4: filter @ B,F2: B > A,X3: A,G: B > A,Y3: A] :
( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ Y3 ) @ F4 )
=> ( ( eventually @ B
@ ^ [X: B] : ( ord_less_eq @ A @ ( G @ X ) @ ( F2 @ X ) )
@ F4 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ) ).
% tendsto_le
thf(fact_5801_tendsto__lowerbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: B > A,X3: A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ F4 )
=> ( ( eventually @ B
@ ^ [I4: B] : ( ord_less_eq @ A @ A2 @ ( F2 @ I4 ) )
@ F4 )
=> ( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ A2 @ X3 ) ) ) ) ) ).
% tendsto_lowerbound
thf(fact_5802_tendsto__upperbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: B > A,X3: A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ F4 )
=> ( ( eventually @ B
@ ^ [I4: B] : ( ord_less_eq @ A @ ( F2 @ I4 ) @ A2 )
@ F4 )
=> ( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ X3 @ A2 ) ) ) ) ) ).
% tendsto_upperbound
thf(fact_5803_order__tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,X3: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ F4 )
= ( ! [L2: A] :
( ( ord_less @ A @ L2 @ X3 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ A @ L2 @ ( F2 @ X ) )
@ F4 ) )
& ! [U2: A] :
( ( ord_less @ A @ X3 @ U2 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ A @ ( F2 @ X ) @ U2 )
@ F4 ) ) ) ) ) ).
% order_tendsto_iff
thf(fact_5804_order__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [Y3: A,F2: B > A,F4: filter @ B] :
( ! [A6: A] :
( ( ord_less @ A @ A6 @ Y3 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ A @ A6 @ ( F2 @ X ) )
@ F4 ) )
=> ( ! [A6: A] :
( ( ord_less @ A @ Y3 @ A6 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ A @ ( F2 @ X ) @ A6 )
@ F4 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y3 ) @ F4 ) ) ) ) ).
% order_tendstoI
thf(fact_5805_order__tendstoD_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,Y3: A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y3 ) @ F4 )
=> ( ( ord_less @ A @ A2 @ Y3 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ A @ A2 @ ( F2 @ X ) )
@ F4 ) ) ) ) ).
% order_tendstoD(1)
thf(fact_5806_order__tendstoD_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,Y3: A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y3 ) @ F4 )
=> ( ( ord_less @ A @ Y3 @ A2 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ A @ ( F2 @ X ) @ A2 )
@ F4 ) ) ) ) ).
% order_tendstoD(2)
thf(fact_5807_eventually__at__right__less,axiom,
! [A: $tType] :
( ( ( no_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X3: A] : ( eventually @ A @ ( ord_less @ A @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ ( set_ord_greaterThan @ A @ X3 ) ) ) ) ).
% eventually_at_right_less
thf(fact_5808_real__tendsto__sandwich,axiom,
! [B: $tType,F2: B > real,G: B > real,Net: filter @ B,H: 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 ) @ ( H @ N3 ) )
@ Net )
=> ( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ Net )
=> ( ( filterlim @ B @ real @ H @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ Net )
=> ( filterlim @ B @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ Net ) ) ) ) ) ).
% real_tendsto_sandwich
thf(fact_5809_eventually__at,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A2: A,S2: set @ A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ S2 ) )
= ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X: A] :
( ( member @ A @ X @ S2 )
=> ( ( ( X != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ A2 ) @ D4 ) )
=> ( P @ X ) ) ) ) ) ) ) ).
% eventually_at
thf(fact_5810_eventually__nhds__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A2: A] :
( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ A2 ) )
= ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ A2 ) @ D4 )
=> ( P @ X ) ) ) ) ) ) ).
% eventually_nhds_metric
thf(fact_5811_eventually__at__leftI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A,P: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
=> ( P @ X4 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ) ).
% eventually_at_leftI
thf(fact_5812_eventually__at__rightI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A,P: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
=> ( P @ X4 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ).
% eventually_at_rightI
thf(fact_5813_eventually__at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P: A > $o,A2: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( eventually @ A
@ ^ [X: A] : ( P @ ( plus_plus @ A @ X @ A2 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% eventually_at_to_0
thf(fact_5814_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 )
=> ( ! [X4: A] :
( ( ord_less @ A @ X4 @ L )
=> ( eventually @ B
@ ^ [N3: B] : ( ord_less @ A @ X4 @ ( F2 @ N3 ) )
@ F4 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% increasing_tendsto
thf(fact_5815_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 )
=> ( ! [X4: A] :
( ( ord_less @ A @ L @ X4 )
=> ( eventually @ B
@ ^ [N3: B] : ( ord_less @ A @ ( F2 @ N3 ) @ X4 )
@ F4 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% decreasing_tendsto
thf(fact_5816_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
@ ^ [X: C] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( G @ X ) @ B2 ) @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X ) @ A2 ) )
@ F4 )
=> ( filterlim @ C @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ F4 ) ) ) ) ).
% metric_tendsto_imp_tendsto
thf(fact_5817_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
@ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F4 )
=> ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_top
thf(fact_5818_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
@ ^ [X: A] : ( ord_less @ real @ ( F2 @ X ) @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ real @ F2 @ ( at_bot @ real ) @ F4 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_bot
thf(fact_5819_eventually__at__right__to__0,axiom,
! [P: real > $o,A2: real] :
( ( eventually @ real @ P @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
= ( eventually @ real
@ ^ [X: real] : ( P @ ( plus_plus @ real @ X @ A2 ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% eventually_at_right_to_0
thf(fact_5820_continuous__arcosh__strong,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F2 @ X ) )
@ F4 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X: A] : ( arcosh @ real @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_arcosh_strong
thf(fact_5821_eventually__at__right__real,axiom,
! [A2: real,B2: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( eventually @ real
@ ^ [X: real] : ( member @ real @ X @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) ) ) ).
% eventually_at_right_real
thf(fact_5822_eventually__at__left__real,axiom,
! [B2: real,A2: real] :
( ( ord_less @ real @ B2 @ A2 )
=> ( eventually @ real
@ ^ [X: real] : ( member @ real @ X @ ( set_or5935395276787703475ssThan @ real @ B2 @ A2 ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ).
% eventually_at_left_real
thf(fact_5823_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 ) )
= ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X: A] :
( ( member @ A @ X @ S2 )
=> ( ( ( X != A2 )
& ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ A2 ) @ D4 ) )
=> ( P @ X ) ) ) ) ) ) ) ).
% eventually_at_le
thf(fact_5824_eventually__at__infinity__pos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P6: A > $o] :
( ( eventually @ A @ P6 @ ( at_infinity @ A ) )
= ( ? [B4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B4 )
& ! [X: A] :
( ( ord_less_eq @ real @ B4 @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( P6 @ X ) ) ) ) ) ) ).
% eventually_at_infinity_pos
thf(fact_5825_tendsto__imp__filterlim__at__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F2: A > B,L5: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X: A] : ( ord_less @ B @ ( F2 @ X ) @ L5 )
@ F4 )
=> ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L5 @ ( set_ord_lessThan @ B @ L5 ) ) @ F4 ) ) ) ) ).
% tendsto_imp_filterlim_at_left
thf(fact_5826_tendsto__imp__filterlim__at__right,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F2: A > B,L5: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X: A] : ( ord_less @ B @ L5 @ ( F2 @ X ) )
@ F4 )
=> ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L5 @ ( set_ord_greaterThan @ B @ L5 ) ) @ F4 ) ) ) ) ).
% tendsto_imp_filterlim_at_right
thf(fact_5827_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
@ ^ [X: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X ) @ L ) @ E2 )
@ F4 ) ) ) ) ).
% tendstoD
thf(fact_5828_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
@ ^ [X: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X ) @ L ) @ E )
@ F4 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ).
% tendstoI
thf(fact_5829_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
@ ^ [X: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X ) @ L ) @ E3 )
@ F4 ) ) ) ) ) ).
% tendsto_iff
thf(fact_5830_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_5831_eventually__at__top__to__right,axiom,
! [P: real > $o] :
( ( eventually @ real @ P @ ( at_top @ real ) )
= ( eventually @ real
@ ^ [X: real] : ( P @ ( inverse_inverse @ real @ X ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% eventually_at_top_to_right
thf(fact_5832_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
@ ^ [X: real] : ( P @ ( inverse_inverse @ real @ X ) )
@ ( at_top @ real ) ) ) ).
% eventually_at_right_to_top
thf(fact_5833_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
@ ^ [X: B] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F2 @ X ) )
@ F4 )
=> ( filterlim @ B @ real
@ ^ [X: B] : ( arcosh @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_arcosh_strong
thf(fact_5834_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] :
( ! [X4: A,Y4: A] :
( ( Q @ X4 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) )
=> ( ! [X4: B] :
( ( P @ X4 )
=> ( ( F2 @ ( G @ X4 ) )
= X4 ) )
=> ( ! [X4: B] :
( ( P @ X4 )
=> ( Q @ ( G @ X4 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) )
=> ( ! [B5: A] :
( ( Q @ B5 )
=> ( ord_less @ A @ B5 @ A2 ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_left
thf(fact_5835_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] :
( ! [X4: A,Y4: A] :
( ( Q @ X4 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) )
=> ( ! [X4: B] :
( ( P @ X4 )
=> ( ( F2 @ ( G @ X4 ) )
= X4 ) )
=> ( ! [X4: B] :
( ( P @ X4 )
=> ( Q @ ( G @ X4 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) )
=> ( ! [B5: A] :
( ( Q @ B5 )
=> ( ord_less @ A @ A2 @ B5 ) )
=> ( ( eventually @ B @ P @ ( at_bot @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_bot_at_right
thf(fact_5836_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,K5: real] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G @ X ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) @ K5 ) )
@ F4 )
=> ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) ) @ F4 ) ) ) ) ).
% tendsto_0_le
thf(fact_5837_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
@ ^ [X: C] : ( ord_less_eq @ real @ R5 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X ) ) )
@ F4 ) ) ) ) ) ) ).
% filterlim_at_infinity
thf(fact_5838_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
@ ^ [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ) ) ).
% tendsto_zero_powrI
thf(fact_5839_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
@ ^ [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F4 ) ) ) ) ) ).
% tendsto_powr2
thf(fact_5840_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
@ ^ [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F4 ) ) )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_powr'
thf(fact_5841_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
@ ^ [X: A] : ( ord_less @ B @ ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ B @ L ) ) @ ( F2 @ X ) )
@ F4 ) ) ) ) ).
% eventually_floor_less
thf(fact_5842_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
@ ^ [X: A] : ( ord_less @ B @ ( F2 @ X ) @ ( ring_1_of_int @ B @ ( archimedean_ceiling @ B @ L ) ) )
@ F4 ) ) ) ) ).
% eventually_less_ceiling
thf(fact_5843_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
@ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ) ).
% LIM_at_top_divide
thf(fact_5844_filterlim__inverse__at__top__iff,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( eventually @ A
@ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F4 )
=> ( ( filterlim @ A @ real
@ ^ [X: A] : ( inverse_inverse @ real @ ( F2 @ X ) )
@ ( at_top @ real )
@ F4 )
= ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ) ).
% filterlim_inverse_at_top_iff
thf(fact_5845_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
@ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( inverse_inverse @ real @ ( F2 @ X ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_inverse_at_top
thf(fact_5846_filterlim__at__top__iff__inverse__0,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( eventually @ A
@ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ 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_5847_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
@ ^ [X: A] : ( ord_less @ real @ ( F2 @ X ) @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( inverse_inverse @ real @ ( F2 @ X ) )
@ ( at_bot @ real )
@ F4 ) ) ) ).
% filterlim_inverse_at_bot
thf(fact_5848_lhopital,axiom,
! [F2: real > real,X3: real,G: real > real,G4: real > real,F6: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G4 @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F6 @ X ) @ ( G4 @ X ) )
@ F4
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ F4
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ) ).
% lhopital
thf(fact_5849_lhopital__at__top,axiom,
! [G: real > real,X3: real,G4: real > real,F2: real > real,F6: real > real,Y3: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G4 @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F6 @ X ) @ ( G4 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y3 )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y3 )
@ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top
thf(fact_5850_lhospital__at__top__at__top,axiom,
! [G: real > real,G4: real > real,F2: real > real,F6: real > real,X3: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G4 @ X )
!= ( zero_zero @ real ) )
@ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( at_top @ real ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F6 @ X ) @ ( G4 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ X3 )
@ ( at_top @ real ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ X3 )
@ ( at_top @ real ) ) ) ) ) ) ) ).
% lhospital_at_top_at_top
thf(fact_5851_lhopital__right,axiom,
! [F2: real > real,X3: real,G: real > real,G4: real > real,F6: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_greaterThan @ real @ X3 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_greaterThan @ real @ X3 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_greaterThan @ real @ X3 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G4 @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_greaterThan @ real @ X3 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_greaterThan @ real @ X3 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_greaterThan @ real @ X3 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F6 @ X ) @ ( G4 @ X ) )
@ F4
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_greaterThan @ real @ X3 ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ F4
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_greaterThan @ real @ X3 ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right
thf(fact_5852_lhopital__right__0,axiom,
! [F0: real > real,G0: real > real,G4: real > real,F6: 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
@ ^ [X: real] :
( ( G0 @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G4 @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ F0 @ ( F6 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G0 @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F6 @ X ) @ ( G4 @ X ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F0 @ X ) @ ( G0 @ X ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0
thf(fact_5853_lhopital__left,axiom,
! [F2: real > real,X3: real,G: real > real,G4: real > real,F6: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_lessThan @ real @ X3 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_lessThan @ real @ X3 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_lessThan @ real @ X3 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G4 @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_lessThan @ real @ X3 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_lessThan @ real @ X3 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_lessThan @ real @ X3 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F6 @ X ) @ ( G4 @ X ) )
@ F4
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_lessThan @ real @ X3 ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ F4
@ ( topolo174197925503356063within @ real @ X3 @ ( set_ord_lessThan @ real @ X3 ) ) ) ) ) ) ) ) ) ) ).
% lhopital_left
thf(fact_5854_lhopital__right__0__at__top,axiom,
! [G: real > real,G4: real > real,F2: real > real,F6: real > real,X3: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G4 @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F6 @ X ) @ ( G4 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ X3 )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ X3 )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0_at_top
thf(fact_5855_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 )
= ( ! [Z8: B] :
( ( ord_less @ B @ Z8 @ C2 )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ Z8 )
@ F4 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_5856_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 )
= ( ! [Z8: B] :
( ( ord_less @ B @ C2 @ Z8 )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ Z8 @ ( F2 @ X ) )
@ F4 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_5857_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 )
= ( ! [Z8: B] :
( eventually @ A
@ ^ [X: A] : ( ord_less @ B @ ( F2 @ X ) @ Z8 )
@ F4 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_5858_filterlim__int__sequentially,axiom,
filterlim @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( at_top @ int ) @ ( at_top @ nat ) ).
% filterlim_int_sequentially
thf(fact_5859_filterlim__nat__sequentially,axiom,
filterlim @ int @ nat @ nat2 @ ( at_top @ nat ) @ ( at_top @ int ) ).
% filterlim_nat_sequentially
thf(fact_5860_filterlim__int__of__nat__at__topD,axiom,
! [A: $tType,F2: int > A,F4: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X: nat] : ( F2 @ ( semiring_1_of_nat @ int @ X ) )
@ F4
@ ( at_top @ nat ) )
=> ( filterlim @ int @ A @ F2 @ F4 @ ( at_top @ int ) ) ) ).
% filterlim_int_of_nat_at_topD
thf(fact_5861_eventually__sequentiallyI,axiom,
! [C2: nat,P: nat > $o] :
( ! [X4: nat] :
( ( ord_less_eq @ nat @ C2 @ X4 )
=> ( P @ X4 ) )
=> ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentiallyI
thf(fact_5862_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
= ( ? [N6: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( P @ N3 ) ) ) ) ).
% eventually_sequentially
thf(fact_5863_le__sequentially,axiom,
! [F4: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F4 @ ( at_top @ nat ) )
= ( ! [N6: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N6 ) @ F4 ) ) ) ).
% le_sequentially
thf(fact_5864_eventually__at__top__linorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,P: A > $o] :
( ! [X4: A] :
( ( ord_less_eq @ A @ C2 @ X4 )
=> ( P @ X4 ) )
=> ( eventually @ A @ P @ ( at_top @ A ) ) ) ) ).
% eventually_at_top_linorderI
thf(fact_5865_eventually__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N6: A] :
! [N3: A] :
( ( ord_less_eq @ A @ N6 @ N3 )
=> ( P @ N3 ) ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_5866_eventually__at__top__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N6: A] :
! [N3: A] :
( ( ord_less @ A @ N6 @ N3 )
=> ( P @ N3 ) ) ) ) ) ).
% eventually_at_top_dense
thf(fact_5867_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N6: A] :
! [N3: A] :
( ( ord_less_eq @ A @ N3 @ N6 )
=> ( P @ N3 ) ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_5868_eventually__at__bot__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N6: A] :
! [N3: A] :
( ( ord_less @ A @ N3 @ N6 )
=> ( P @ N3 ) ) ) ) ) ).
% eventually_at_bot_dense
thf(fact_5869_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_5870_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_5871_eventually__le__at__bot,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ A @ X @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_le_at_bot
thf(fact_5872_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X: A] : ( ord_less @ A @ X @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_5873_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] :
( ! [X4: A,Y4: A] :
( ( Q @ X4 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) )
=> ( ! [X4: B] :
( ( P @ X4 )
=> ( ( F2 @ ( G @ X4 ) )
= X4 ) )
=> ( ! [X4: B] :
( ( P @ X4 )
=> ( Q @ ( G @ X4 ) ) )
=> ( ( 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_5874_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
@ ^ [X: B] : ( ord_less_eq @ A @ ( F2 @ X ) @ ( G @ X ) )
@ F4 )
=> ( filterlim @ B @ A @ G @ ( at_top @ A ) @ F4 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_5875_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 )
= ( ! [Z8: B] :
( ( ord_less_eq @ B @ C2 @ Z8 )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ Z8 @ ( F2 @ X ) )
@ F4 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_5876_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F4 )
= ( ! [Z8: B] :
( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ Z8 @ ( F2 @ X ) )
@ F4 ) ) ) ) ).
% filterlim_at_top
thf(fact_5877_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 )
= ( ! [Z8: B] :
( eventually @ A
@ ^ [X: A] : ( ord_less @ B @ Z8 @ ( F2 @ X ) )
@ F4 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_5878_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F4 )
= ( ! [Z8: B] :
( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ Z8 )
@ F4 ) ) ) ) ).
% filterlim_at_bot
thf(fact_5879_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 )
= ( ! [Z8: B] :
( ( ord_less_eq @ B @ Z8 @ C2 )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ Z8 )
@ F4 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_5880_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_5881_Bfun__metric__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F3: A > B,F8: filter @ A] :
? [Y: B,K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X ) @ Y ) @ K6 )
@ F8 ) ) ) ) ) ).
% Bfun_metric_def
thf(fact_5882_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_5883_decseq__bounded,axiom,
! [X8: nat > real,B6: real] :
( ( order_antimono @ nat @ real @ X8 )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ B6 @ ( X8 @ I3 ) )
=> ( bfun @ nat @ real @ X8 @ ( at_top @ nat ) ) ) ) ).
% decseq_bounded
thf(fact_5884_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F3: A > B] :
! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ Y ) @ ( F3 @ X ) ) ) ) ) ) ).
% antimono_def
thf(fact_5885_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) )
=> ( order_antimono @ A @ B @ F2 ) ) ) ).
% antimonoI
thf(fact_5886_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y3: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) ) ) ) ).
% antimonoE
thf(fact_5887_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y3: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) ) ) ) ).
% antimonoD
thf(fact_5888_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_5889_decseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) )
=> ( order_antimono @ nat @ A @ X8 ) ) ) ).
% decseq_SucI
thf(fact_5890_decseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: nat > A,I: nat] :
( ( order_antimono @ nat @ A @ A4 )
=> ( ord_less_eq @ A @ ( A4 @ ( suc @ I ) ) @ ( A4 @ I ) ) ) ) ).
% decseq_SucD
thf(fact_5891_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_5892_decseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [X9: nat > A] :
! [M3: nat,N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
=> ( ord_less_eq @ A @ ( X9 @ N3 ) @ ( X9 @ M3 ) ) ) ) ) ) ).
% decseq_def
thf(fact_5893_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X: A] :
! [Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( P @ Y ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_5894_BseqI_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,K5: real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N2 ) ) @ K5 )
=> ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) ) ) ).
% BseqI'
thf(fact_5895_Bseq__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( bfun @ nat @ A
@ ^ [X: nat] : ( times_times @ A @ C2 @ ( F2 @ X ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_cmult_iff
thf(fact_5896_decseq__ge,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,L5: A,N: nat] :
( ( order_antimono @ nat @ A @ X8 )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ L5 @ ( X8 @ N ) ) ) ) ) ).
% decseq_ge
thf(fact_5897_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_5898_decseq__convergent,axiom,
! [X8: nat > real,B6: real] :
( ( order_antimono @ nat @ real @ X8 )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ B6 @ ( X8 @ I3 ) )
=> ~ ! [L6: real] :
( ( filterlim @ nat @ real @ X8 @ ( topolo7230453075368039082e_nhds @ real @ L6 ) @ ( at_top @ nat ) )
=> ~ ! [I2: nat] : ( ord_less_eq @ real @ L6 @ ( X8 @ I2 ) ) ) ) ) ).
% decseq_convergent
thf(fact_5899_Bseq__def,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N3 ) ) @ K6 ) ) ) ) ) ).
% Bseq_def
thf(fact_5900_BseqI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [K5: real,X8: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K5 )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N2 ) ) @ K5 )
=> ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) ) ) ) ).
% BseqI
thf(fact_5901_BseqE,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
=> ~ ! [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
=> ~ ! [N9: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N9 ) ) @ K9 ) ) ) ) ).
% BseqE
thf(fact_5902_BseqD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
=> ? [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
& ! [N9: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N9 ) ) @ K9 ) ) ) ) ).
% BseqD
thf(fact_5903_Bseq__iff1a,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [N6: nat] :
! [N3: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) ) ) ) ).
% Bseq_iff1a
thf(fact_5904_Bseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [N6: nat] :
! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) ) ) ) ).
% Bseq_iff
thf(fact_5905_Bseq__realpow,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( bfun @ nat @ real @ ( power_power @ real @ X3 ) @ ( at_top @ nat ) ) ) ) ).
% Bseq_realpow
thf(fact_5906_BfunI,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,K5: real,F4: filter @ A] :
( ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) @ K5 )
@ F4 )
=> ( bfun @ A @ B @ F2 @ F4 ) ) ) ).
% BfunI
thf(fact_5907_Bseq__iff3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [N6: nat] :
! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X8 @ N3 ) @ ( uminus_uminus @ A @ ( X8 @ N6 ) ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff3
thf(fact_5908_Bseq__iff2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [X: A] :
! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X8 @ N3 ) @ ( uminus_uminus @ A @ X ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff2
thf(fact_5909_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
@ ^ [X: B] : ( inverse_inverse @ A @ ( F2 @ X ) )
@ F4 ) ) ) ) ).
% Bfun_inverse
thf(fact_5910_tendsto__at__right__sequentially,axiom,
! [C: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: B,B2: B,X8: B > C,L5: C] :
( ( ord_less @ B @ A2 @ B2 )
=> ( ! [S5: nat > B] :
( ! [N9: nat] : ( ord_less @ B @ A2 @ ( S5 @ N9 ) )
=> ( ! [N9: nat] : ( ord_less @ B @ ( S5 @ N9 ) @ B2 )
=> ( ( order_antimono @ nat @ B @ S5 )
=> ( ( filterlim @ nat @ B @ S5 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C
@ ^ [N3: nat] : ( X8 @ ( S5 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L5 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ C @ X8 @ ( topolo7230453075368039082e_nhds @ C @ L5 ) @ ( topolo174197925503356063within @ B @ A2 @ ( set_ord_greaterThan @ B @ A2 ) ) ) ) ) ) ).
% tendsto_at_right_sequentially
thf(fact_5911_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
@ ^ [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) @ B7 )
@ F4 ) ) ) ) ).
% BfunE
thf(fact_5912_Bfun__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F3: A > B,F8: filter @ A] :
? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X ) ) @ K6 )
@ F8 ) ) ) ) ) ).
% Bfun_def
thf(fact_5913_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 )
=> ! [B4: nat] :
( ( ord_less @ nat @ A5 @ B4 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or3652927894154168847AtMost @ nat @ A5 @ B4 ) ) ) @ ( 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_5914_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P3: A > $o] :
( the @ A
@ ^ [X: A] :
( ( P3 @ X )
& ! [Y: A] :
( ( P3 @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_5915_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_5916_greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% greaterThanAtMost_iff
thf(fact_5917_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_5918_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_5919_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_5920_infinite__Ioc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite @ A @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ioc_iff
thf(fact_5921_cSup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less @ A @ Y3 @ X3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ Y3 @ X3 ) )
= X3 ) ) ) ).
% cSup_greaterThanAtMost
thf(fact_5922_Sup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X3 @ Y3 ) )
= Y3 ) ) ) ).
% Sup_greaterThanAtMost
thf(fact_5923_cInf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y3: A,X3: A] :
( ( ord_less @ A @ Y3 @ X3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ Y3 @ X3 ) )
= Y3 ) ) ) ).
% cInf_greaterThanAtMost
thf(fact_5924_Inf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X3 @ Y3 ) )
= X3 ) ) ) ).
% Inf_greaterThanAtMost
thf(fact_5925_Ioc__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% Ioc_subset_iff
thf(fact_5926_Ioc__inj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ( set_or3652927894154168847AtMost @ A @ A2 @ B2 )
= ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ D3 @ C2 ) )
| ( ( A2 = C2 )
& ( B2 = D3 ) ) ) ) ) ).
% Ioc_inj
thf(fact_5927_infinite__Ioc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite @ A @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ioc
thf(fact_5928_GreatestI__ex__nat,axiom,
! [P: nat > $o,B2: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_5929_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B2 ) )
=> ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_5930_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_5931_open__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S2: set @ A,X3: A,Y3: A] :
( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( member @ A @ X3 @ S2 )
=> ( ( ord_less @ A @ Y3 @ X3 )
=> ? [B5: A] :
( ( ord_less @ A @ B5 @ X3 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ B5 @ X3 ) @ S2 ) ) ) ) ) ) ).
% open_left
thf(fact_5932_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ( order_Greatest @ A @ P )
= X3 ) ) ) ) ).
% Greatest_equality
thf(fact_5933_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A,Q: A > $o] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_5934_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_5935_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_5936_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_5937_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_5938_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_5939_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_5940_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_5941_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_5942_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_5943_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S2: set @ A] :
( ! [A6: A,B5: A,X4: A] :
( ( member @ A @ A6 @ S2 )
=> ( ( member @ A @ B5 @ S2 )
=> ( ( ord_less_eq @ A @ A6 @ X4 )
=> ( ( ord_less_eq @ A @ X4 @ B5 )
=> ( member @ A @ X4 @ S2 ) ) ) ) )
=> ? [A6: A,B5: A] :
( ( S2
= ( bot_bot @ ( set @ A ) ) )
| ( S2
= ( top_top @ ( set @ A ) ) )
| ( S2
= ( set_ord_lessThan @ A @ B5 ) )
| ( S2
= ( set_ord_atMost @ A @ B5 ) )
| ( S2
= ( set_ord_greaterThan @ A @ A6 ) )
| ( S2
= ( set_ord_atLeast @ A @ A6 ) )
| ( S2
= ( set_or5935395276787703475ssThan @ A @ A6 @ B5 ) )
| ( S2
= ( set_or3652927894154168847AtMost @ A @ A6 @ B5 ) )
| ( S2
= ( set_or7035219750837199246ssThan @ A @ A6 @ B5 ) )
| ( S2
= ( set_or1337092689740270186AtMost @ A @ A6 @ B5 ) ) ) ) ) ).
% interval_cases
thf(fact_5944_cauchy__filter__metric,axiom,
! [A: $tType] :
( ( ( real_V768167426530841204y_dist @ A )
& ( topolo7287701948861334536_space @ A ) )
=> ( ( topolo6773858410816713723filter @ A )
= ( ^ [F8: filter @ A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [P3: A > $o] :
( ( eventually @ A @ P3 @ F8 )
& ! [X: A,Y: A] :
( ( ( P3 @ X )
& ( P3 @ Y ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E3 ) ) ) ) ) ) ) ).
% cauchy_filter_metric
thf(fact_5945_GMVT,axiom,
! [A2: real,B2: real,F2: real > real,G: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ( ! [X4: real] :
( ( ( ord_less @ real @ A2 @ X4 )
& ( ord_less @ real @ X4 @ B2 ) )
=> ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) @ G ) )
=> ( ! [X4: real] :
( ( ( ord_less @ real @ A2 @ X4 )
& ( ord_less @ real @ X4 @ B2 ) )
=> ( differentiable @ real @ real @ G @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [G_c: real,F_c: real,C3: real] :
( ( has_field_derivative @ real @ G @ G_c @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) )
& ( has_field_derivative @ real @ F2 @ F_c @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ A2 @ C3 )
& ( ord_less @ real @ C3 @ B2 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) @ G_c )
= ( times_times @ real @ ( minus_minus @ real @ ( G @ B2 ) @ ( G @ A2 ) ) @ F_c ) ) ) ) ) ) ) ) ).
% GMVT
thf(fact_5946_atLeast__0,axiom,
( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
= ( top_top @ ( set @ nat ) ) ) ).
% atLeast_0
thf(fact_5947_atLeast__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_atLeast @ A @ K ) )
= ( ord_less_eq @ A @ K @ I ) ) ) ).
% atLeast_iff
thf(fact_5948_cInf__atLeast,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_atLeast @ A @ X3 ) )
= X3 ) ) ).
% cInf_atLeast
thf(fact_5949_atLeast__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X3 ) @ ( set_ord_atLeast @ A @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% atLeast_subset_iff
thf(fact_5950_card__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or3652927894154168847AtMost @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ L ) ) ) ).
% card_greaterThanAtMost_int
thf(fact_5951_Icc__subset__Ici__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H: A,L3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H ) @ ( set_ord_atLeast @ A @ L3 ) )
= ( ~ ( ord_less_eq @ A @ L @ H )
| ( ord_less_eq @ A @ L3 @ L ) ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_5952_differentiable__cmult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Q3: B > A,C2: A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ ( Q3 @ T3 ) @ C2 )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q3 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_right_iff
thf(fact_5953_differentiable__cmult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: A,Q3: B > A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ C2 @ ( Q3 @ T3 ) )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q3 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_left_iff
thf(fact_5954_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less_eq @ A @ L2 ) ) ) ) ) ).
% atLeast_def
thf(fact_5955_differentiable__power,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X3: A,S: set @ A,N: nat] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( differentiable @ A @ B
@ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ).
% differentiable_power
thf(fact_5956_Ici__subset__Ioi__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A2 ) @ ( set_ord_greaterThan @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% Ici_subset_Ioi_iff
thf(fact_5957_differentiable__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X3: A,S: set @ A,G: A > B] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( differentiable @ A @ B @ G @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ( G @ X3 )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ).
% differentiable_divide
thf(fact_5958_differentiable__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X3: A,S: set @ A] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ( F2 @ X3 )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X: A] : ( inverse_inverse @ B @ ( F2 @ X ) )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% differentiable_inverse
thf(fact_5959_normalize__negative,axiom,
! [Q3: int,P6: int] :
( ( ord_less @ int @ Q3 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P6 @ Q3 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P6 ) @ ( uminus_uminus @ int @ Q3 ) ) ) ) ) ).
% normalize_negative
thf(fact_5960_MVT,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A2 @ X4 )
=> ( ( ord_less @ real @ X4 @ B2 )
=> ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [L4: real,Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
& ( ord_less @ real @ Z3 @ B2 )
& ( has_field_derivative @ real @ F2 @ L4 @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) )
& ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ L4 ) ) ) ) ) ) ).
% MVT
thf(fact_5961_normalize__denom__zero,axiom,
! [P6: int] :
( ( normalize @ ( product_Pair @ int @ int @ P6 @ ( zero_zero @ int ) ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% normalize_denom_zero
thf(fact_5962_IVT2_H,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,B2: A,Y3: B,A2: A] :
( ( ord_less_eq @ B @ ( F2 @ B2 ) @ Y3 )
=> ( ( ord_less_eq @ B @ Y3 @ ( F2 @ A2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 )
=> ? [X4: A] :
( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less_eq @ A @ X4 @ B2 )
& ( ( F2 @ X4 )
= Y3 ) ) ) ) ) ) ) ).
% IVT2'
thf(fact_5963_IVT_H,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,A2: A,Y3: B,B2: A] :
( ( ord_less_eq @ B @ ( F2 @ A2 ) @ Y3 )
=> ( ( ord_less_eq @ B @ Y3 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 )
=> ? [X4: A] :
( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less_eq @ A @ X4 @ B2 )
& ( ( F2 @ X4 )
= Y3 ) ) ) ) ) ) ) ).
% IVT'
thf(fact_5964_continuous__on__real__root,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real,N: nat] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X: A] : ( root @ N @ ( F2 @ X ) ) ) ) ) ).
% continuous_on_real_root
thf(fact_5965_continuous__on__power_H,axiom,
! [B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo1898628316856586783d_mult @ B ) )
=> ! [A4: set @ C,F2: C > B,G: C > nat] :
( ( topolo81223032696312382ous_on @ C @ B @ A4 @ F2 )
=> ( ( topolo81223032696312382ous_on @ C @ nat @ A4 @ G )
=> ( topolo81223032696312382ous_on @ C @ B @ A4
@ ^ [X: C] : ( power_power @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_on_power'
thf(fact_5966_continuous__on__power,axiom,
! [B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [S: set @ C,F2: C > B,N: nat] :
( ( topolo81223032696312382ous_on @ C @ B @ S @ F2 )
=> ( topolo81223032696312382ous_on @ C @ B @ S
@ ^ [X: C] : ( power_power @ B @ ( F2 @ X ) @ N ) ) ) ) ).
% continuous_on_power
thf(fact_5967_continuous__on__real__sqrt,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X: A] : ( sqrt @ ( F2 @ X ) ) ) ) ) ).
% continuous_on_real_sqrt
thf(fact_5968_continuous__on__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( F2 @ X4 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X: A] : ( sgn_sgn @ B @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_sgn
thf(fact_5969_continuous__on__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( F2 @ X4 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X: A] : ( inverse_inverse @ B @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_inverse
thf(fact_5970_continuous__on__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S: set @ A,F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S @ G )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( G @ X4 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% continuous_on_divide
thf(fact_5971_continuous__on__powr,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S: set @ C,F2: C > real,G: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ C @ real @ S @ G )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ S )
=> ( ( F2 @ X4 )
!= ( zero_zero @ real ) ) )
=> ( topolo81223032696312382ous_on @ C @ real @ S
@ ^ [X: C] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% continuous_on_powr
thf(fact_5972_continuous__on__ln,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( F2 @ X4 )
!= ( zero_zero @ real ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X: A] : ( ln_ln @ real @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_ln
thf(fact_5973_continuous__onI__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( dense_order @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( topolo1002775350975398744n_open @ B @ ( image @ A @ B @ F2 @ A4 ) )
=> ( ! [X4: A,Y4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ( member @ A @ Y4 @ A4 )
=> ( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ A4 @ F2 ) ) ) ) ).
% continuous_onI_mono
thf(fact_5974_open__Collect__less,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% open_Collect_less
thf(fact_5975_continuous__on__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: set @ A,F2: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S @ F2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( cos @ A @ ( F2 @ X4 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S
@ ^ [X: A] : ( tan @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_tan
thf(fact_5976_continuous__on__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: set @ A,F2: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S @ F2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( sin @ A @ ( F2 @ X4 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S
@ ^ [X: A] : ( cot @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_cot
thf(fact_5977_continuous__on__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A4: set @ C,F2: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A4 @ F2 )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ A4 )
=> ( ( cosh @ A @ ( F2 @ X4 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ C @ A @ A4
@ ^ [X: C] : ( tanh @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_tanh
thf(fact_5978_continuous__on__arcosh_H,axiom,
! [A4: set @ real,F2: real > real] :
( ( topolo81223032696312382ous_on @ real @ real @ A4 @ F2 )
=> ( ! [X4: real] :
( ( member @ real @ X4 @ A4 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( F2 @ X4 ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A4
@ ^ [X: real] : ( arcosh @ real @ ( F2 @ X ) ) ) ) ) ).
% continuous_on_arcosh'
thf(fact_5979_continuous__image__closed__interval,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ? [C3: real,D6: real] :
( ( ( image @ real @ real @ F2 @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ real @ C3 @ D6 ) )
& ( ord_less_eq @ real @ C3 @ D6 ) ) ) ) ).
% continuous_image_closed_interval
thf(fact_5980_continuous__on__powr_H,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S: set @ C,F2: C > real,G: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ C @ real @ S @ G )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ S )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X4 ) )
& ( ( ( F2 @ X4 )
= ( zero_zero @ real ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X4 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ C @ real @ S
@ ^ [X: C] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% continuous_on_powr'
thf(fact_5981_continuous__on__log,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real,G: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ real @ S @ G )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X4 ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( F2 @ X4 )
!= ( one_one @ real ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X4 ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ).
% continuous_on_log
thf(fact_5982_continuous__on__arccos,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( F2 @ X4 ) )
& ( ord_less_eq @ real @ ( F2 @ X4 ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X: A] : ( arccos @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_arccos
thf(fact_5983_continuous__on__arcsin,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( F2 @ X4 ) )
& ( ord_less_eq @ real @ ( F2 @ X4 ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X: A] : ( arcsin @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_arcsin
thf(fact_5984_DERIV__atLeastAtMost__imp__continuous__on,axiom,
! [A: $tType] :
( ( ( ord @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,B2: A,F2: A > A] :
( ! [X4: A] :
( ( ord_less_eq @ A @ A2 @ X4 )
=> ( ( ord_less_eq @ A @ X4 @ B2 )
=> ? [Y5: A] : ( has_field_derivative @ A @ F2 @ Y5 @ ( topolo174197925503356063within @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 ) ) ) ).
% DERIV_atLeastAtMost_imp_continuous_on
thf(fact_5985_Rolle__deriv,axiom,
! [A2: real,B2: real,F2: real > real,F6: real > real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( ( F2 @ A2 )
= ( F2 @ B2 ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A2 @ X4 )
=> ( ( ord_less @ real @ X4 @ B2 )
=> ( has_derivative @ real @ real @ F2 @ ( F6 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
& ( ord_less @ real @ Z3 @ B2 )
& ( ( F6 @ Z3 )
= ( ^ [V5: real] : ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% Rolle_deriv
thf(fact_5986_mvt,axiom,
! [A2: real,B2: real,F2: real > real,F6: real > real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A2 @ X4 )
=> ( ( ord_less @ real @ X4 @ B2 )
=> ( has_derivative @ real @ real @ F2 @ ( F6 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ~ ! [Xi: real] :
( ( ord_less @ real @ A2 @ Xi )
=> ( ( ord_less @ real @ Xi @ B2 )
=> ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
!= ( F6 @ Xi @ ( minus_minus @ real @ B2 @ A2 ) ) ) ) ) ) ) ) ).
% mvt
thf(fact_5987_normalize__denom__pos,axiom,
! [R2: product_prod @ int @ int,P6: int,Q3: int] :
( ( ( normalize @ R2 )
= ( product_Pair @ int @ int @ P6 @ Q3 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q3 ) ) ).
% normalize_denom_pos
thf(fact_5988_normalize__crossproduct,axiom,
! [Q3: int,S: int,P6: int,R2: int] :
( ( Q3
!= ( zero_zero @ int ) )
=> ( ( S
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P6 @ Q3 ) )
= ( normalize @ ( product_Pair @ int @ int @ R2 @ S ) ) )
=> ( ( times_times @ int @ P6 @ S )
= ( times_times @ int @ R2 @ Q3 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_5989_continuous__on__of__int__floor,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ A )
& ( topolo2564578578187576103pology @ A )
& ( ring_1 @ B )
& ( topolo4958980785337419405_space @ B ) )
=> ( topolo81223032696312382ous_on @ A @ B @ ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( ring_1_Ints @ A ) )
@ ^ [X: A] : ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ).
% continuous_on_of_int_floor
thf(fact_5990_continuous__on__of__int__ceiling,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ A )
& ( topolo2564578578187576103pology @ A )
& ( ring_1 @ B )
& ( topolo4958980785337419405_space @ B ) )
=> ( topolo81223032696312382ous_on @ A @ B @ ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( ring_1_Ints @ A ) )
@ ^ [X: A] : ( ring_1_of_int @ B @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% continuous_on_of_int_ceiling
thf(fact_5991_continuous__on__Icc__at__leftD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,B2: A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ B2 ) ) @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ) ).
% continuous_on_Icc_at_leftD
thf(fact_5992_continuous__on__Icc__at__rightD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,B2: A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ).
% continuous_on_Icc_at_rightD
thf(fact_5993_DERIV__pos__imp__increasing__open,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A2 @ X4 )
=> ( ( ord_less @ real @ X4 @ B2 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) ) ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ord_less @ real @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) ) ) ) ).
% DERIV_pos_imp_increasing_open
thf(fact_5994_DERIV__neg__imp__decreasing__open,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A2 @ X4 )
=> ( ( ord_less @ real @ X4 @ B2 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y5 @ ( zero_zero @ real ) ) ) ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ord_less @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) ) ) ) ).
% DERIV_neg_imp_decreasing_open
thf(fact_5995_DERIV__isconst__end,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A2 @ X4 )
=> ( ( ord_less @ real @ X4 @ B2 )
=> ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ( F2 @ B2 )
= ( F2 @ A2 ) ) ) ) ) ).
% DERIV_isconst_end
thf(fact_5996_DERIV__isconst2,axiom,
! [A2: real,B2: real,F2: real > real,X3: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A2 @ X4 )
=> ( ( ord_less @ real @ X4 @ B2 )
=> ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ( ( F2 @ X3 )
= ( F2 @ A2 ) ) ) ) ) ) ) ).
% DERIV_isconst2
thf(fact_5997_continuous__on__IccI,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A2: A,B2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) )
=> ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ B2 ) ) @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) )
=> ( ! [X4: A] :
( ( ord_less @ A @ A2 @ X4 )
=> ( ( ord_less @ A @ X4 @ B2 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ X4 ) ) @ ( topolo174197925503356063within @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 ) ) ) ) ) ) ).
% continuous_on_IccI
thf(fact_5998_Rolle,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( ( F2 @ A2 )
= ( F2 @ B2 ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A2 @ X4 )
=> ( ( ord_less @ real @ X4 @ B2 )
=> ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
& ( ord_less @ real @ Z3 @ B2 )
& ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% Rolle
thf(fact_5999_normalize__def,axiom,
( normalize
= ( ^ [P5: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P5 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P5 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P5 ) @ ( product_snd @ int @ int @ P5 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P5 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P5 ) @ ( product_snd @ int @ int @ P5 ) ) ) )
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_snd @ int @ int @ P5 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P5 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P5 ) @ ( product_snd @ int @ int @ P5 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P5 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P5 ) @ ( product_snd @ int @ int @ P5 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_6000_ord_OLeast__def,axiom,
! [A: $tType] :
( ( least @ A )
= ( ^ [Less_eq: A > A > $o,P3: A > $o] :
( the @ A
@ ^ [X: A] :
( ( P3 @ X )
& ! [Y: A] :
( ( P3 @ Y )
=> ( Less_eq @ X @ Y ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_6001_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_6002_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_6003_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_6004_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_6005_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_6006_gcd__neg__numeral__2__int,axiom,
! [X3: int,N: num] :
( ( gcd_gcd @ int @ X3 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( gcd_gcd @ int @ X3 @ ( numeral_numeral @ int @ N ) ) ) ).
% gcd_neg_numeral_2_int
thf(fact_6007_gcd__neg__numeral__1__int,axiom,
! [N: num,X3: int] :
( ( gcd_gcd @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) @ X3 )
= ( gcd_gcd @ int @ ( numeral_numeral @ int @ N ) @ X3 ) ) ).
% gcd_neg_numeral_1_int
thf(fact_6008_gcd__0__int,axiom,
! [X3: int] :
( ( gcd_gcd @ int @ X3 @ ( zero_zero @ int ) )
= ( abs_abs @ int @ X3 ) ) ).
% gcd_0_int
thf(fact_6009_gcd__0__left__int,axiom,
! [X3: int] :
( ( gcd_gcd @ int @ ( zero_zero @ int ) @ X3 )
= ( abs_abs @ int @ X3 ) ) ).
% gcd_0_left_int
thf(fact_6010_ord_OLeast_Ocong,axiom,
! [A: $tType] :
( ( least @ A )
= ( least @ A ) ) ).
% ord.Least.cong
thf(fact_6011_gcd__ge__0__int,axiom,
! [X3: int,Y3: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ X3 @ Y3 ) ) ).
% gcd_ge_0_int
thf(fact_6012_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_6013_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_6014_gcd__cases__int,axiom,
! [X3: int,Y3: int,P: int > $o] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( P @ ( gcd_gcd @ int @ X3 @ Y3 ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( ( ord_less_eq @ int @ Y3 @ ( zero_zero @ int ) )
=> ( P @ ( gcd_gcd @ int @ X3 @ ( uminus_uminus @ int @ Y3 ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ X3 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X3 ) @ Y3 ) ) ) )
=> ( ( ( ord_less_eq @ int @ X3 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ Y3 @ ( zero_zero @ int ) )
=> ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X3 ) @ ( uminus_uminus @ int @ Y3 ) ) ) ) )
=> ( P @ ( gcd_gcd @ int @ X3 @ Y3 ) ) ) ) ) ) ).
% gcd_cases_int
thf(fact_6015_gcd__unique__int,axiom,
! [D3: int,A2: int,B2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D3 )
& ( dvd_dvd @ int @ D3 @ A2 )
& ( dvd_dvd @ int @ D3 @ B2 )
& ! [E3: int] :
( ( ( dvd_dvd @ int @ E3 @ A2 )
& ( dvd_dvd @ int @ E3 @ B2 ) )
=> ( dvd_dvd @ int @ E3 @ D3 ) ) )
= ( D3
= ( gcd_gcd @ int @ A2 @ B2 ) ) ) ).
% gcd_unique_int
thf(fact_6016_gcd__non__0__int,axiom,
! [Y3: int,X3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( gcd_gcd @ int @ X3 @ Y3 )
= ( gcd_gcd @ int @ Y3 @ ( modulo_modulo @ int @ X3 @ Y3 ) ) ) ) ).
% gcd_non_0_int
thf(fact_6017_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_6018_gcd__is__Max__divisors__int,axiom,
! [N: int,M: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( gcd_gcd @ int @ M @ N )
= ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D4: int] :
( ( dvd_dvd @ int @ D4 @ M )
& ( dvd_dvd @ int @ D4 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_int
thf(fact_6019_sequentially__imp__eventually__at__left,axiom,
! [A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: A,A2: A,P: A > $o] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ! [F5: nat > A] :
( ! [N9: nat] : ( ord_less @ A @ B2 @ ( F5 @ N9 ) )
=> ( ! [N9: nat] : ( ord_less @ A @ ( F5 @ N9 ) @ A2 )
=> ( ( order_mono @ 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_lessThan @ A @ A2 ) ) ) ) ) ) ).
% sequentially_imp_eventually_at_left
thf(fact_6020_range__abs__Nats,axiom,
( ( image @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) )
= ( semiring_1_Nats @ int ) ) ).
% range_abs_Nats
thf(fact_6021_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_6022_gcd__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ A2 )
= A2 ) ).
% gcd_nat.left_neutral
thf(fact_6023_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_6024_gcd__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( gcd_gcd @ nat @ A2 @ ( zero_zero @ nat ) )
= A2 ) ).
% gcd_nat.right_neutral
thf(fact_6025_gcd__0__nat,axiom,
! [X3: nat] :
( ( gcd_gcd @ nat @ X3 @ ( zero_zero @ nat ) )
= X3 ) ).
% gcd_0_nat
thf(fact_6026_gcd__0__left__nat,axiom,
! [X3: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ X3 )
= X3 ) ).
% gcd_0_left_nat
thf(fact_6027_gcd__Suc__0,axiom,
! [M: nat] :
( ( gcd_gcd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% gcd_Suc_0
thf(fact_6028_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_6029_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_6030_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_6031_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_6032_gcd__non__0__nat,axiom,
! [Y3: nat,X3: nat] :
( ( Y3
!= ( zero_zero @ nat ) )
=> ( ( gcd_gcd @ nat @ X3 @ Y3 )
= ( gcd_gcd @ nat @ Y3 @ ( modulo_modulo @ nat @ X3 @ Y3 ) ) ) ) ).
% gcd_non_0_nat
thf(fact_6033_gcd__nat_Osimps,axiom,
( ( gcd_gcd @ nat )
= ( ^ [X: nat,Y: nat] :
( if @ nat
@ ( Y
= ( zero_zero @ nat ) )
@ X
@ ( gcd_gcd @ nat @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) ) ) ).
% gcd_nat.simps
thf(fact_6034_gcd__nat_Oelims,axiom,
! [X3: nat,Xa: nat,Y3: nat] :
( ( ( gcd_gcd @ nat @ X3 @ Xa )
= Y3 )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3 = X3 ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y3
= ( gcd_gcd @ nat @ Xa @ ( modulo_modulo @ nat @ X3 @ Xa ) ) ) ) ) ) ).
% gcd_nat.elims
thf(fact_6035_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_6036_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_6037_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_6038_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_6039_funpow__mono,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: A > A,A4: A,B6: A,N: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ A4 @ B6 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F2 @ A4 ) @ ( compow @ ( A > A ) @ N @ F2 @ B6 ) ) ) ) ) ).
% funpow_mono
thf(fact_6040_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y3: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ).
% mono_invE
thf(fact_6041_incseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [X9: nat > A] :
! [M3: nat,N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
=> ( ord_less_eq @ A @ ( X9 @ M3 ) @ ( X9 @ N3 ) ) ) ) ) ) ).
% incseq_def
thf(fact_6042_incseqD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,I: nat,J: nat] :
( ( order_mono @ nat @ A @ F2 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ) ).
% incseqD
thf(fact_6043_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_mono @ A @ B )
= ( ^ [F3: A > B] :
! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ) ).
% mono_def
thf(fact_6044_monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( order_mono @ A @ B @ F2 ) ) ) ).
% monoI
thf(fact_6045_monoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y3: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% monoE
thf(fact_6046_monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y3: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% monoD
thf(fact_6047_incseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: nat > A,I: nat] :
( ( order_mono @ nat @ A @ A4 )
=> ( ord_less_eq @ A @ ( A4 @ I ) @ ( A4 @ ( suc @ I ) ) ) ) ) ).
% incseq_SucD
thf(fact_6048_incseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
=> ( order_mono @ nat @ A @ X8 ) ) ) ).
% incseq_SucI
thf(fact_6049_incseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [F3: nat > A] :
! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) ) ) ) ) ).
% incseq_Suc_iff
thf(fact_6050_Nats__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [W: num] : ( member @ A @ ( numeral_numeral @ A @ W ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_numeral
thf(fact_6051_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y3: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
=> ( ord_less @ A @ X3 @ Y3 ) ) ) ) ).
% mono_strict_invE
thf(fact_6052_Nats__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_0
thf(fact_6053_mono__Suc,axiom,
order_mono @ nat @ nat @ suc ).
% mono_Suc
thf(fact_6054_Nats__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( member @ A @ ( times_times @ A @ A2 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% Nats_mult
thf(fact_6055_of__nat__in__Nats,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_Nats @ A ) ) ) ).
% of_nat_in_Nats
thf(fact_6056_Nats__induct,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X3: A,P: A > $o] :
( ( member @ A @ X3 @ ( semiring_1_Nats @ A ) )
=> ( ! [N2: nat] : ( P @ ( semiring_1_of_nat @ A @ N2 ) )
=> ( P @ X3 ) ) ) ) ).
% Nats_induct
thf(fact_6057_Nats__cases,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X3: A] :
( ( member @ A @ X3 @ ( semiring_1_Nats @ A ) )
=> ~ ! [N2: nat] :
( X3
!= ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% Nats_cases
thf(fact_6058_mono__pow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,N: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( order_mono @ A @ A @ ( compow @ ( A > A ) @ N @ F2 ) ) ) ) ).
% mono_pow
thf(fact_6059_Nats__1,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_1
thf(fact_6060_Nats__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% Nats_add
thf(fact_6061_mono__add,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A2 ) ) ) ).
% mono_add
thf(fact_6062_mono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_mono @ nat @ A
@ ^ [I4: nat] : ( compow @ ( A > A ) @ I4 @ Q @ ( bot_bot @ A ) ) ) ) ) ).
% mono_funpow
thf(fact_6063_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_6064_mono__times__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( order_mono @ nat @ nat @ ( times_times @ nat @ N ) ) ) ).
% mono_times_nat
thf(fact_6065_mono__mult,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( order_mono @ A @ A @ ( times_times @ A @ A2 ) ) ) ) ).
% mono_mult
thf(fact_6066_mono__image__least,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [F2: A > B,M: A,N: A,M6: B,N5: B] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ( image @ A @ B @ F2 @ ( set_or7035219750837199246ssThan @ A @ M @ N ) )
= ( set_or7035219750837199246ssThan @ B @ M6 @ N5 ) )
=> ( ( ord_less @ A @ M @ N )
=> ( ( F2 @ M )
= M6 ) ) ) ) ) ).
% mono_image_least
thf(fact_6067_Kleene__iter__gpfp,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [F2: A > A,P6: A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ P6 @ ( F2 @ P6 ) )
=> ( ord_less_eq @ A @ P6 @ ( compow @ ( A > A ) @ K @ F2 @ ( top_top @ A ) ) ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_6068_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [F2: A > A,P6: A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ ( F2 @ P6 ) @ P6 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K @ F2 @ ( bot_bot @ A ) ) @ P6 ) ) ) ) ).
% Kleene_iter_lpfp
thf(fact_6069_funpow__mono2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: A > A,I: nat,J: nat,X3: A,Y3: A] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ ( F2 @ X3 ) )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ I @ F2 @ X3 ) @ ( compow @ ( A > A ) @ J @ F2 @ Y3 ) ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_6070_bezout__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ? [X4: nat,Y4: nat] :
( ( times_times @ nat @ A2 @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ).
% bezout_nat
thf(fact_6071_bezout__gcd__nat_H,axiom,
! [B2: nat,A2: nat] :
? [X4: nat,Y4: nat] :
( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ ( times_times @ nat @ A2 @ X4 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ A2 @ X4 ) @ ( times_times @ nat @ B2 @ Y4 ) )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) )
| ( ( ord_less_eq @ nat @ ( times_times @ nat @ A2 @ Y4 ) @ ( times_times @ nat @ B2 @ X4 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X4 ) @ ( times_times @ nat @ A2 @ Y4 ) )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_6072_incseq__bounded,axiom,
! [X8: nat > real,B6: real] :
( ( order_mono @ nat @ real @ X8 )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ ( X8 @ I3 ) @ B6 )
=> ( bfun @ nat @ real @ X8 @ ( at_top @ nat ) ) ) ) ).
% incseq_bounded
thf(fact_6073_Nats__diff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( member @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ) ).
% Nats_diff
thf(fact_6074_mono__Sup,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image @ A @ B @ F2 @ A4 ) ) @ ( F2 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% mono_Sup
thf(fact_6075_mono__SUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F2: A > B,A4: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image @ C @ B
@ ^ [X: C] : ( F2 @ ( A4 @ X ) )
@ I6 ) )
@ ( F2 @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ A4 @ I6 ) ) ) ) ) ) ).
% mono_SUP
thf(fact_6076_mono__Inf,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ A4 ) ) @ ( complete_Inf_Inf @ B @ ( image @ A @ B @ F2 @ A4 ) ) ) ) ) ).
% mono_Inf
thf(fact_6077_mono__INF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F2: A > B,A4: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ A4 @ I6 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image @ C @ B
@ ^ [X: C] : ( F2 @ ( A4 @ X ) )
@ I6 ) ) ) ) ) ).
% mono_INF
thf(fact_6078_Nats__subset__Ints,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ord_less_eq @ ( set @ A ) @ ( semiring_1_Nats @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Nats_subset_Ints
thf(fact_6079_antimono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_antimono @ nat @ A
@ ^ [I4: nat] : ( compow @ ( A > A ) @ I4 @ Q @ ( top_top @ A ) ) ) ) ) ).
% antimono_funpow
thf(fact_6080_incseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,L5: A,N: nat] :
( ( order_mono @ nat @ A @ X8 )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ ( X8 @ N ) @ L5 ) ) ) ) ).
% incseq_le
thf(fact_6081_funpow__increasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [M: nat,N: nat,F2: A > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F2 @ ( top_top @ A ) ) @ ( compow @ ( A > A ) @ M @ F2 @ ( top_top @ A ) ) ) ) ) ) ).
% funpow_increasing
thf(fact_6082_funpow__decreasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [M: nat,N: nat,F2: A > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ M @ F2 @ ( bot_bot @ A ) ) @ ( compow @ ( A > A ) @ N @ F2 @ ( bot_bot @ A ) ) ) ) ) ) ).
% funpow_decreasing
thf(fact_6083_incseq__convergent,axiom,
! [X8: nat > real,B6: real] :
( ( order_mono @ nat @ real @ X8 )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ ( X8 @ I3 ) @ B6 )
=> ~ ! [L6: real] :
( ( filterlim @ nat @ real @ X8 @ ( topolo7230453075368039082e_nhds @ real @ L6 ) @ ( at_top @ nat ) )
=> ~ ! [I2: nat] : ( ord_less_eq @ real @ ( X8 @ I2 ) @ L6 ) ) ) ) ).
% incseq_convergent
thf(fact_6084_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F2 @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( lattic643756798349783984er_Max @ B @ ( image @ A @ B @ F2 @ A4 ) ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_6085_gcd__int__def,axiom,
( ( gcd_gcd @ int )
= ( ^ [X: int,Y: int] : ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ X ) ) @ ( nat2 @ ( abs_abs @ int @ Y ) ) ) ) ) ) ).
% gcd_int_def
thf(fact_6086_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) @ ( dvd_dvd @ nat )
@ ^ [M3: nat,N3: nat] :
( ( dvd_dvd @ nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_6087_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
@ ^ [D4: nat] :
( ( dvd_dvd @ nat @ D4 @ M )
& ( dvd_dvd @ nat @ D4 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_6088_Nats__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( image @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% Nats_def
thf(fact_6089_Nats__altdef2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [N3: A] :
( ( member @ A @ N3 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ N3 ) ) ) ) ) ).
% Nats_altdef2
thf(fact_6090_mono__ge2__power__minus__self,axiom,
! [K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( order_mono @ nat @ nat
@ ^ [M3: nat] : ( minus_minus @ nat @ ( power_power @ nat @ K @ M3 ) @ M3 ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_6091_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A] :
( ( finite_finite @ A @ ( image @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( order_mono @ nat @ A @ F2 )
=> ( ! [N2: nat] :
( ( ( F2 @ N2 )
= ( F2 @ ( suc @ N2 ) ) )
=> ( ( F2 @ ( suc @ N2 ) )
= ( F2 @ ( suc @ ( suc @ N2 ) ) ) ) )
=> ? [N8: nat] :
( ! [N9: nat] :
( ( ord_less_eq @ nat @ N9 @ N8 )
=> ! [M2: nat] :
( ( ord_less_eq @ nat @ M2 @ N8 )
=> ( ( ord_less @ nat @ M2 @ N9 )
=> ( ord_less @ A @ ( F2 @ M2 ) @ ( F2 @ N9 ) ) ) ) )
& ! [N9: nat] :
( ( ord_less_eq @ nat @ N8 @ N9 )
=> ( ( F2 @ N8 )
= ( F2 @ N9 ) ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_6092_bezw__aux,axiom,
! [X3: nat,Y3: nat] :
( ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ X3 @ Y3 ) )
= ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ ( bezw @ X3 @ Y3 ) ) @ ( semiring_1_of_nat @ int @ X3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ X3 @ Y3 ) ) @ ( semiring_1_of_nat @ int @ Y3 ) ) ) ) ).
% bezw_aux
thf(fact_6093_tendsto__at__left__sequentially,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [B2: B,A2: B,X8: B > A,L5: A] :
( ( ord_less @ B @ B2 @ A2 )
=> ( ! [S5: nat > B] :
( ! [N9: nat] : ( ord_less @ B @ ( S5 @ N9 ) @ A2 )
=> ( ! [N9: nat] : ( ord_less @ B @ B2 @ ( S5 @ N9 ) )
=> ( ( order_mono @ nat @ B @ S5 )
=> ( ( filterlim @ nat @ B @ S5 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( X8 @ ( S5 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L5 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( topolo174197925503356063within @ B @ A2 @ ( set_ord_lessThan @ B @ A2 ) ) ) ) ) ) ).
% tendsto_at_left_sequentially
thf(fact_6094_gcd__nat_Opelims,axiom,
! [X3: nat,Xa: nat,Y3: nat] :
( ( ( gcd_gcd @ nat @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa ) )
=> ~ ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3 = X3 ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y3
= ( gcd_gcd @ nat @ Xa @ ( modulo_modulo @ nat @ X3 @ Xa ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_6095_remdups__adj__altdef,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( ( remdups_adj @ A @ Xs2 )
= Ys )
= ( ? [F3: nat > nat] :
( ( order_mono @ nat @ nat @ F3 )
& ( ( image @ nat @ nat @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys ) ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I4 )
= ( nth @ A @ Ys @ ( F3 @ I4 ) ) ) )
& ! [I4: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( nth @ A @ Xs2 @ I4 )
= ( nth @ A @ Xs2 @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) ) )
= ( ( F3 @ I4 )
= ( F3 @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% remdups_adj_altdef
thf(fact_6096_remdups__adj__length,axiom,
! [A: $tType,Xs2: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% remdups_adj_length
thf(fact_6097_remdups__adj__adjacent,axiom,
! [A: $tType,I: nat,Xs2: list @ A] :
( ( ord_less @ nat @ ( suc @ I ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs2 ) ) )
=> ( ( nth @ A @ ( remdups_adj @ A @ Xs2 ) @ I )
!= ( nth @ A @ ( remdups_adj @ A @ Xs2 ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_6098_complex__is__Nat__iff,axiom,
! [Z: complex] :
( ( member @ complex @ Z @ ( semiring_1_Nats @ complex ) )
= ( ( ( im @ Z )
= ( zero_zero @ real ) )
& ? [I4: nat] :
( ( re @ Z )
= ( semiring_1_of_nat @ real @ I4 ) ) ) ) ).
% complex_is_Nat_iff
thf(fact_6099_tendsto__at__topI__sequentially__real,axiom,
! [F2: real > real,Y3: real] :
( ( order_mono @ real @ real @ F2 )
=> ( ( filterlim @ nat @ real
@ ^ [N3: nat] : ( F2 @ ( semiring_1_of_nat @ real @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y3 )
@ ( at_top @ nat ) )
=> ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ Y3 ) @ ( at_top @ real ) ) ) ) ).
% tendsto_at_topI_sequentially_real
thf(fact_6100_nonneg__incseq__Bseq__subseq__iff,axiom,
! [F2: nat > real,G: nat > nat] :
( ! [X4: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X4 ) )
=> ( ( order_mono @ nat @ real @ F2 )
=> ( ( order_strict_mono @ nat @ nat @ G )
=> ( ( bfun @ nat @ real
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ real @ F2 @ ( at_top @ nat ) ) ) ) ) ) ).
% nonneg_incseq_Bseq_subseq_iff
thf(fact_6101_inj__sgn__power,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( inj_on @ real @ real
@ ^ [Y: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N ) )
@ ( top_top @ ( set @ real ) ) ) ) ).
% inj_sgn_power
thf(fact_6102_inj__mult__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A2: A] :
( ( inj_on @ A @ A @ ( times_times @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% inj_mult_left
thf(fact_6103_inj__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( inj_on @ A @ A
@ ^ [B4: A] : ( divide_divide @ A @ B4 @ A2 )
@ ( top_top @ ( set @ A ) ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% inj_divide_right
thf(fact_6104_strict__mono__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( order_mono @ A @ B @ F2 ) ) ) ).
% strict_mono_mono
thf(fact_6105_strict__mono__imp__increasing,axiom,
! [F2: nat > nat,N: nat] :
( ( order_strict_mono @ nat @ nat @ F2 )
=> ( ord_less_eq @ nat @ N @ ( F2 @ N ) ) ) ).
% strict_mono_imp_increasing
thf(fact_6106_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y3: A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
= ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ).
% strict_mono_less_eq
thf(fact_6107_strict__mono__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [R2: A > B,M: A,N: A] :
( ( order_strict_mono @ A @ B @ R2 )
=> ( ( ord_less_eq @ A @ M @ N )
=> ( ord_less_eq @ B @ ( R2 @ M ) @ ( R2 @ N ) ) ) ) ) ).
% strict_mono_leD
thf(fact_6108_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A4: set @ A,F2: A > B] :
( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ( member @ A @ X4 @ A4 )
=> ( ( member @ A @ Y4 @ A4 )
=> ( ( F2 @ X4 )
!= ( F2 @ Y4 ) ) ) ) )
=> ( ! [X4: A,Y4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ( member @ A @ Y4 @ A4 )
=> ( ( ord_less_eq @ A @ X4 @ Y4 )
| ( ord_less_eq @ A @ Y4 @ X4 ) ) ) )
=> ( inj_on @ A @ B @ F2 @ A4 ) ) ) ) ).
% linorder_inj_onI
thf(fact_6109_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [F2: A > B] :
( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ( F2 @ X4 )
!= ( F2 @ Y4 ) ) )
=> ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% linorder_injI
thf(fact_6110_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,A4: set @ A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A2 ) @ A4 ) ) ) ).
% inj_on_mult
thf(fact_6111_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y3: A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
= ( ord_less @ A @ X3 @ Y3 ) ) ) ) ).
% strict_mono_less
thf(fact_6112_strict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_strict_mono @ A @ B )
= ( ^ [F3: A > B] :
! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ) ).
% strict_mono_def
thf(fact_6113_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ B @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( order_strict_mono @ A @ B @ F2 ) ) ) ).
% strict_monoI
thf(fact_6114_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y3: A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% strict_monoD
thf(fact_6115_inj__fn,axiom,
! [A: $tType,F2: A > A,N: nat] :
( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ A @ ( compow @ ( A > A ) @ N @ F2 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_fn
thf(fact_6116_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X3: A,Y3: A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
= ( X3 = Y3 ) ) ) ) ).
% strict_mono_eq
thf(fact_6117_strict__mono__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_strict_mono @ nat @ A )
= ( ^ [F3: nat > A] :
! [N3: nat] : ( ord_less @ A @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) ) ) ) ) ).
% strict_mono_Suc_iff
thf(fact_6118_pigeonhole,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( ord_less @ nat @ ( finite_card @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( finite_card @ B @ A4 ) )
=> ~ ( inj_on @ B @ A @ F2 @ A4 ) ) ).
% pigeonhole
thf(fact_6119_continuous__inj__imp__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo8458572112393995274pology @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [A2: A,X3: A,B2: A,F2: A > B] :
( ( ord_less @ A @ A2 @ X3 )
=> ( ( ord_less @ A @ X3 @ B2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 )
=> ( ( inj_on @ A @ B @ F2 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
=> ( ( ( ord_less @ B @ ( F2 @ A2 ) @ ( F2 @ X3 ) )
& ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ B2 ) ) )
| ( ( ord_less @ B @ ( F2 @ B2 ) @ ( F2 @ X3 ) )
& ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ A2 ) ) ) ) ) ) ) ) ) ).
% continuous_inj_imp_mono
thf(fact_6120_injective__scaleR,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [C2: real] :
( ( C2
!= ( zero_zero @ real ) )
=> ( inj_on @ A @ A @ ( real_V8093663219630862766scaleR @ A @ C2 ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% injective_scaleR
thf(fact_6121_card__le__inj,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
( ( finite_finite @ A @ A4 )
=> ( ( finite_finite @ B @ B6 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B6 ) )
=> ? [F5: A > B] :
( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F5 @ A4 ) @ B6 )
& ( inj_on @ A @ B @ F5 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_6122_card__inj__on__le,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B6: set @ B] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A4 ) @ B6 )
=> ( ( finite_finite @ B @ B6 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B6 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_6123_inj__on__iff__card__le,axiom,
! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
( ( finite_finite @ A @ A4 )
=> ( ( finite_finite @ B @ B6 )
=> ( ( ? [F3: A > B] :
( ( inj_on @ A @ B @ F3 @ A4 )
& ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A4 ) @ B6 ) ) )
= ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B6 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_6124_log__inj,axiom,
! [B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( inj_on @ real @ real @ ( log @ B2 ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ).
% log_inj
thf(fact_6125_summable__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G: nat > nat,F2: nat > A] :
( ( order_strict_mono @ nat @ nat @ G )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ ( image @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( ( summable @ A
@ ^ [N3: nat] : ( F2 @ ( G @ N3 ) ) )
= ( summable @ A @ F2 ) ) ) ) ) ).
% summable_mono_reindex
thf(fact_6126_sums__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G: nat > nat,F2: nat > A,C2: A] :
( ( order_strict_mono @ nat @ nat @ G )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ ( image @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [N3: nat] : ( F2 @ ( G @ N3 ) )
@ C2 )
= ( sums @ A @ F2 @ C2 ) ) ) ) ) ).
% sums_mono_reindex
thf(fact_6127_suminf__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G: nat > nat,F2: nat > A] :
( ( order_strict_mono @ nat @ nat @ G )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ ( image @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( F2 @ ( G @ N3 ) ) )
= ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_mono_reindex
thf(fact_6128_increasing__Bseq__subseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > nat] :
( ! [X4: nat,Y4: nat] :
( ( ord_less_eq @ nat @ X4 @ Y4 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X4 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ Y4 ) ) ) )
=> ( ( order_strict_mono @ nat @ nat @ G )
=> ( ( bfun @ nat @ A
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ) ) ).
% increasing_Bseq_subseq_iff
thf(fact_6129_has__derivative__power__int_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X3: A,N: int,S2: set @ A] :
( ( X3
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A
@ ^ [X: A] : ( power_int @ A @ X @ N )
@ ^ [Y: A] : ( times_times @ A @ Y @ ( times_times @ A @ ( ring_1_of_int @ A @ N ) @ ( power_int @ A @ X3 @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X3 @ S2 ) ) ) ) ).
% has_derivative_power_int'
thf(fact_6130_has__derivative__power__int,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,X3: C,F6: C > A,S2: set @ C,N: int] :
( ( ( F2 @ X3 )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F2 @ F6 @ ( topolo174197925503356063within @ C @ X3 @ S2 ) )
=> ( has_derivative @ C @ A
@ ^ [X: C] : ( power_int @ A @ ( F2 @ X ) @ N )
@ ^ [H2: C] : ( times_times @ A @ ( F6 @ H2 ) @ ( times_times @ A @ ( ring_1_of_int @ A @ N ) @ ( power_int @ A @ ( F2 @ X3 ) @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) ) )
@ ( topolo174197925503356063within @ C @ X3 @ S2 ) ) ) ) ) ).
% has_derivative_power_int
thf(fact_6131_power__int__1__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int] :
( ( power_int @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_int_1_left
thf(fact_6132_power__int__1__right,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( monoid_mult @ A ) )
=> ! [Y3: A] :
( ( power_int @ A @ Y3 @ ( one_one @ int ) )
= Y3 ) ) ).
% power_int_1_right
thf(fact_6133_power__int__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,N: int] :
( ( sgn_sgn @ A @ ( power_int @ A @ A2 @ N ) )
= ( power_int @ A @ ( sgn_sgn @ A @ A2 ) @ N ) ) ) ).
% power_int_sgn
thf(fact_6134_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [W: num,Y3: A,M: int] :
( ( power_int @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Y3 ) @ M )
= ( times_times @ A @ ( power_int @ A @ ( numeral_numeral @ A @ W ) @ M ) @ ( power_int @ A @ Y3 @ M ) ) ) ) ).
% power_int_mult_distrib_numeral1
thf(fact_6135_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,W: num,M: int] :
( ( power_int @ A @ ( times_times @ A @ X3 @ ( numeral_numeral @ A @ W ) ) @ M )
= ( times_times @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ ( numeral_numeral @ A @ W ) @ M ) ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_6136_power__int__0__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M: int] :
( ( M
!= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M )
= ( zero_zero @ A ) ) ) ) ).
% power_int_0_left
thf(fact_6137_power__int__eq__0__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,N: int] :
( ( ( power_int @ A @ X3 @ N )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( N
!= ( zero_zero @ int ) ) ) ) ) ).
% power_int_eq_0_iff
thf(fact_6138_power__int__0__right,axiom,
! [B: $tType] :
( ( ( inverse @ B )
& ( power @ B ) )
=> ! [X3: B] :
( ( power_int @ B @ X3 @ ( zero_zero @ int ) )
= ( one_one @ B ) ) ) ).
% power_int_0_right
thf(fact_6139_abs__power__int__minus,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,N: int] :
( ( abs_abs @ A @ ( power_int @ A @ ( uminus_uminus @ A @ A2 ) @ N ) )
= ( abs_abs @ A @ ( power_int @ A @ A2 @ N ) ) ) ) ).
% abs_power_int_minus
thf(fact_6140_power__int__of__nat,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ! [X3: A,N: nat] :
( ( power_int @ A @ X3 @ ( semiring_1_of_nat @ int @ N ) )
= ( power_power @ A @ X3 @ N ) ) ) ).
% power_int_of_nat
thf(fact_6141_power__int__mult__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: num,N: num] :
( ( power_int @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( power_int @ A @ X3 @ ( numeral_numeral @ int @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% power_int_mult_numeral
thf(fact_6142_power__int__numeral,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ! [X3: A,N: num] :
( ( power_int @ A @ X3 @ ( numeral_numeral @ int @ N ) )
= ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% power_int_numeral
thf(fact_6143_numeral__power__int__eq__of__real__cancel__iff,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [X3: num,N: int,Y3: real] :
( ( ( power_int @ A @ ( numeral_numeral @ A @ X3 ) @ N )
= ( real_Vector_of_real @ A @ Y3 ) )
= ( ( power_int @ real @ ( numeral_numeral @ real @ X3 ) @ N )
= Y3 ) ) ) ).
% numeral_power_int_eq_of_real_cancel_iff
thf(fact_6144_of__real__eq__numeral__power__int__cancel__iff,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [Y3: real,X3: num,N: int] :
( ( ( real_Vector_of_real @ A @ Y3 )
= ( power_int @ A @ ( numeral_numeral @ A @ X3 ) @ N ) )
= ( Y3
= ( power_int @ real @ ( numeral_numeral @ real @ X3 ) @ N ) ) ) ) ).
% of_real_eq_numeral_power_int_cancel_iff
thf(fact_6145_power__int__minus1__right,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( monoid_mult @ A ) )
=> ! [Y3: A] :
( ( power_int @ A @ Y3 @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( inverse_inverse @ A @ Y3 ) ) ) ).
% power_int_minus1_right
thf(fact_6146_power__int__add__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: num,N: num] :
( ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ M ) ) @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ N ) ) )
= ( power_int @ A @ X3 @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% power_int_add_numeral
thf(fact_6147_power__int__add__numeral2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: num,N: num,B2: A] :
( ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ M ) ) @ ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ N ) ) @ B2 ) )
= ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N ) ) ) @ B2 ) ) ) ).
% power_int_add_numeral2
thf(fact_6148_power__int__mono__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ A2 @ N ) @ ( power_int @ A @ B2 @ N ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ) ).
% power_int_mono_iff
thf(fact_6149_power__int__minus__left__odd,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int,A2: A] :
( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( uminus_uminus @ A @ ( power_int @ A @ A2 @ N ) ) ) ) ) ).
% power_int_minus_left_odd
thf(fact_6150_power__int__minus__left__even,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int,A2: A] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( power_int @ A @ A2 @ N ) ) ) ) ).
% power_int_minus_left_even
thf(fact_6151_power__int__numeral__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [M: num,N: num] :
( ( power_int @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( inverse_inverse @ A @ ( numeral_numeral @ A @ ( pow @ M @ N ) ) ) ) ) ).
% power_int_numeral_neg_numeral
thf(fact_6152_inj__on__diff__nat,axiom,
! [N4: set @ nat,K: nat] :
( ! [N2: nat] :
( ( member @ nat @ N2 @ N4 )
=> ( ord_less_eq @ nat @ K @ N2 ) )
=> ( inj_on @ nat @ nat
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ K )
@ N4 ) ) ).
% inj_on_diff_nat
thf(fact_6153_inj__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ).
% inj_of_nat
thf(fact_6154_inj__Some,axiom,
! [A: $tType,A4: set @ A] : ( inj_on @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ).
% inj_Some
thf(fact_6155_inj__Suc,axiom,
! [N4: set @ nat] : ( inj_on @ nat @ nat @ suc @ N4 ) ).
% inj_Suc
thf(fact_6156_inj__on__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N4: set @ nat] : ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ N4 ) ) ).
% inj_on_of_nat
thf(fact_6157_power__int__divide__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,Y3: A,M: int] :
( ( power_int @ A @ ( divide_divide @ A @ X3 @ Y3 ) @ M )
= ( divide_divide @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ Y3 @ M ) ) ) ) ).
% power_int_divide_distrib
thf(fact_6158_power__int__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,N: int] :
( ( power_int @ A @ ( divide_divide @ A @ ( one_one @ A ) @ X3 ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_int @ A @ X3 @ N ) ) ) ) ).
% power_int_one_over
thf(fact_6159_power__int__abs,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,N: int] :
( ( abs_abs @ A @ ( power_int @ A @ A2 @ N ) )
= ( power_int @ A @ ( abs_abs @ A @ A2 ) @ N ) ) ) ).
% power_int_abs
thf(fact_6160_power__int__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,N: int] :
( ( power_int @ A @ ( inverse_inverse @ A @ X3 ) @ N )
= ( inverse_inverse @ A @ ( power_int @ A @ X3 @ N ) ) ) ) ).
% power_int_inverse
thf(fact_6161_power__int__commutes,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,N: int] :
( ( times_times @ A @ ( power_int @ A @ X3 @ N ) @ X3 )
= ( times_times @ A @ X3 @ ( power_int @ A @ X3 @ N ) ) ) ) ).
% power_int_commutes
thf(fact_6162_power__int__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,Y3: A,M: int] :
( ( power_int @ A @ ( times_times @ A @ X3 @ Y3 ) @ M )
= ( times_times @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ Y3 @ M ) ) ) ) ).
% power_int_mult_distrib
thf(fact_6163_power__int__mult,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: int,N: int] :
( ( power_int @ A @ X3 @ ( times_times @ int @ M @ N ) )
= ( power_int @ A @ ( power_int @ A @ X3 @ M ) @ N ) ) ) ).
% power_int_mult
thf(fact_6164_power__int__not__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,N: int] :
( ( ( X3
!= ( zero_zero @ A ) )
| ( N
= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X3 @ N )
!= ( zero_zero @ A ) ) ) ) ).
% power_int_not_zero
thf(fact_6165_zero__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,N: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X3 @ N ) ) ) ) ).
% zero_less_power_int
thf(fact_6166_power__int__minus,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,N: int] :
( ( power_int @ A @ X3 @ ( uminus_uminus @ int @ N ) )
= ( inverse_inverse @ A @ ( power_int @ A @ X3 @ N ) ) ) ) ).
% power_int_minus
thf(fact_6167_zero__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X3 @ N ) ) ) ) ).
% zero_le_power_int
thf(fact_6168_continuous__on__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S: set @ A,F2: A > B,N: int] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( F2 @ X4 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X: A] : ( power_int @ B @ ( F2 @ X ) @ N ) ) ) ) ) ).
% continuous_on_power_int
thf(fact_6169_power__int__0__left__If,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M: int] :
( ( ( M
= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M )
= ( one_one @ A ) ) )
& ( ( M
!= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M )
= ( zero_zero @ A ) ) ) ) ) ).
% power_int_0_left_If
thf(fact_6170_power__int__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N4: int,A2: A] :
( ( ord_less_eq @ int @ N @ N4 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_int @ A @ A2 @ N ) @ ( power_int @ A @ A2 @ N4 ) ) ) ) ) ).
% power_int_increasing
thf(fact_6171_power__int__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N4: int,A2: A] :
( ( ord_less @ int @ N @ N4 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_int @ A @ A2 @ N ) @ ( power_int @ A @ A2 @ N4 ) ) ) ) ) ).
% power_int_strict_increasing
thf(fact_6172_power__int__diff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,M: int,N: int] :
( ( ( X3
!= ( zero_zero @ A ) )
| ( M != N ) )
=> ( ( power_int @ A @ X3 @ ( minus_minus @ int @ M @ N ) )
= ( divide_divide @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ X3 @ N ) ) ) ) ) ).
% power_int_diff
thf(fact_6173_tendsto__power__int,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > A,A2: A,F4: filter @ B,N: int] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( power_int @ A @ ( F2 @ X ) @ N )
@ ( topolo7230453075368039082e_nhds @ A @ ( power_int @ A @ A2 @ N ) )
@ F4 ) ) ) ) ).
% tendsto_power_int
thf(fact_6174_continuous__at__within__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A2: A,S: set @ A,F2: A > B,N: int] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X: A] : ( power_int @ B @ ( F2 @ X ) @ N ) ) ) ) ) ).
% continuous_at_within_power_int
thf(fact_6175_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ? [N2: nat,F5: nat > A] :
( ( A4
= ( image @ nat @ A @ F5
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N2 ) ) ) )
& ( inj_on @ nat @ A @ F5
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N2 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_6176_differentiable__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X3: A,S: set @ A,N: int] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ( F2 @ X3 )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X: A] : ( power_int @ B @ ( F2 @ X ) @ N )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% differentiable_power_int
thf(fact_6177_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ? [F5: A > nat,N2: nat] :
( ( ( image @ A @ nat @ F5 @ A4 )
= ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N2 ) ) )
& ( inj_on @ A @ nat @ F5 @ A4 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_6178_continuous__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [F4: filter @ A,F2: A > B,N: int] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X: A] : X ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X: A] : ( power_int @ B @ ( F2 @ X ) @ N ) ) ) ) ) ).
% continuous_power_int
thf(fact_6179_power__int__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N4: int,A2: A] :
( ( ord_less @ int @ N @ N4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_int @ A @ A2 @ N4 ) @ ( power_int @ A @ A2 @ N ) ) ) ) ) ) ).
% power_int_strict_decreasing
thf(fact_6180_power__int__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y3: A,N: int] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( power_int @ A @ X3 @ N ) @ ( power_int @ A @ Y3 @ N ) ) ) ) ) ) ).
% power_int_mono
thf(fact_6181_power__int__strict__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,N: int] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
=> ( ord_less @ A @ ( power_int @ A @ B2 @ N ) @ ( power_int @ A @ A2 @ N ) ) ) ) ) ) ).
% power_int_strict_antimono
thf(fact_6182_one__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,N: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ X3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X3 @ N ) ) ) ) ) ).
% one_le_power_int
thf(fact_6183_one__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A2 @ N ) ) ) ) ) ).
% one_less_power_int
thf(fact_6184_power__int__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: int,N: int] :
( ( ( X3
!= ( zero_zero @ A ) )
| ( ( plus_plus @ int @ M @ N )
!= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X3 @ ( plus_plus @ int @ M @ N ) )
= ( times_times @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ X3 @ N ) ) ) ) ) ).
% power_int_add
thf(fact_6185_inj__on__nth,axiom,
! [A: $tType,Xs2: list @ A,I6: set @ nat] :
( ( distinct @ A @ Xs2 )
=> ( ! [X4: nat] :
( ( member @ nat @ X4 @ I6 )
=> ( ord_less @ nat @ X4 @ ( size_size @ ( list @ A ) @ Xs2 ) ) )
=> ( inj_on @ nat @ A @ ( nth @ A @ Xs2 ) @ I6 ) ) ) ).
% inj_on_nth
thf(fact_6186_summable__reindex,axiom,
! [F2: nat > real,G: nat > nat] :
( ( summable @ real @ F2 )
=> ( ( inj_on @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X4: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X4 ) )
=> ( summable @ real @ ( comp @ nat @ real @ nat @ F2 @ G ) ) ) ) ) ).
% summable_reindex
thf(fact_6187_inj__on__funpow__least,axiom,
! [A: $tType,N: nat,F2: A > A,S: A] :
( ( ( compow @ ( A > A ) @ N @ F2 @ S )
= S )
=> ( ! [M4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M4 )
=> ( ( ord_less @ nat @ M4 @ N )
=> ( ( compow @ ( A > A ) @ M4 @ F2 @ S )
!= S ) ) )
=> ( inj_on @ nat @ A
@ ^ [K3: nat] : ( compow @ ( A > A ) @ K3 @ F2 @ S )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% inj_on_funpow_least
thf(fact_6188_power__int__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,N: int] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ B2 @ N ) @ ( power_int @ A @ A2 @ N ) ) ) ) ) ) ).
% power_int_antimono
thf(fact_6189_power__int__strict__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,N: int] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( power_int @ A @ A2 @ N ) @ ( power_int @ A @ B2 @ N ) ) ) ) ) ) ).
% power_int_strict_mono
thf(fact_6190_power__int__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N4: int,A2: A] :
( ( ord_less_eq @ int @ N @ N4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( A2
!= ( zero_zero @ A ) )
| ( N4
!= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ int ) ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ A2 @ N4 ) @ ( power_int @ A @ A2 @ N ) ) ) ) ) ) ) ).
% power_int_decreasing
thf(fact_6191_power__int__le__one,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ X3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ X3 @ N ) @ ( one_one @ A ) ) ) ) ) ) ).
% power_int_le_one
thf(fact_6192_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,M: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ X3 @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ M @ N ) ) ) ) ) ).
% power_int_le_imp_le_exp
thf(fact_6193_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,M: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X3 )
=> ( ( ord_less @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ X3 @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ int @ M @ N ) ) ) ) ) ).
% power_int_le_imp_less_exp
thf(fact_6194_power__int__minus__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int,A2: A] :
( ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( power_int @ A @ A2 @ N ) ) )
& ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( uminus_uminus @ A @ ( power_int @ A @ A2 @ N ) ) ) ) ) ) ).
% power_int_minus_left
thf(fact_6195_power__int__minus__mult,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,N: int] :
( ( ( X3
!= ( zero_zero @ A ) )
| ( N
!= ( zero_zero @ int ) ) )
=> ( ( times_times @ A @ ( power_int @ A @ X3 @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) @ X3 )
= ( power_int @ A @ X3 @ N ) ) ) ) ).
% power_int_minus_mult
thf(fact_6196_power__int__add__1_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: int] :
( ( ( X3
!= ( zero_zero @ A ) )
| ( M
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X3 @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
= ( times_times @ A @ X3 @ ( power_int @ A @ X3 @ M ) ) ) ) ) ).
% power_int_add_1'
thf(fact_6197_power__int__add__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M: int] :
( ( ( X3
!= ( zero_zero @ A ) )
| ( M
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X3 @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
= ( times_times @ A @ ( power_int @ A @ X3 @ M ) @ X3 ) ) ) ) ).
% power_int_add_1
thf(fact_6198_suminf__reindex__mono,axiom,
! [F2: nat > real,G: nat > nat] :
( ( summable @ real @ F2 )
=> ( ( inj_on @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X4: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X4 ) )
=> ( ord_less_eq @ real @ ( suminf @ real @ ( comp @ nat @ real @ nat @ F2 @ G ) ) @ ( suminf @ real @ F2 ) ) ) ) ) ).
% suminf_reindex_mono
thf(fact_6199_power__int__def,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ( ( power_int @ A )
= ( ^ [X: A,N3: int] : ( if @ A @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N3 ) @ ( power_power @ A @ X @ ( nat2 @ N3 ) ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ ( nat2 @ ( uminus_uminus @ int @ N3 ) ) ) ) ) ) ) ).
% power_int_def
thf(fact_6200_powr__real__of__int_H,axiom,
! [X3: real,N: int] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ( X3
!= ( zero_zero @ real ) )
| ( ord_less @ int @ ( zero_zero @ int ) @ N ) )
=> ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ N ) )
= ( power_int @ real @ X3 @ N ) ) ) ) ).
% powr_real_of_int'
thf(fact_6201_inj__on__char__of__nat,axiom,
inj_on @ nat @ char @ ( unique5772411509450598832har_of @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% inj_on_char_of_nat
thf(fact_6202_suminf__reindex,axiom,
! [F2: nat > real,G: nat > nat] :
( ( summable @ real @ F2 )
=> ( ( inj_on @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X4: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X4 ) )
=> ( ! [X4: nat] :
( ~ ( member @ nat @ X4 @ ( image @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F2 @ X4 )
= ( zero_zero @ real ) ) )
=> ( ( suminf @ real @ ( comp @ nat @ real @ nat @ F2 @ G ) )
= ( suminf @ real @ F2 ) ) ) ) ) ) ).
% suminf_reindex
thf(fact_6203_pos__deriv__imp__strict__mono,axiom,
! [F2: real > real,F6: real > real] :
( ! [X4: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
=> ( ! [X4: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F6 @ X4 ) )
=> ( order_strict_mono @ real @ real @ F2 ) ) ) ).
% pos_deriv_imp_strict_mono
thf(fact_6204_DERIV__power__int,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D3: A,X3: A,S: set @ A,N: int] :
( ( has_field_derivative @ A @ F2 @ D3 @ ( topolo174197925503356063within @ A @ X3 @ S ) )
=> ( ( ( F2 @ X3 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( power_int @ A @ ( F2 @ X ) @ N )
@ ( times_times @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ N ) @ ( power_int @ A @ ( F2 @ X3 ) @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) ) @ D3 )
@ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ).
% DERIV_power_int
thf(fact_6205_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_6206_length__n__lists,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( n_lists @ A @ N @ Xs2 ) )
= ( power_power @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ).
% length_n_lists
thf(fact_6207_integer__of__num__triv_I1_J,axiom,
( ( code_integer_of_num @ one2 )
= ( one_one @ code_integer ) ) ).
% integer_of_num_triv(1)
thf(fact_6208_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_6209_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_6210_sum__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A4: set @ B,P: B > $o] :
( ( finite_finite @ B @ A4 )
=> ( ( finite_finite @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( zero_neq_one_of_bool @ A @ ( P @ X ) )
@ A4 )
= ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_6211_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_6212_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ Y3 @ Z ) )
= ( ( ord_less_eq @ A @ X3 @ Y3 )
& ( ord_less_eq @ A @ X3 @ Z ) ) ) ) ).
% le_inf_iff
thf(fact_6213_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_6214_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ Y3 ) @ Z )
= ( ( ord_less_eq @ A @ X3 @ Z )
& ( ord_less_eq @ A @ Y3 @ Z ) ) ) ) ).
% le_sup_iff
thf(fact_6215_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A4: set @ B,B6: set @ B,F2: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( finite_finite @ B @ B6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) )
=> ( ( F2 @ X4 )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ B6 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_6216_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_inf @ A )
& ( semilattice_inf @ B ) )
=> ! [F2: A > B,A4: A,B6: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( F2 @ ( inf_inf @ A @ A4 @ B6 ) ) @ ( inf_inf @ B @ ( F2 @ A4 ) @ ( F2 @ B6 ) ) ) ) ) ).
% mono_inf
thf(fact_6217_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_sup @ A )
& ( semilattice_sup @ B ) )
=> ! [F2: A > B,A4: A,B6: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F2 @ A4 ) @ ( F2 @ B6 ) ) @ ( F2 @ ( sup_sup @ A @ A4 @ B6 ) ) ) ) ) ).
% mono_sup
thf(fact_6218_ivl__disj__un__two_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_6219_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B6: set @ B,G: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( finite_finite @ B @ B6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B6 ) ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_6220_less__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,X3: A,B2: A] :
( ( ord_less @ A @ A2 @ X3 )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% less_infI1
thf(fact_6221_less__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X3: A,A2: A] :
( ( ord_less @ A @ B2 @ X3 )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% less_infI2
thf(fact_6222_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_6223_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_6224_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_6225_inf_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( A5
= ( inf_inf @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% inf.strict_order_iff
thf(fact_6226_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_6227_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_6228_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,A2: A,B2: A] :
( ( ord_less @ A @ X3 @ A2 )
=> ( ord_less @ A @ X3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% less_supI1
thf(fact_6229_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,B2: A,A2: A] :
( ( ord_less @ A @ X3 @ B2 )
=> ( ord_less @ A @ X3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% less_supI2
thf(fact_6230_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_6231_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_6232_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_6233_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( A5
= ( sup_sup @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_6234_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_6235_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_6236_ivl__disj__un__two__touch_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_6237_Sup__inter__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B6: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ).
% Sup_inter_less_eq
thf(fact_6238_less__eq__Inf__inter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B6: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B6 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ) ).
% less_eq_Inf_inter
thf(fact_6239_distrib__sup__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y3: A,Z: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y3 @ Z ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y3 ) @ ( sup_sup @ A @ X3 @ Z ) ) ) ) ).
% distrib_sup_le
thf(fact_6240_distrib__inf__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y3: A,Z: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ ( inf_inf @ A @ X3 @ Z ) ) @ ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y3 @ Z ) ) ) ) ).
% distrib_inf_le
thf(fact_6241_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_6242_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_6243_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( inf_inf @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_6244_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( inf_inf @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_6245_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_6246_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_6247_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( A5
= ( inf_inf @ A @ A5 @ B4 ) ) ) ) ) ).
% inf.order_iff
thf(fact_6248_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Z )
=> ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ Y3 @ Z ) ) ) ) ) ).
% inf_greatest
thf(fact_6249_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_6250_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_6251_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( inf_inf @ A @ X3 @ Y3 )
= Y3 ) ) ) ).
% inf_absorb2
thf(fact_6252_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( inf_inf @ A @ X3 @ Y3 )
= X3 ) ) ) ).
% inf_absorb1
thf(fact_6253_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_6254_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_6255_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y: A] :
( ( inf_inf @ A @ X @ Y )
= X ) ) ) ) ).
% le_iff_inf
thf(fact_6256_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F2: A > A > A,X3: A,Y3: A] :
( ! [X4: A,Y4: A] : ( ord_less_eq @ A @ ( F2 @ X4 @ Y4 ) @ X4 )
=> ( ! [X4: A,Y4: A] : ( ord_less_eq @ A @ ( F2 @ X4 @ Y4 ) @ Y4 )
=> ( ! [X4: A,Y4: A,Z3: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ( ord_less_eq @ A @ X4 @ Z3 )
=> ( ord_less_eq @ A @ X4 @ ( F2 @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf @ A @ X3 @ Y3 )
= ( F2 @ X3 @ Y3 ) ) ) ) ) ) ).
% inf_unique
thf(fact_6257_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_6258_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_6259_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X3: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ X3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% le_infI2
thf(fact_6260_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,X3: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% le_infI1
thf(fact_6261_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ ( inf_inf @ A @ C2 @ D3 ) ) ) ) ) ).
% inf_mono
thf(fact_6262_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X3 @ A2 )
=> ( ( ord_less_eq @ A @ X3 @ B2 )
=> ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% le_infI
thf(fact_6263_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq @ A @ X3 @ A2 )
=> ~ ( ord_less_eq @ A @ X3 @ B2 ) ) ) ) ).
% le_infE
thf(fact_6264_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ Y3 ) ) ).
% inf_le2
thf(fact_6265_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ X3 ) ) ).
% inf_le1
thf(fact_6266_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ X3 ) ) ).
% inf_sup_ord(1)
thf(fact_6267_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ Y3 ) ) ).
% inf_sup_ord(2)
thf(fact_6268_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_6269_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_6270_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( sup_sup @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_6271_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( sup_sup @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_6272_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_6273_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_6274_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( A5
= ( sup_sup @ A @ A5 @ B4 ) ) ) ) ) ).
% sup.order_iff
thf(fact_6275_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_6276_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_6277_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( sup_sup @ A @ X3 @ Y3 )
= Y3 ) ) ) ).
% sup_absorb2
thf(fact_6278_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( sup_sup @ A @ X3 @ Y3 )
= X3 ) ) ) ).
% sup_absorb1
thf(fact_6279_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_6280_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_6281_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F2: A > A > A,X3: A,Y3: A] :
( ! [X4: A,Y4: A] : ( ord_less_eq @ A @ X4 @ ( F2 @ X4 @ Y4 ) )
=> ( ! [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ ( F2 @ X4 @ Y4 ) )
=> ( ! [X4: A,Y4: A,Z3: A] :
( ( ord_less_eq @ A @ Y4 @ X4 )
=> ( ( ord_less_eq @ A @ Z3 @ X4 )
=> ( ord_less_eq @ A @ ( F2 @ Y4 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup @ A @ X3 @ Y3 )
= ( F2 @ X3 @ Y3 ) ) ) ) ) ) ).
% sup_unique
thf(fact_6282_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_6283_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_6284_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y: A] :
( ( sup_sup @ A @ X @ Y )
= Y ) ) ) ) ).
% le_iff_sup
thf(fact_6285_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y3: A,X3: A,Z: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ Z @ X3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y3 @ Z ) @ X3 ) ) ) ) ).
% sup_least
thf(fact_6286_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ ( sup_sup @ A @ C2 @ D3 ) ) ) ) ) ).
% sup_mono
thf(fact_6287_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A2: A,D3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ A @ D3 @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D3 ) @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_6288_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ X3 @ B2 )
=> ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_6289_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X3 @ A2 )
=> ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_6290_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y3: A,X3: A] : ( ord_less_eq @ A @ Y3 @ ( sup_sup @ A @ X3 @ Y3 ) ) ) ).
% sup_ge2
thf(fact_6291_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ X3 @ Y3 ) ) ) ).
% sup_ge1
thf(fact_6292_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,X3: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X3 )
=> ( ( ord_less_eq @ A @ B2 @ X3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X3 ) ) ) ) ).
% le_supI
thf(fact_6293_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A,X3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq @ A @ A2 @ X3 )
=> ~ ( ord_less_eq @ A @ B2 @ X3 ) ) ) ) ).
% le_supE
thf(fact_6294_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ X3 @ Y3 ) ) ) ).
% inf_sup_ord(3)
thf(fact_6295_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y3: A,X3: A] : ( ord_less_eq @ A @ Y3 @ ( sup_sup @ A @ X3 @ Y3 ) ) ) ).
% inf_sup_ord(4)
thf(fact_6296_shunt1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ Z )
= ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ ( uminus_uminus @ A @ Y3 ) @ Z ) ) ) ) ).
% shunt1
thf(fact_6297_shunt2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ ( uminus_uminus @ A @ Y3 ) ) @ Z )
= ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ Y3 @ Z ) ) ) ) ).
% shunt2
thf(fact_6298_sup__neg__inf,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [P6: A,Q3: A,R2: A] :
( ( ord_less_eq @ A @ P6 @ ( sup_sup @ A @ Q3 @ R2 ) )
= ( ord_less_eq @ A @ ( inf_inf @ A @ P6 @ ( uminus_uminus @ A @ Q3 ) ) @ R2 ) ) ) ).
% sup_neg_inf
thf(fact_6299_ivl__disj__un__two_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_6300_sup__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y3: A] :
( ( ( sup_sup @ A @ X3 @ Y3 )
= ( top_top @ A ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y3 ) ) ) ).
% sup_shunt
thf(fact_6301_inf__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y3: A] :
( ( ( inf_inf @ A @ X3 @ Y3 )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% inf_shunt
thf(fact_6302_ivl__disj__un__two_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_6303_card__Un__le,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) ) ) ).
% card_Un_le
thf(fact_6304_ivl__disj__un__one_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( set_ord_lessThan @ A @ U ) ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_6305_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_6306_Ioc__disjoint,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ord_less_eq @ A @ D3 @ C2 )
| ( ord_less_eq @ A @ B2 @ C2 )
| ( ord_less_eq @ A @ D3 @ A2 ) ) ) ) ).
% Ioc_disjoint
thf(fact_6307_ivl__disj__un__two_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_6308_ivl__disj__un__one_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( set_ord_atLeast @ A @ L ) ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_6309_ivl__disj__un__one_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( set_ord_atMost @ A @ U ) ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_6310_ivl__disj__un__one_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% ivl_disj_un_one(5)
thf(fact_6311_sum_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A,B6: set @ B] :
( ( finite_finite @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( if @ A @ ( member @ B @ X @ B6 ) @ ( G @ X ) @ ( zero_zero @ A ) )
@ A4 ) ) ) ) ).
% sum.inter_restrict
thf(fact_6312_open__Collect__less__Int,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real,G: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ real @ S @ G )
=> ? [A7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ A7 )
& ( ( inf_inf @ ( set @ A ) @ A7 @ S )
= ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ S )
& ( ord_less @ real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ).
% open_Collect_less_Int
thf(fact_6313_ivl__disj__un__two__touch_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_6314_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S2: set @ B,H: B > A,G: B > A] :
( ( finite_finite @ B @ T6 )
=> ( ( finite_finite @ B @ S2 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( H @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ S2 @ T6 ) )
=> ( ( G @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( inf_inf @ ( set @ B ) @ S2 @ T6 ) )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ T6 ) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_6315_Iio__Int__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,K: A] :
( ( ( ord_less @ A @ X3 @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( ord_less @ A @ X3 @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Iio_Int_singleton
thf(fact_6316_ivl__disj__un__two__touch_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_6317_ivl__disj__un__two_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_6318_ivl__disj__un__one_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( set_ord_atMost @ A @ U ) ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_6319_less__separate,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ? [A6: A,B5: A] :
( ( member @ A @ X3 @ ( set_ord_lessThan @ A @ A6 ) )
& ( member @ A @ Y3 @ ( set_ord_greaterThan @ A @ B5 ) )
& ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A6 ) @ ( set_ord_greaterThan @ A @ B5 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% less_separate
thf(fact_6320_inf__top_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ( semila1105856199041335345_order @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_6321_Max_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B6 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
= ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ ( lattic643756798349783984er_Max @ A @ B6 ) ) ) ) ) ) ) ) ).
% Max.union
thf(fact_6322_ivl__disj__un__two_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_6323_ivl__disj__un__one_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( set_ord_lessThan @ A @ U ) ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_6324_ivl__disj__un__one_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( set_ord_atLeast @ A @ L ) ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_6325_ivl__disj__un__two__touch_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_6326_ivl__disj__un__one_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_6327_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X: A,Y: A] : ( ord_less_eq @ A @ Y @ X )
@ ^ [X: A,Y: A] : ( ord_less @ A @ Y @ X ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_6328_open__Collect__positive,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ? [A7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ A7 )
& ( ( inf_inf @ ( set @ A ) @ A7 @ S )
= ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ S )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) ) ) ) ) ) ) ) ).
% open_Collect_positive
thf(fact_6329_ivl__disj__un__singleton_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_6330_ivl__disj__un__singleton_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_6331_ivl__disj__un__two_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_6332_ivl__disj__un__singleton_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_6333_ivl__disj__un__two_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_6334_ivl__disj__un__singleton_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_6335_INF__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: A,B6: A] :
( ( inf_inf @ A @ A4
@ ( complete_Inf_Inf @ A
@ ( image @ nat @ A
@ ^ [X: nat] : B6
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( inf_inf @ A @ A4 @ B6 ) ) ) ).
% INF_nat_binary
thf(fact_6336_SUP__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: A,B6: A] :
( ( sup_sup @ A @ A4
@ ( complete_Sup_Sup @ A
@ ( image @ nat @ A
@ ^ [X: nat] : B6
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( sup_sup @ A @ A4 @ B6 ) ) ) ).
% SUP_nat_binary
thf(fact_6337_sup__nat__def,axiom,
( ( sup_sup @ nat )
= ( ord_max @ nat ) ) ).
% sup_nat_def
thf(fact_6338_sup__enat__def,axiom,
( ( sup_sup @ extended_enat )
= ( ord_max @ extended_enat ) ) ).
% sup_enat_def
thf(fact_6339_sup__int__def,axiom,
( ( sup_sup @ int )
= ( ord_max @ int ) ) ).
% sup_int_def
thf(fact_6340_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: $o] :
( ( ( vEBT_VEBT_valid @ X3 @ Xa )
= Y3 )
=> ( ( ? [Uu: $o,Uv: $o] :
( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( Y3
= ( Xa
!= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( Y3
= ( ~ ( ( Deg2 = Xa )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X9 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X @ X9 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_6341_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_valid @ X3 @ Xa )
=> ( ( ? [Uu: $o,Uv: $o] :
( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( Xa
!= ( one_one @ nat ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ~ ( ( Deg2 = Xa )
& ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X6 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X9 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X @ X9 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_6342_Ball__def__raw,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A9: set @ A,P3: A > $o] :
! [X: A] :
( ( member @ A @ X @ A9 )
=> ( P3 @ X ) ) ) ) ).
% Ball_def_raw
thf(fact_6343_option_Odisc__eq__case_I1_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
= ( none @ A ) )
= ( case_option @ $o @ A @ $true
@ ^ [Uu3: A] : $false
@ Option ) ) ).
% option.disc_eq_case(1)
thf(fact_6344_option_Odisc__eq__case_I2_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
= ( case_option @ $o @ A @ $false
@ ^ [Uu3: A] : $true
@ Option ) ) ).
% option.disc_eq_case(2)
thf(fact_6345_option_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > B,X2: A] :
( ( case_option @ B @ A @ F1 @ F22 @ ( some @ A @ X2 ) )
= ( F22 @ X2 ) ) ).
% option.simps(5)
thf(fact_6346_option_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: A > B] :
( ( case_option @ B @ A @ F1 @ F22 @ ( none @ A ) )
= F1 ) ).
% option.simps(4)
thf(fact_6347_option_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H: B > C,F1: B,F22: A > B,Option: option @ A] :
( ( H @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( case_option @ C @ A @ ( H @ F1 )
@ ^ [X: A] : ( H @ ( F22 @ X ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_6348_open__subdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X: A,Y: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X @ Y ) )
& ( ord_less @ A @ X @ Y ) ) ) ) ) ).
% open_subdiagonal
thf(fact_6349_open__superdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X: A,Y: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X @ Y ) )
& ( ord_less @ A @ Y @ X ) ) ) ) ) ).
% open_superdiagonal
thf(fact_6350_option_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_option @ B @ A )
= ( ^ [F12: B,F23: A > B,Option3: option @ A] :
( if @ B
@ ( Option3
= ( none @ A ) )
@ F12
@ ( F23 @ ( the2 @ A @ Option3 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_6351_case__optionE,axiom,
! [A: $tType,P: $o,Q: A > $o,X3: option @ A] :
( ( case_option @ $o @ A @ P @ Q @ X3 )
=> ( ( ( X3
= ( none @ A ) )
=> ~ P )
=> ~ ! [Y4: A] :
( ( X3
= ( some @ A @ Y4 ) )
=> ~ ( Q @ Y4 ) ) ) ) ).
% case_optionE
thf(fact_6352_Inf__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A )
= ( ^ [A9: set @ A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [B4: A] :
! [X: A] :
( ( member @ A @ X @ A9 )
=> ( ord_less_eq @ A @ B4 @ X ) ) ) ) ) ) ) ).
% Inf_eq_Sup
thf(fact_6353_Sup__eq__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A )
= ( ^ [A9: set @ A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [B4: A] :
! [X: A] :
( ( member @ A @ X @ A9 )
=> ( ord_less_eq @ A @ X @ B4 ) ) ) ) ) ) ) ).
% Sup_eq_Inf
thf(fact_6354_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs: list @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [I4: nat] :
( ( Uu3
= ( nth @ A @ Xs @ I4 ) )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_6355_option_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option
= ( none @ A ) )
& ~ ( P @ F1 ) )
| ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
& ~ ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_6356_option_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ( ( Option
= ( none @ A ) )
=> ( P @ F1 ) )
& ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
=> ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_6357_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
! [Mima2: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg3 )
= ( ( Deg = Deg3 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X9 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X @ X9 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_6358_funpow__inj__finite,axiom,
! [A: $tType,P6: A > A,X3: A] :
( ( inj_on @ A @ A @ P6 @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite @ A
@ ( collect @ A
@ ^ [Y: A] :
? [N3: nat] :
( Y
= ( compow @ ( A > A ) @ N3 @ P6 @ X3 ) ) ) )
=> ~ ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( compow @ ( A > A ) @ N2 @ P6 @ X3 )
!= X3 ) ) ) ) ).
% funpow_inj_finite
thf(fact_6359_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_valid @ X3 @ Xa )
=> ( ( ? [Uu: $o,Uv: $o] :
( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( Xa
= ( one_one @ nat ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( Deg2 = Xa )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X9 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X @ X9 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_6360_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa: nat,Y3: $o] :
( ( ( vEBT_VEBT_valid @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Y3
= ( Xa
= ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( Y3
= ( ( Deg2 = Xa )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X9 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X @ X9 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
thf(fact_6361_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_valid @ X3 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa ) )
=> ( Xa
!= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) )
=> ~ ( ( Deg2 = Xa )
& ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X6 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X9 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X @ X9 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_6362_Inf__Sup__le,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A4: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A4 ) )
@ ( complete_Sup_Sup @ A
@ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F3: ( set @ A ) > A] :
( ( Uu3
= ( image @ ( set @ A ) @ A @ F3 @ A4 ) )
& ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A4 )
=> ( member @ A @ ( F3 @ X ) @ X ) ) ) ) ) ) ) ) ).
% Inf_Sup_le
thf(fact_6363_Sup__Inf__le,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ ( set @ A )] :
( ord_less_eq @ A
@ ( complete_Sup_Sup @ A
@ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F3: ( set @ A ) > A] :
( ( Uu3
= ( image @ ( set @ A ) @ A @ F3 @ A4 ) )
& ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A4 )
=> ( member @ A @ ( F3 @ X ) @ X ) ) ) ) ) )
@ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A4 ) ) ) ) ).
% Sup_Inf_le
thf(fact_6364_Nats__altdef1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [Uu3: A] :
? [N3: int] :
( ( Uu3
= ( ring_1_of_int @ A @ N3 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ N3 ) ) ) ) ) ).
% Nats_altdef1
thf(fact_6365_Sup__int__def,axiom,
( ( complete_Sup_Sup @ int )
= ( ^ [X9: set @ int] :
( the @ int
@ ^ [X: int] :
( ( member @ int @ X @ X9 )
& ! [Y: int] :
( ( member @ int @ Y @ X9 )
=> ( ord_less_eq @ int @ Y @ X ) ) ) ) ) ) ).
% Sup_int_def
thf(fact_6366_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_valid @ X3 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X3
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa ) )
=> ( Xa
= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) )
=> ( ( Deg2 = Xa )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X9 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X @ X9 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_6367_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 )
@ ^ [Q5: 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 @ Q5 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_6368_finite__Inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A4: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A4 ) )
@ ( complete_Sup_Sup @ A
@ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F3: ( set @ A ) > A] :
( ( Uu3
= ( image @ ( set @ A ) @ A @ F3 @ A4 ) )
& ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A4 )
=> ( member @ A @ ( F3 @ X ) @ X ) ) ) ) ) ) ) ) ).
% finite_Inf_Sup
thf(fact_6369_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_6370_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_6371_take__bit__num__simps_I5_J,axiom,
! [R2: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R2 ) @ one2 )
= ( some @ num @ one2 ) ) ).
% take_bit_num_simps(5)
thf(fact_6372_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 )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ N @ M ) ) ) ).
% take_bit_num_simps(3)
thf(fact_6373_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_6374_take__bit__num__simps_I6_J,axiom,
! [R2: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R2 ) @ ( bit0 @ M ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).
% take_bit_num_simps(6)
thf(fact_6375_take__bit__num__simps_I7_J,axiom,
! [R2: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R2 ) @ ( bit1 @ M ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).
% take_bit_num_simps(7)
thf(fact_6376_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_6377_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 )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ N3 @ M ) )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_6378_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_6379_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_6380_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_6381_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_6382_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_6383_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_6384_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_6385_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_6386_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_6387_and__not__num_Osimps_I1_J,axiom,
( ( bit_and_not_num @ one2 @ one2 )
= ( none @ num ) ) ).
% and_not_num.simps(1)
thf(fact_6388_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_6389_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_6390_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_6391_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_6392_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_6393_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_6394_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_6395_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_6396_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_6397_Bit__Operations_Otake__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N3: nat,M3: num] :
( product_case_prod @ nat @ num @ ( option @ num )
@ ^ [A5: nat,X: num] :
( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [O: nat] :
( case_num @ ( option @ num ) @ ( some @ num @ one2 )
@ ^ [P5: num] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ O @ P5 ) )
@ ^ [P5: num] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ O @ P5 ) ) )
@ X )
@ A5 )
@ ( product_Pair @ nat @ num @ N3 @ M3 ) ) ) ) ).
% Bit_Operations.take_bit_num_code
thf(fact_6398_mlex__eq,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F3: A > nat,R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X: A,Y: A] :
( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
| ( ( ord_less_eq @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R6 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_6399_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_6400_num_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H: A > B,F1: A,F22: num > A,F32: num > A,Num: num] :
( ( H @ ( case_num @ A @ F1 @ F22 @ F32 @ Num ) )
= ( case_num @ B @ ( H @ F1 )
@ ^ [X: num] : ( H @ ( F22 @ X ) )
@ ^ [X: num] : ( H @ ( F32 @ X ) )
@ Num ) ) ).
% num.case_distrib
thf(fact_6401_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_6402_verit__eq__simplify_I17_J,axiom,
! [A: $tType,F1: A,F22: num > A,F32: num > A,X2: num] :
( ( case_num @ A @ F1 @ F22 @ F32 @ ( bit0 @ X2 ) )
= ( F22 @ X2 ) ) ).
% verit_eq_simplify(17)
thf(fact_6403_mlex__leq,axiom,
! [A: $tType,F2: A > nat,X3: A,Y3: A,R3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( mlex_prod @ A @ F2 @ R3 ) ) ) ) ).
% mlex_leq
thf(fact_6404_mlex__less,axiom,
! [A: $tType,F2: A > nat,X3: A,Y3: A,R3: set @ ( product_prod @ A @ A )] :
( ( ord_less @ nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( mlex_prod @ A @ F2 @ R3 ) ) ) ).
% mlex_less
thf(fact_6405_mlex__iff,axiom,
! [A: $tType,X3: A,Y3: A,F2: A > nat,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( mlex_prod @ A @ F2 @ R3 ) )
= ( ( ord_less @ nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
| ( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ).
% mlex_iff
thf(fact_6406_in__measure,axiom,
! [A: $tType,X3: A,Y3: A,F2: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( measure @ A @ F2 ) )
= ( ord_less @ nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ).
% in_measure
thf(fact_6407_Rats__eq__int__div__nat,axiom,
( ( field_char_0_Rats @ real )
= ( collect @ real
@ ^ [Uu3: real] :
? [I4: int,N3: nat] :
( ( Uu3
= ( divide_divide @ real @ ( ring_1_of_int @ real @ I4 ) @ ( semiring_1_of_nat @ real @ N3 ) ) )
& ( N3
!= ( zero_zero @ nat ) ) ) ) ) ).
% Rats_eq_int_div_nat
thf(fact_6408_Rats__abs__iff,axiom,
! [X3: real] :
( ( member @ real @ ( abs_abs @ real @ X3 ) @ ( field_char_0_Rats @ real ) )
= ( member @ real @ X3 @ ( field_char_0_Rats @ real ) ) ) ).
% Rats_abs_iff
thf(fact_6409_Rats__power,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N: nat] :
( ( member @ A @ A2 @ ( field_char_0_Rats @ A ) )
=> ( member @ A @ ( power_power @ A @ A2 @ N ) @ ( field_char_0_Rats @ A ) ) ) ) ).
% Rats_power
thf(fact_6410_Rats__of__int,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z: int] : ( member @ A @ ( ring_1_of_int @ A @ Z ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_of_int
thf(fact_6411_Rats__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_of_nat
thf(fact_6412_Rats__number__of,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [W: num] : ( member @ A @ ( numeral_numeral @ A @ W ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_number_of
thf(fact_6413_Rats__0,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_0
thf(fact_6414_Rats__dense__in__real,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ X3 @ Y3 )
=> ? [X4: real] :
( ( member @ real @ X4 @ ( field_char_0_Rats @ real ) )
& ( ord_less @ real @ X3 @ X4 )
& ( ord_less @ real @ X4 @ Y3 ) ) ) ).
% Rats_dense_in_real
thf(fact_6415_Rats__no__bot__less,axiom,
! [X3: real] :
? [X4: real] :
( ( member @ real @ X4 @ ( field_char_0_Rats @ real ) )
& ( ord_less @ real @ X4 @ X3 ) ) ).
% Rats_no_bot_less
thf(fact_6416_Rats__no__top__le,axiom,
! [X3: real] :
? [X4: real] :
( ( member @ real @ X4 @ ( field_char_0_Rats @ real ) )
& ( ord_less_eq @ real @ X3 @ X4 ) ) ).
% Rats_no_top_le
thf(fact_6417_Rats__eq__int__div__int,axiom,
( ( field_char_0_Rats @ real )
= ( collect @ real
@ ^ [Uu3: real] :
? [I4: int,J3: int] :
( ( Uu3
= ( divide_divide @ real @ ( ring_1_of_int @ real @ I4 ) @ ( ring_1_of_int @ real @ J3 ) ) )
& ( J3
!= ( zero_zero @ int ) ) ) ) ) ).
% Rats_eq_int_div_int
thf(fact_6418_and__not__num_Oelims,axiom,
! [X3: num,Xa: num,Y3: option @ num] :
( ( ( bit_and_not_num @ X3 @ Xa )
= Y3 )
=> ( ( ( X3 = one2 )
=> ( ( Xa = one2 )
=> ( Y3
!= ( none @ num ) ) ) )
=> ( ( ( X3 = one2 )
=> ( ? [N2: num] :
( Xa
= ( bit0 @ N2 ) )
=> ( Y3
!= ( some @ num @ one2 ) ) ) )
=> ( ( ( X3 = one2 )
=> ( ? [N2: num] :
( Xa
= ( bit1 @ N2 ) )
=> ( Y3
!= ( none @ num ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ( ( Xa = one2 )
=> ( Y3
!= ( some @ num @ ( bit0 @ M4 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y3
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y3
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ( ( Xa = one2 )
=> ( Y3
!= ( some @ num @ ( bit0 @ M4 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y3
!= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N10: num] : ( some @ num @ ( bit1 @ N10 ) )
@ ( bit_and_not_num @ M4 @ N2 ) ) ) ) )
=> ~ ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y3
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.elims
thf(fact_6419_xor__num_Osimps_I6_J,axiom,
! [M: num,N: num] :
( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% xor_num.simps(6)
thf(fact_6420_map__option__eq__Some,axiom,
! [A: $tType,B: $tType,F2: B > A,Xo: option @ B,Y3: A] :
( ( ( map_option @ B @ A @ F2 @ Xo )
= ( some @ A @ Y3 ) )
= ( ? [Z6: B] :
( ( Xo
= ( some @ B @ Z6 ) )
& ( ( F2 @ Z6 )
= Y3 ) ) ) ) ).
% map_option_eq_Some
thf(fact_6421_option_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: option @ A] :
( ( ( map_option @ A @ B @ F2 @ A2 )
= ( none @ B ) )
= ( A2
= ( none @ A ) ) ) ).
% option.map_disc_iff
thf(fact_6422_map__option__is__None,axiom,
! [A: $tType,B: $tType,F2: B > A,Opt: option @ B] :
( ( ( map_option @ B @ A @ F2 @ Opt )
= ( none @ A ) )
= ( Opt
= ( none @ B ) ) ) ).
% map_option_is_None
thf(fact_6423_None__eq__map__option__iff,axiom,
! [A: $tType,B: $tType,F2: B > A,X3: option @ B] :
( ( ( none @ A )
= ( map_option @ B @ A @ F2 @ X3 ) )
= ( X3
= ( none @ B ) ) ) ).
% None_eq_map_option_iff
thf(fact_6424_case__map__option,axiom,
! [B: $tType,A: $tType,C: $tType,G: A,H: B > A,F2: C > B,X3: option @ C] :
( ( case_option @ A @ B @ G @ H @ ( map_option @ C @ B @ F2 @ X3 ) )
= ( case_option @ A @ C @ G @ ( comp @ B @ A @ C @ H @ F2 ) @ X3 ) ) ).
% case_map_option
thf(fact_6425_option_Omap__ident,axiom,
! [A: $tType,T2: option @ A] :
( ( map_option @ A @ A
@ ^ [X: A] : X
@ T2 )
= T2 ) ).
% option.map_ident
thf(fact_6426_xor__num_Osimps_I9_J,axiom,
! [M: num,N: num] :
( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% xor_num.simps(9)
thf(fact_6427_xor__num_Osimps_I5_J,axiom,
! [M: num,N: num] :
( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% xor_num.simps(5)
thf(fact_6428_map__option_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F2: B > C,G: A > B] :
( ( comp @ ( option @ B ) @ ( option @ C ) @ ( option @ A ) @ ( map_option @ B @ C @ F2 ) @ ( map_option @ A @ B @ G ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ F2 @ G ) ) ) ).
% map_option.comp
thf(fact_6429_option_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F2: A > B,V2: option @ A] :
( ( map_option @ B @ C @ G @ ( map_option @ A @ B @ F2 @ V2 ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ G @ F2 ) @ V2 ) ) ).
% option.map_comp
thf(fact_6430_map__option_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F2: B > C,G: A > B,Option: option @ A] :
( ( map_option @ B @ C @ F2 @ ( map_option @ A @ B @ G @ Option ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ F2 @ G ) @ Option ) ) ).
% map_option.compositionality
thf(fact_6431_option_Omap__sel,axiom,
! [B: $tType,A: $tType,A2: option @ A,F2: A > B] :
( ( A2
!= ( none @ A ) )
=> ( ( the2 @ B @ ( map_option @ A @ B @ F2 @ A2 ) )
= ( F2 @ ( the2 @ A @ A2 ) ) ) ) ).
% option.map_sel
thf(fact_6432_option_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ( map_option @ A @ B @ F2 @ ( none @ A ) )
= ( none @ B ) ) ).
% option.simps(8)
thf(fact_6433_and__not__num_Osimps_I5_J,axiom,
! [M: num,N: num] :
( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(5)
thf(fact_6434_map__option__cong,axiom,
! [B: $tType,A: $tType,X3: option @ A,Y3: option @ A,F2: A > B,G: A > B] :
( ( X3 = Y3 )
=> ( ! [A6: A] :
( ( Y3
= ( some @ A @ A6 ) )
=> ( ( F2 @ A6 )
= ( G @ A6 ) ) )
=> ( ( map_option @ A @ B @ F2 @ X3 )
= ( map_option @ A @ B @ G @ Y3 ) ) ) ) ).
% map_option_cong
thf(fact_6435_option_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F2: A > B,X2: A] :
( ( map_option @ A @ B @ F2 @ ( some @ A @ X2 ) )
= ( some @ B @ ( F2 @ X2 ) ) ) ).
% option.simps(9)
thf(fact_6436_and__not__num_Osimps_I6_J,axiom,
! [M: num,N: num] :
( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(6)
thf(fact_6437_and__not__num_Osimps_I9_J,axiom,
! [M: num,N: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(9)
thf(fact_6438_xor__num_Osimps_I1_J,axiom,
( ( bit_un2480387367778600638or_num @ one2 @ one2 )
= ( none @ num ) ) ).
% xor_num.simps(1)
thf(fact_6439_option_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F2: B > nat,G: A > B] :
( ( comp @ ( option @ B ) @ nat @ ( option @ A ) @ ( size_option @ B @ F2 ) @ ( map_option @ A @ B @ G ) )
= ( size_option @ A @ ( comp @ B @ nat @ A @ F2 @ G ) ) ) ).
% option.size_gen_o_map
thf(fact_6440_option_Oinj__map,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( option @ A ) @ ( option @ B ) @ ( map_option @ A @ B @ F2 ) @ ( top_top @ ( set @ ( option @ A ) ) ) ) ) ).
% option.inj_map
thf(fact_6441_xor__num_Oelims,axiom,
! [X3: num,Xa: num,Y3: option @ num] :
( ( ( bit_un2480387367778600638or_num @ X3 @ Xa )
= Y3 )
=> ( ( ( X3 = one2 )
=> ( ( Xa = one2 )
=> ( Y3
!= ( none @ num ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y3
!= ( some @ num @ ( bit1 @ N2 ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y3
!= ( some @ num @ ( bit0 @ N2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ( ( Xa = one2 )
=> ( Y3
!= ( some @ num @ ( bit1 @ M4 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y3
!= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y3
!= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ( ( Xa = one2 )
=> ( Y3
!= ( some @ num @ ( bit0 @ M4 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y3
!= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) ) )
=> ~ ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y3
!= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.elims
thf(fact_6442_map__option__case,axiom,
! [A: $tType,B: $tType] :
( ( map_option @ B @ A )
= ( ^ [F3: B > A] :
( case_option @ ( option @ A ) @ B @ ( none @ A )
@ ^ [X: B] : ( some @ A @ ( F3 @ X ) ) ) ) ) ).
% map_option_case
thf(fact_6443_xor__num__eq__Some__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: num,Q3: num] :
( ( ( bit_un2480387367778600638or_num @ M @ N )
= ( some @ num @ Q3 ) )
= ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ Q3 ) ) ) ) ).
% xor_num_eq_Some_iff
thf(fact_6444_xor__num_Osimps_I7_J,axiom,
! [M: num] :
( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one2 )
= ( some @ num @ ( bit0 @ M ) ) ) ).
% xor_num.simps(7)
thf(fact_6445_xor__num_Osimps_I4_J,axiom,
! [M: num] :
( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one2 )
= ( some @ num @ ( bit1 @ M ) ) ) ).
% xor_num.simps(4)
thf(fact_6446_xor__num_Osimps_I3_J,axiom,
! [N: num] :
( ( bit_un2480387367778600638or_num @ one2 @ ( bit1 @ N ) )
= ( some @ num @ ( bit0 @ N ) ) ) ).
% xor_num.simps(3)
thf(fact_6447_xor__num_Osimps_I2_J,axiom,
! [N: num] :
( ( bit_un2480387367778600638or_num @ one2 @ ( bit0 @ N ) )
= ( some @ num @ ( bit1 @ N ) ) ) ).
% xor_num.simps(2)
thf(fact_6448_xor__num__eq__None__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: num] :
( ( ( bit_un2480387367778600638or_num @ M @ N )
= ( none @ num ) )
= ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% xor_num_eq_None_iff
thf(fact_6449_numeral__xor__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% numeral_xor_num
thf(fact_6450_xor__num_Osimps_I8_J,axiom,
! [M: num,N: num] :
( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% xor_num.simps(8)
thf(fact_6451_and__num_Oelims,axiom,
! [X3: num,Xa: num,Y3: option @ num] :
( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa )
= Y3 )
=> ( ( ( X3 = one2 )
=> ( ( Xa = one2 )
=> ( Y3
!= ( some @ num @ one2 ) ) ) )
=> ( ( ( X3 = one2 )
=> ( ? [N2: num] :
( Xa
= ( bit0 @ N2 ) )
=> ( Y3
!= ( none @ num ) ) ) )
=> ( ( ( X3 = one2 )
=> ( ? [N2: num] :
( Xa
= ( bit1 @ N2 ) )
=> ( Y3
!= ( some @ num @ one2 ) ) ) )
=> ( ( ? [M4: num] :
( X3
= ( bit0 @ M4 ) )
=> ( ( Xa = one2 )
=> ( Y3
!= ( none @ num ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y3
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y3
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) )
=> ( ( ? [M4: num] :
( X3
= ( bit1 @ M4 ) )
=> ( ( Xa = one2 )
=> ( Y3
!= ( some @ num @ one2 ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y3
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) )
=> ~ ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y3
!= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N10: num] : ( some @ num @ ( bit1 @ N10 ) )
@ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.elims
thf(fact_6452_xor__num__dict,axiom,
bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).
% xor_num_dict
thf(fact_6453_and__num_Osimps_I1_J,axiom,
( ( bit_un7362597486090784418nd_num @ one2 @ one2 )
= ( some @ num @ one2 ) ) ).
% and_num.simps(1)
thf(fact_6454_and__num_Osimps_I5_J,axiom,
! [M: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% and_num.simps(5)
thf(fact_6455_and__num_Osimps_I7_J,axiom,
! [M: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one2 )
= ( some @ num @ one2 ) ) ).
% and_num.simps(7)
thf(fact_6456_and__num_Osimps_I3_J,axiom,
! [N: num] :
( ( bit_un7362597486090784418nd_num @ one2 @ ( bit1 @ N ) )
= ( some @ num @ one2 ) ) ).
% and_num.simps(3)
thf(fact_6457_and__num_Osimps_I2_J,axiom,
! [N: num] :
( ( bit_un7362597486090784418nd_num @ one2 @ ( bit0 @ N ) )
= ( none @ num ) ) ).
% and_num.simps(2)
thf(fact_6458_and__num_Osimps_I4_J,axiom,
! [M: num] :
( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one2 )
= ( none @ num ) ) ).
% and_num.simps(4)
thf(fact_6459_and__num__eq__Some__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: num,Q3: num] :
( ( ( bit_un7362597486090784418nd_num @ M @ N )
= ( some @ num @ Q3 ) )
= ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ Q3 ) ) ) ) ).
% and_num_eq_Some_iff
thf(fact_6460_and__num_Osimps_I8_J,axiom,
! [M: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% and_num.simps(8)
thf(fact_6461_and__num_Osimps_I6_J,axiom,
! [M: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% and_num.simps(6)
thf(fact_6462_and__num__eq__None__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: num] :
( ( ( bit_un7362597486090784418nd_num @ M @ N )
= ( none @ num ) )
= ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% and_num_eq_None_iff
thf(fact_6463_numeral__and__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ) ).
% numeral_and_num
thf(fact_6464_and__num_Osimps_I9_J,axiom,
! [M: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N10: num] : ( some @ num @ ( bit1 @ N10 ) )
@ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% and_num.simps(9)
thf(fact_6465_and__num__dict,axiom,
bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).
% and_num_dict
thf(fact_6466_option_Orec__o__map,axiom,
! [B: $tType,C: $tType,A: $tType,G: C,Ga: B > C,F2: A > B] :
( ( comp @ ( option @ B ) @ C @ ( option @ A ) @ ( rec_option @ C @ B @ G @ Ga ) @ ( map_option @ A @ B @ F2 ) )
= ( rec_option @ C @ A @ G
@ ^ [X: A] : ( Ga @ ( F2 @ X ) ) ) ) ).
% option.rec_o_map
thf(fact_6467_option_Osimps_I7_J,axiom,
! [C: $tType,A: $tType,F1: C,F22: A > C,X2: A] :
( ( rec_option @ C @ A @ F1 @ F22 @ ( some @ A @ X2 ) )
= ( F22 @ X2 ) ) ).
% option.simps(7)
thf(fact_6468_option_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,F1: C,F22: A > C] :
( ( rec_option @ C @ A @ F1 @ F22 @ ( none @ A ) )
= F1 ) ).
% option.simps(6)
thf(fact_6469_set__nths,axiom,
! [A: $tType,Xs2: list @ A,I6: set @ nat] :
( ( set2 @ A @ ( nths @ A @ Xs2 @ I6 ) )
= ( collect @ A
@ ^ [Uu3: A] :
? [I4: nat] :
( ( Uu3
= ( nth @ A @ Xs2 @ I4 ) )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( member @ nat @ I4 @ I6 ) ) ) ) ).
% set_nths
thf(fact_6470_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( ^ [A9: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X: A,Y: option @ A] : ( some @ A @ ( case_option @ A @ A @ X @ ( ord_max @ A @ X ) @ Y ) )
@ ( none @ A )
@ A9 ) ) ) ) ) ).
% Max.eq_fold'
thf(fact_6471_nths__all,axiom,
! [A: $tType,Xs2: list @ A,I6: set @ nat] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( member @ nat @ I3 @ I6 ) )
=> ( ( nths @ A @ Xs2 @ I6 )
= Xs2 ) ) ).
% nths_all
thf(fact_6472_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_6473_Max_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X3 @ A4 ) )
= ( finite_fold @ A @ A @ ( ord_max @ A ) @ X3 @ A4 ) ) ) ) ).
% Max.eq_fold
thf(fact_6474_length__nths,axiom,
! [A: $tType,Xs2: list @ A,I6: set @ nat] :
( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs2 @ I6 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( member @ nat @ I4 @ I6 ) ) ) ) ) ).
% length_nths
thf(fact_6475_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( ^ [A9: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X: A,Y: option @ A] : ( some @ A @ ( case_option @ A @ A @ X @ ( sup_sup @ A @ X ) @ Y ) )
@ ( none @ A )
@ A9 ) ) ) ) ) ).
% Sup_fin.eq_fold'
thf(fact_6476_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( ^ [A9: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X: A,Y: option @ A] : ( some @ A @ ( case_option @ A @ A @ X @ ( inf_inf @ A @ X ) @ Y ) )
@ ( none @ A )
@ A9 ) ) ) ) ) ).
% Inf_fin.eq_fold'
thf(fact_6477_Sup__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A] :
( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% Sup_fin.singleton
thf(fact_6478_Inf__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A] :
( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% Inf_fin.singleton
thf(fact_6479_inf__Sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A,A2: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ A2 @ A4 )
=> ( ( inf_inf @ A @ A2 @ ( lattic5882676163264333800up_fin @ A @ A4 ) )
= A2 ) ) ) ) ).
% inf_Sup_absorb
thf(fact_6480_sup__Inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A,A2: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ A2 @ A4 )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ A2 )
= A2 ) ) ) ) ).
% sup_Inf_absorb
thf(fact_6481_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X3 @ A4 ) )
= ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_6482_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X3 @ A4 ) )
= ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_6483_Sup__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ A4 ) )
= ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_6484_Inf__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,A2: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ A2 @ A4 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ A2 ) ) ) ) ).
% Inf_fin.coboundedI
thf(fact_6485_Sup__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,A2: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ A2 @ A4 )
=> ( ord_less_eq @ A @ A2 @ ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_6486_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_6487_Sup__fin__Max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% Sup_fin_Max
thf(fact_6488_Inf__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
= ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_6489_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ X3 )
= ( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_6490_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_6491_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ( ord_less_eq @ A @ A6 @ X3 ) )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ X3 ) ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_6492_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ X3 )
=> ! [A8: A] :
( ( member @ A @ A8 @ A4 )
=> ( ord_less_eq @ A @ A8 @ X3 ) ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_6493_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ( ord_less_eq @ A @ X3 @ A6 ) )
=> ( ord_less_eq @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_6494_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
=> ! [A8: A] :
( ( member @ A @ A8 @ A4 )
=> ( ord_less_eq @ A @ X3 @ A8 ) ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_6495_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu3: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_6496_Sup__fin__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% Sup_fin_Sup
thf(fact_6497_cSup__eq__Sup__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A] :
( ( finite_finite @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= ( lattic5882676163264333800up_fin @ A @ X8 ) ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_6498_Inf__fin__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= ( complete_Inf_Inf @ A @ A4 ) ) ) ) ) ).
% Inf_fin_Inf
thf(fact_6499_cInf__eq__Inf__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A] :
( ( finite_finite @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= ( lattic7752659483105999362nf_fin @ A @ X8 ) ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_6500_Sup__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite @ A @ A4 )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Sup_fin.infinite
thf(fact_6501_Inf__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite @ A @ A4 )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Inf_fin.infinite
thf(fact_6502_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B6 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B6 ) @ ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_6503_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B6 )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ ( lattic5882676163264333800up_fin @ A @ B6 ) ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_6504_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [H: A > A,N4: set @ A] :
( ! [X4: A,Y4: A] :
( ( H @ ( inf_inf @ A @ X4 @ Y4 ) )
= ( inf_inf @ A @ ( H @ X4 ) @ ( H @ Y4 ) ) )
=> ( ( finite_finite @ A @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic7752659483105999362nf_fin @ A @ N4 ) )
= ( lattic7752659483105999362nf_fin @ A @ ( image @ A @ A @ H @ N4 ) ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_6505_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [H: A > A,N4: set @ A] :
( ! [X4: A,Y4: A] :
( ( H @ ( sup_sup @ A @ X4 @ Y4 ) )
= ( sup_sup @ A @ ( H @ X4 ) @ ( H @ Y4 ) ) )
=> ( ( finite_finite @ A @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic5882676163264333800up_fin @ A @ N4 ) )
= ( lattic5882676163264333800up_fin @ A @ ( image @ A @ A @ H @ N4 ) ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_6506_Inf__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
=> ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B6 ) @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
= ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_6507_Sup__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
=> ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B6 ) @ ( lattic5882676163264333800up_fin @ A @ A4 ) )
= ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_6508_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ~ ( member @ A @ X3 @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X3 @ A4 ) )
= ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_6509_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y4: A] : ( member @ A @ ( inf_inf @ A @ X4 @ Y4 ) @ ( insert @ A @ X4 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ A4 ) ) ) ) ) ).
% Inf_fin.closed
thf(fact_6510_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ~ ( member @ A @ X3 @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X3 @ A4 ) )
= ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_6511_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y4: A] : ( member @ A @ ( sup_sup @ A @ X4 @ Y4 ) @ ( insert @ A @ X4 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ A4 ) ) ) ) ) ).
% Sup_fin.closed
thf(fact_6512_Inf__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B6 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
= ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ ( lattic7752659483105999362nf_fin @ A @ B6 ) ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_6513_Sup__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B6 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
= ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ ( lattic5882676163264333800up_fin @ A @ B6 ) ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_6514_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X3 @ A4 ) )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ X3 @ A4 ) ) ) ) ).
% Inf_fin.eq_fold
thf(fact_6515_Sup__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X3 @ A4 ) )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ X3 @ A4 ) ) ) ) ).
% Sup_fin.eq_fold
thf(fact_6516_inf__Sup2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B6 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ ( lattic5882676163264333800up_fin @ A @ B6 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A5: A,B4: A] :
( ( Uu3
= ( inf_inf @ A @ A5 @ B4 ) )
& ( member @ A @ A5 @ A4 )
& ( member @ A @ B4 @ B6 ) ) ) ) ) ) ) ) ) ) ).
% inf_Sup2_distrib
thf(fact_6517_inf__Sup1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ A4 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A5: A] :
( ( Uu3
= ( inf_inf @ A @ X3 @ A5 ) )
& ( member @ A @ A5 @ A4 ) ) ) ) ) ) ) ) ).
% inf_Sup1_distrib
thf(fact_6518_sup__Inf2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B6 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ ( lattic7752659483105999362nf_fin @ A @ B6 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A5: A,B4: A] :
( ( Uu3
= ( sup_sup @ A @ A5 @ B4 ) )
& ( member @ A @ A5 @ A4 )
& ( member @ A @ B4 @ B6 ) ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_6519_sup__Inf1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A5: A] :
( ( Uu3
= ( sup_sup @ A @ X3 @ A5 ) )
& ( member @ A @ A5 @ A4 ) ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_6520_Inf__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_6521_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X3 @ A4 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X3 @ A4 ) )
= ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_6522_Sup__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_6523_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X3 @ A4 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X3 @ A4 ) )
= ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_6524_ring__1__class_Oof__int__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ A @ A @ rep_Integ @ ( id @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I4: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ) ).
% ring_1_class.of_int_def
thf(fact_6525_dual__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( min @ A
@ ^ [X: A,Y: A] : ( ord_less_eq @ A @ Y @ X ) )
= ( ord_max @ A ) ) ) ).
% dual_min
thf(fact_6526_of__nat__eq__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( id @ nat ) ) ).
% of_nat_eq_id
thf(fact_6527_id__funpow,axiom,
! [A: $tType,N: nat] :
( ( compow @ ( A > A ) @ N @ ( id @ A ) )
= ( id @ A ) ) ).
% id_funpow
thf(fact_6528_push__bit__0__id,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A @ ( zero_zero @ nat ) )
= ( id @ A ) ) ) ).
% push_bit_0_id
thf(fact_6529_drop__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A @ ( zero_zero @ nat ) )
= ( id @ A ) ) ) ).
% drop_bit_0
thf(fact_6530_comp__the__Some,axiom,
! [A: $tType] :
( ( comp @ ( option @ A ) @ A @ A @ ( the2 @ A ) @ ( some @ A ) )
= ( id @ A ) ) ).
% comp_the_Some
thf(fact_6531_ord_Omin_Ocong,axiom,
! [A: $tType] :
( ( min @ A )
= ( min @ A ) ) ).
% ord.min.cong
thf(fact_6532_ord_Omin__def,axiom,
! [A: $tType] :
( ( min @ A )
= ( ^ [Less_eq: A > A > $o,A5: A,B4: A] : ( if @ A @ ( Less_eq @ A5 @ B4 ) @ A5 @ B4 ) ) ) ).
% ord.min_def
thf(fact_6533_funpow__simps__right_I1_J,axiom,
! [A: $tType,F2: A > A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F2 )
= ( id @ A ) ) ).
% funpow_simps_right(1)
thf(fact_6534_less__eq__int__def,axiom,
( ( ord_less_eq @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_6535_less__int__def,axiom,
( ( ord_less @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y ) ) ) ) ) ) ).
% less_int_def
thf(fact_6536_option_Omap__id0,axiom,
! [A: $tType] :
( ( map_option @ A @ A @ ( id @ A ) )
= ( id @ ( option @ A ) ) ) ).
% option.map_id0
thf(fact_6537_option_Omap__id,axiom,
! [A: $tType,T2: option @ A] :
( ( map_option @ A @ A @ ( id @ A ) @ T2 )
= T2 ) ).
% option.map_id
thf(fact_6538_map__option_Oidentity,axiom,
! [A: $tType] :
( ( map_option @ A @ A
@ ^ [X: A] : X )
= ( id @ ( option @ A ) ) ) ).
% map_option.identity
thf(fact_6539_nat__def,axiom,
( nat2
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ nat @ nat @ rep_Integ @ ( id @ nat ) @ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ) ).
% nat_def
thf(fact_6540_nth__image,axiom,
! [A: $tType,L: nat,Xs2: list @ A] :
( ( ord_less_eq @ nat @ L @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( image @ nat @ A @ ( nth @ A @ Xs2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ L ) )
= ( set2 @ A @ ( take @ A @ L @ Xs2 ) ) ) ) ).
% nth_image
thf(fact_6541_card__Min__le__sum,axiom,
! [A: $tType,A4: set @ A,F2: A > nat] :
( ( finite_finite @ A @ A4 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( finite_card @ A @ A4 ) @ ( lattic643756798350308766er_Min @ nat @ ( image @ A @ nat @ F2 @ A4 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) ) ) ).
% card_Min_le_sum
thf(fact_6542_take__all__iff,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( take @ A @ N @ Xs2 )
= Xs2 )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ).
% take_all_iff
thf(fact_6543_take__all,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N )
=> ( ( take @ A @ N @ Xs2 )
= Xs2 ) ) ).
% take_all
thf(fact_6544_nth__take,axiom,
! [A: $tType,I: nat,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( take @ A @ N @ Xs2 ) @ I )
= ( nth @ A @ Xs2 @ I ) ) ) ).
% nth_take
thf(fact_6545_Min__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A] :
( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% Min_singleton
thf(fact_6546_take__update__cancel,axiom,
! [A: $tType,N: nat,M: nat,Xs2: list @ A,Y3: A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( take @ A @ N @ ( list_update @ A @ Xs2 @ M @ Y3 ) )
= ( take @ A @ N @ Xs2 ) ) ) ).
% take_update_cancel
thf(fact_6547_Min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_6548_Min__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( ord_less @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_6549_Min__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ B,C2: A] :
( ( finite_finite @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image @ B @ A
@ ^ [Uu3: B] : C2
@ A4 ) )
= C2 ) ) ) ) ).
% Min_const
thf(fact_6550_minus__Max__eq__Min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S2: set @ A] :
( ( finite_finite @ 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_6551_minus__Min__eq__Max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S2: set @ A] :
( ( finite_finite @ 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_6552_Min_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,A2: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ A2 @ A4 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ A2 ) ) ) ) ).
% Min.coboundedI
thf(fact_6553_Min__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ A4 )
=> ( ord_less_eq @ A @ X3 @ Y4 ) )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= X3 ) ) ) ) ) ).
% Min_eqI
thf(fact_6554_Min__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ X3 ) ) ) ) ).
% Min_le
thf(fact_6555_Min__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ A4 ) ) ) ) ).
% Min_in
thf(fact_6556_Inf__fin__Min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% Inf_fin_Min
thf(fact_6557_set__take__subset__set__take,axiom,
! [A: $tType,M: nat,N: nat,Xs2: list @ A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ M @ Xs2 ) ) @ ( set2 @ A @ ( take @ A @ N @ Xs2 ) ) ) ) ).
% set_take_subset_set_take
thf(fact_6558_Min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ( ord_less_eq @ A @ X3 @ A6 ) )
=> ( ord_less_eq @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ).
% Min.boundedI
thf(fact_6559_Min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A4 ) )
=> ! [A8: A] :
( ( member @ A @ A8 @ A4 )
=> ( ord_less_eq @ A @ X3 @ A8 ) ) ) ) ) ) ).
% Min.boundedE
thf(fact_6560_eq__Min__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,M: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( ( member @ A @ M @ A4 )
& ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ M @ X ) ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_6561_Min__le__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ X3 )
= ( ? [X: A] :
( ( member @ A @ X @ A4 )
& ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_6562_Min__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,M: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798350308766er_Min @ A @ A4 )
= M )
= ( ( member @ A @ M @ A4 )
& ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ M @ X ) ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_6563_Min__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ X3 )
= ( ? [X: A] :
( ( member @ A @ X @ A4 )
& ( ord_less @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_6564_Min__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,A2: A] :
( ( finite_finite @ A @ A4 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ A4 )
=> ( ord_less_eq @ A @ A2 @ B5 ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ A2 @ A4 ) )
= A2 ) ) ) ) ).
% Min_insert2
thf(fact_6565_Min__Inf,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= ( complete_Inf_Inf @ A @ A4 ) ) ) ) ) ).
% Min_Inf
thf(fact_6566_cInf__eq__Min,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A] :
( ( finite_finite @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= ( lattic643756798350308766er_Min @ A @ X8 ) ) ) ) ) ).
% cInf_eq_Min
thf(fact_6567_Min_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite @ A @ A4 )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Min.infinite
thf(fact_6568_nth__take__lemma,axiom,
! [A: $tType,K: nat,Xs2: list @ A,Ys: list @ A] :
( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ( ( nth @ A @ Xs2 @ I3 )
= ( nth @ A @ Ys @ I3 ) ) )
=> ( ( take @ A @ K @ Xs2 )
= ( take @ A @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_6569_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_finite @ A @ N4 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N4 ) @ ( lattic643756798350308766er_Min @ A @ M7 ) ) ) ) ) ) ).
% Min_antimono
thf(fact_6570_Min_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B6 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B6 ) @ ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ).
% Min.subset_imp
thf(fact_6571_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F2 @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( lattic643756798350308766er_Min @ B @ ( image @ A @ B @ F2 @ A4 ) ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_6572_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S2: set @ B,F2: B > A,K: A] :
( ( finite_finite @ B @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image @ B @ A
@ ^ [X: B] : ( plus_plus @ A @ ( F2 @ X ) @ K )
@ S2 ) )
= ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image @ B @ A @ F2 @ S2 ) ) @ K ) ) ) ) ) ).
% Min_add_commute
thf(fact_6573_arg__min__SOME__Min,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S2: set @ A,F2: A > B] :
( ( finite_finite @ A @ S2 )
=> ( ( lattic7623131987881927897min_on @ A @ B @ F2 @ S2 )
= ( fChoice @ A
@ ^ [Y: A] :
( ( member @ A @ Y @ S2 )
& ( ( F2 @ Y )
= ( lattic643756798350308766er_Min @ B @ ( image @ A @ B @ F2 @ S2 ) ) ) ) ) ) ) ) ).
% arg_min_SOME_Min
thf(fact_6574_take__bit__horner__sum__bit__eq,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,Bs: list @ $o] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs ) )
= ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( take @ $o @ N @ Bs ) ) ) ) ).
% take_bit_horner_sum_bit_eq
thf(fact_6575_dual__Max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Max @ A
@ ^ [X: A,Y: A] : ( ord_less_eq @ A @ Y @ X ) )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% dual_Max
thf(fact_6576_lex__take__index,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( lex @ A @ R2 ) )
=> ~ ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( take @ A @ I3 @ Xs2 )
= ( take @ A @ I3 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs2 @ I3 ) @ ( nth @ A @ Ys @ I3 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_6577_linorder_OMax_Ocong,axiom,
! [A: $tType] :
( ( lattices_Max @ A )
= ( lattices_Max @ A ) ) ).
% linorder.Max.cong
thf(fact_6578_Min_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798350308766er_Min @ A )
= ( ^ [A9: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X: A,Y: option @ A] : ( some @ A @ ( case_option @ A @ A @ X @ ( ord_min @ A @ X ) @ Y ) )
@ ( none @ A )
@ A9 ) ) ) ) ) ).
% Min.eq_fold'
thf(fact_6579_listrel1__iff__update,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( listrel1 @ A @ R2 ) )
= ( ? [Y: A,N3: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs2 @ N3 ) @ Y ) @ R2 )
& ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( Ys
= ( list_update @ A @ Xs2 @ N3 @ Y ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_6580_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min @ nat @ M @ N ) ) ) ).
% min_Suc_Suc
thf(fact_6581_min__0L,axiom,
! [N: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_6582_min__0R,axiom,
! [N: nat] :
( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_6583_take__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( ord_min @ nat @ M @ N ) @ A2 ) ) ) ).
% take_bit_take_bit
thf(fact_6584_signed__take__bit__signed__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ M @ ( bit_ri4674362597316999326ke_bit @ A @ N @ A2 ) )
= ( bit_ri4674362597316999326ke_bit @ A @ ( ord_min @ nat @ M @ N ) @ A2 ) ) ) ).
% signed_take_bit_signed_take_bit
thf(fact_6585_min__enat__simps_I3_J,axiom,
! [Q3: extended_enat] :
( ( ord_min @ extended_enat @ ( zero_zero @ extended_enat ) @ Q3 )
= ( zero_zero @ extended_enat ) ) ).
% min_enat_simps(3)
thf(fact_6586_min__enat__simps_I2_J,axiom,
! [Q3: extended_enat] :
( ( ord_min @ extended_enat @ Q3 @ ( zero_zero @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) ).
% min_enat_simps(2)
thf(fact_6587_min_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= A2 ) ) ) ).
% min.absorb1
thf(fact_6588_min_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= B2 ) ) ) ).
% min.absorb2
thf(fact_6589_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% min.bounded_iff
thf(fact_6590_min__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X3: A,Y3: A] :
( ( ord_less @ A @ Z @ ( ord_min @ A @ X3 @ Y3 ) )
= ( ( ord_less @ A @ Z @ X3 )
& ( ord_less @ A @ Z @ Y3 ) ) ) ) ).
% min_less_iff_conj
thf(fact_6591_min_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= B2 ) ) ) ).
% min.absorb4
thf(fact_6592_min_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= A2 ) ) ) ).
% min.absorb3
thf(fact_6593_min__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X3: A] :
( ( ord_min @ A @ X3 @ ( top_top @ A ) )
= X3 ) ) ).
% min_top2
thf(fact_6594_min__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X3: A] :
( ( ord_min @ A @ ( top_top @ A ) @ X3 )
= X3 ) ) ).
% min_top
thf(fact_6595_min__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ord_min @ A @ X3 @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% min_bot2
thf(fact_6596_min__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ord_min @ A @ ( bot_bot @ A ) @ X3 )
= ( bot_bot @ A ) ) ) ).
% min_bot
thf(fact_6597_max__min__same_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y3: A,X3: A] :
( ( ord_max @ A @ Y3 @ ( ord_min @ A @ X3 @ Y3 ) )
= Y3 ) ) ).
% max_min_same(4)
thf(fact_6598_max__min__same_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_max @ A @ ( ord_min @ A @ X3 @ Y3 ) @ Y3 )
= Y3 ) ) ).
% max_min_same(3)
thf(fact_6599_max__min__same_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_max @ A @ ( ord_min @ A @ X3 @ Y3 ) @ X3 )
= X3 ) ) ).
% max_min_same(2)
thf(fact_6600_max__min__same_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_max @ A @ X3 @ ( ord_min @ A @ X3 @ Y3 ) )
= X3 ) ) ).
% max_min_same(1)
thf(fact_6601_take__bit__of__mask,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se2239418461657761734s_mask @ A @ N ) )
= ( bit_se2239418461657761734s_mask @ A @ ( ord_min @ nat @ M @ N ) ) ) ) ).
% take_bit_of_mask
thf(fact_6602_min__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% min_number_of(1)
thf(fact_6603_min__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X3 ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(4)
thf(fact_6604_min__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_min @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X3 ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(3)
thf(fact_6605_min__0__1_I1_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_min @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(1)
thf(fact_6606_min__0__1_I2_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_min @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(2)
thf(fact_6607_min__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X3 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% min_0_1(6)
thf(fact_6608_min__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: num] :
( ( ord_min @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X3 ) )
= ( one_one @ A ) ) ) ).
% min_0_1(5)
thf(fact_6609_min__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_min @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_min @ nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% min_numeral_Suc
thf(fact_6610_min__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_min @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_min @ nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% min_Suc_numeral
thf(fact_6611_min__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ) ).
% min_number_of(2)
thf(fact_6612_min__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% min_number_of(3)
thf(fact_6613_min__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ) ).
% min_number_of(4)
thf(fact_6614_Min__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X3 @ A4 ) )
= ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ).
% Min_insert
thf(fact_6615_Min_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ).
% Min.in_idem
thf(fact_6616_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B4 ) @ A5 @ B4 ) ) ) ) ).
% min_def_raw
thf(fact_6617_min__absorb2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_min @ A @ X3 @ Y3 )
= Y3 ) ) ) ).
% min_absorb2
thf(fact_6618_min__absorb1,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_min @ A @ X3 @ Y3 )
= X3 ) ) ) ).
% min_absorb1
thf(fact_6619_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B4 ) @ A5 @ B4 ) ) ) ) ).
% min_def
thf(fact_6620_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ ( ord_min @ A @ C2 @ D3 ) ) ) ) ) ).
% min.mono
thf(fact_6621_min_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2
= ( ord_min @ A @ A2 @ B2 ) ) ) ) ).
% min.orderE
thf(fact_6622_min_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( ord_min @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% min.orderI
thf(fact_6623_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% min.boundedE
thf(fact_6624_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ) ).
% min.boundedI
thf(fact_6625_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( A5
= ( ord_min @ A @ A5 @ B4 ) ) ) ) ) ).
% min.order_iff
thf(fact_6626_min_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ A2 ) ) ).
% min.cobounded1
thf(fact_6627_min_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ B2 ) ) ).
% min.cobounded2
thf(fact_6628_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_min @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% min.absorb_iff1
thf(fact_6629_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_min @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% min.absorb_iff2
thf(fact_6630_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% min.coboundedI1
thf(fact_6631_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% min.coboundedI2
thf(fact_6632_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X3 @ Y3 ) @ Z )
= ( ( ord_less_eq @ A @ X3 @ Z )
| ( ord_less_eq @ A @ Y3 @ Z ) ) ) ) ).
% min_le_iff_disj
thf(fact_6633_concat__bit__assoc__sym,axiom,
! [M: nat,N: nat,K: int,L: int,R2: int] :
( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L ) @ R2 )
= ( bit_concat_bit @ ( ord_min @ nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus @ nat @ M @ N ) @ L @ R2 ) ) ) ).
% concat_bit_assoc_sym
thf(fact_6634_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% min.strict_coboundedI2
thf(fact_6635_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ A2 @ C2 )
=> ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% min.strict_coboundedI1
thf(fact_6636_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( A5
= ( ord_min @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% min.strict_order_iff
thf(fact_6637_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% min.strict_boundedE
thf(fact_6638_min__less__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( ord_less @ A @ ( ord_min @ A @ X3 @ Y3 ) @ Z )
= ( ( ord_less @ A @ X3 @ Z )
| ( ord_less @ A @ Y3 @ Z ) ) ) ) ).
% min_less_iff_disj
thf(fact_6639_minus__max__eq__min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X3: A,Y3: A] :
( ( uminus_uminus @ A @ ( ord_max @ A @ X3 @ Y3 ) )
= ( ord_min @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% minus_max_eq_min
thf(fact_6640_minus__min__eq__max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X3: A,Y3: A] :
( ( uminus_uminus @ A @ ( ord_min @ A @ X3 @ Y3 ) )
= ( ord_max @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% minus_min_eq_max
thf(fact_6641_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X3 @ Y3 ) @ Z )
= ( ord_min @ A @ ( minus_minus @ A @ X3 @ Z ) @ ( minus_minus @ A @ Y3 @ Z ) ) ) ) ).
% min_diff_distrib_left
thf(fact_6642_min__diff,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_min @ nat @ ( minus_minus @ nat @ M @ I ) @ ( minus_minus @ nat @ N @ I ) )
= ( minus_minus @ nat @ ( ord_min @ nat @ M @ N ) @ I ) ) ).
% min_diff
thf(fact_6643_nat__mult__min__right,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( times_times @ nat @ M @ ( ord_min @ nat @ N @ Q3 ) )
= ( ord_min @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q3 ) ) ) ).
% nat_mult_min_right
thf(fact_6644_nat__mult__min__left,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( times_times @ nat @ ( ord_min @ nat @ M @ N ) @ Q3 )
= ( ord_min @ nat @ ( times_times @ nat @ M @ Q3 ) @ ( times_times @ nat @ N @ Q3 ) ) ) ).
% nat_mult_min_left
thf(fact_6645_of__int__min,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: int,Y3: int] :
( ( ring_1_of_int @ A @ ( ord_min @ int @ X3 @ Y3 ) )
= ( ord_min @ A @ ( ring_1_of_int @ A @ X3 ) @ ( ring_1_of_int @ A @ Y3 ) ) ) ) ).
% of_int_min
thf(fact_6646_of__nat__min,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X3: nat,Y3: nat] :
( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X3 @ Y3 ) )
= ( ord_min @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( semiring_1_of_nat @ A @ Y3 ) ) ) ) ).
% of_nat_min
thf(fact_6647_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( plus_plus @ A @ X3 @ ( ord_min @ A @ Y3 @ Z ) )
= ( ord_min @ A @ ( plus_plus @ A @ X3 @ Y3 ) @ ( plus_plus @ A @ X3 @ Z ) ) ) ) ).
% min_add_distrib_right
thf(fact_6648_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X3 @ Y3 ) @ Z )
= ( ord_min @ A @ ( plus_plus @ A @ X3 @ Z ) @ ( plus_plus @ A @ Y3 @ Z ) ) ) ) ).
% min_add_distrib_left
thf(fact_6649_take__bit__concat__bit__eq,axiom,
! [M: nat,N: nat,K: int,L: int] :
( ( bit_se2584673776208193580ke_bit @ int @ M @ ( bit_concat_bit @ N @ K @ L ) )
= ( bit_concat_bit @ ( ord_min @ nat @ M @ N ) @ K @ ( bit_se2584673776208193580ke_bit @ int @ ( minus_minus @ nat @ M @ N ) @ L ) ) ) ).
% take_bit_concat_bit_eq
thf(fact_6650_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P6: A,X3: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ ( ord_min @ A @ X3 @ Y3 ) @ P6 )
= ( ord_min @ A @ ( times_times @ A @ X3 @ P6 ) @ ( times_times @ A @ Y3 @ P6 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ ( ord_min @ A @ X3 @ Y3 ) @ P6 )
= ( ord_max @ A @ ( times_times @ A @ X3 @ P6 ) @ ( times_times @ A @ Y3 @ P6 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_6651_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P6: A,X3: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ ( ord_max @ A @ X3 @ Y3 ) @ P6 )
= ( ord_max @ A @ ( times_times @ A @ X3 @ P6 ) @ ( times_times @ A @ Y3 @ P6 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ ( ord_max @ A @ X3 @ Y3 ) @ P6 )
= ( ord_min @ A @ ( times_times @ A @ X3 @ P6 ) @ ( times_times @ A @ Y3 @ P6 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_6652_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P6: A,X3: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ P6 @ ( ord_min @ A @ X3 @ Y3 ) )
= ( ord_min @ A @ ( times_times @ A @ P6 @ X3 ) @ ( times_times @ A @ P6 @ Y3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ P6 @ ( ord_min @ A @ X3 @ Y3 ) )
= ( ord_max @ A @ ( times_times @ A @ P6 @ X3 ) @ ( times_times @ A @ P6 @ Y3 ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_6653_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P6: A,X3: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ P6 @ ( ord_max @ A @ X3 @ Y3 ) )
= ( ord_max @ A @ ( times_times @ A @ P6 @ X3 ) @ ( times_times @ A @ P6 @ Y3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( times_times @ A @ P6 @ ( ord_max @ A @ X3 @ Y3 ) )
= ( ord_min @ A @ ( times_times @ A @ P6 @ X3 ) @ ( times_times @ A @ P6 @ Y3 ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_6654_max__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P6: A,X3: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X3 @ Y3 ) @ P6 )
= ( ord_max @ A @ ( divide_divide @ A @ X3 @ P6 ) @ ( divide_divide @ A @ Y3 @ P6 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X3 @ Y3 ) @ P6 )
= ( ord_min @ A @ ( divide_divide @ A @ X3 @ P6 ) @ ( divide_divide @ A @ Y3 @ P6 ) ) ) ) ) ) ).
% max_divide_distrib_right
thf(fact_6655_min__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P6: A,X3: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X3 @ Y3 ) @ P6 )
= ( ord_min @ A @ ( divide_divide @ A @ X3 @ P6 ) @ ( divide_divide @ A @ Y3 @ P6 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P6 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X3 @ Y3 ) @ P6 )
= ( ord_max @ A @ ( divide_divide @ A @ X3 @ P6 ) @ ( divide_divide @ A @ Y3 @ P6 ) ) ) ) ) ) ).
% min_divide_distrib_right
thf(fact_6656_min__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_min @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M5: nat] : ( suc @ ( ord_min @ nat @ M5 @ N ) )
@ M ) ) ).
% min_Suc2
thf(fact_6657_min__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_min @ nat @ ( suc @ N ) @ M )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M5: nat] : ( suc @ ( ord_min @ nat @ N @ M5 ) )
@ M ) ) ).
% min_Suc1
thf(fact_6658_Inf__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S2: set @ A,X3: A] :
( ( finite_finite @ A @ S2 )
=> ( ( ( S2
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ X3 @ S2 ) )
= X3 ) )
& ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ X3 @ S2 ) )
= ( ord_min @ A @ X3 @ ( complete_Inf_Inf @ A @ S2 ) ) ) ) ) ) ) ).
% Inf_insert_finite
thf(fact_6659_hom__Min__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H: A > A,N4: set @ A] :
( ! [X4: A,Y4: A] :
( ( H @ ( ord_min @ A @ X4 @ Y4 ) )
= ( ord_min @ A @ ( H @ X4 ) @ ( H @ Y4 ) ) )
=> ( ( finite_finite @ A @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic643756798350308766er_Min @ A @ N4 ) )
= ( lattic643756798350308766er_Min @ A @ ( image @ A @ A @ H @ N4 ) ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_6660_Min_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
=> ( ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ B6 ) @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ).
% Min.subset
thf(fact_6661_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ~ ( member @ A @ X3 @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X3 @ A4 ) )
= ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_6662_Min_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y4: A] : ( member @ A @ ( ord_min @ A @ X4 @ Y4 ) @ ( insert @ A @ X4 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ A4 ) ) ) ) ) ).
% Min.closed
thf(fact_6663_Min_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B6 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
= ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ ( lattic643756798350308766er_Min @ A @ B6 ) ) ) ) ) ) ) ) ).
% Min.union
thf(fact_6664_Min_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X3 @ A4 ) )
= ( finite_fold @ A @ A @ ( ord_min @ A ) @ X3 @ A4 ) ) ) ) ).
% Min.eq_fold
thf(fact_6665_mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N ) ) ) ) ) ).
% mod_exp_eq
thf(fact_6666_Min_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X3 @ A4 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X3 @ A4 ) )
= ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_6667_Min_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X3: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_6668_mask__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat,M: nat] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N ) ) @ ( one_one @ A ) ) ) ) ).
% mask_mod_exp
thf(fact_6669_lexord__take__index__conv,axiom,
! [A: $tType,X3: list @ A,Y3: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y3 ) @ ( lexord @ A @ R2 ) )
= ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X3 ) @ ( size_size @ ( list @ A ) @ Y3 ) )
& ( ( take @ A @ ( size_size @ ( list @ A ) @ X3 ) @ Y3 )
= X3 ) )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X3 ) @ ( size_size @ ( list @ A ) @ Y3 ) ) )
& ( ( take @ A @ I4 @ X3 )
= ( take @ A @ I4 @ Y3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X3 @ I4 ) @ ( nth @ A @ Y3 @ I4 ) ) @ R2 ) ) ) ) ).
% lexord_take_index_conv
thf(fact_6670_lenlex__conv,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs: list @ A,Ys3: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys3 ) )
| ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( lex @ A @ R5 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_6671_inf__int__def,axiom,
( ( inf_inf @ int )
= ( ord_min @ int ) ) ).
% inf_int_def
thf(fact_6672_inf__nat__def,axiom,
( ( inf_inf @ nat )
= ( ord_min @ nat ) ) ).
% inf_nat_def
thf(fact_6673_inf__enat__def,axiom,
( ( inf_inf @ extended_enat )
= ( ord_min @ extended_enat ) ) ).
% inf_enat_def
thf(fact_6674_lenlex__length,axiom,
! [A: $tType,Ms: list @ A,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R2 ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) ) ) ).
% lenlex_length
thf(fact_6675_set__zip,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [I4: nat] :
( ( Uu3
= ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ I4 ) @ ( nth @ B @ Ys @ I4 ) ) )
& ( ord_less @ nat @ I4 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% set_zip
thf(fact_6676_dual__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( max @ A
@ ^ [X: A,Y: A] : ( ord_less_eq @ A @ Y @ X ) )
= ( ord_min @ A ) ) ) ).
% dual_max
thf(fact_6677_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs2: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) @ I )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ I ) @ ( nth @ B @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_6678_ord_Omax__def,axiom,
! [A: $tType] :
( ( max @ A )
= ( ^ [Less_eq: A > A > $o,A5: A,B4: A] : ( if @ A @ ( Less_eq @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% ord.max_def
thf(fact_6679_ord_Omax_Ocong,axiom,
! [A: $tType] :
( ( max @ A )
= ( max @ A ) ) ).
% ord.max.cong
thf(fact_6680_in__set__zip,axiom,
! [B: $tType,A: $tType,P6: product_prod @ A @ B,Xs2: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ P6 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) ) )
= ( ? [N3: nat] :
( ( ( nth @ A @ Xs2 @ N3 )
= ( product_fst @ A @ B @ P6 ) )
& ( ( nth @ B @ Ys @ N3 )
= ( product_snd @ A @ B @ P6 ) )
& ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ord_less @ nat @ N3 @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ).
% in_set_zip
thf(fact_6681_nth__rotate,axiom,
! [A: $tType,N: nat,Xs2: list @ A,M: nat] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( rotate @ A @ M @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ M @ N ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ).
% nth_rotate
thf(fact_6682_find__Some__iff2,axiom,
! [A: $tType,X3: A,P: A > $o,Xs2: list @ A] :
( ( ( some @ A @ X3 )
= ( find @ A @ P @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P @ ( nth @ A @ Xs2 @ I4 ) )
& ( X3
= ( nth @ A @ Xs2 @ I4 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I4 )
=> ~ ( P @ ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_6683_rotate0,axiom,
! [A: $tType] :
( ( rotate @ A @ ( zero_zero @ nat ) )
= ( id @ ( list @ A ) ) ) ).
% rotate0
thf(fact_6684_rotate__length01,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) )
=> ( ( rotate @ A @ N @ Xs2 )
= Xs2 ) ) ).
% rotate_length01
thf(fact_6685_rotate__id,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
= ( zero_zero @ nat ) )
=> ( ( rotate @ A @ N @ Xs2 )
= Xs2 ) ) ).
% rotate_id
thf(fact_6686_find__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A,X3: A] :
( ( ( find @ A @ P @ Xs2 )
= ( some @ A @ X3 ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P @ ( nth @ A @ Xs2 @ I4 ) )
& ( X3
= ( nth @ A @ Xs2 @ I4 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I4 )
=> ~ ( P @ ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_6687_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_6688_and__not__num_Opelims,axiom,
! [X3: num,Xa: num,Y3: option @ num] :
( ( ( bit_and_not_num @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ X3 @ Xa ) )
=> ( ( ( X3 = one2 )
=> ( ( Xa = one2 )
=> ( ( Y3
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y3
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y3
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ( ( Xa = one2 )
=> ( ( Y3
= ( some @ num @ ( bit0 @ M4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ one2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y3
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y3
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ( ( Xa = one2 )
=> ( ( Y3
= ( some @ num @ ( bit0 @ M4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ one2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y3
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N10: num] : ( some @ num @ ( bit1 @ N10 ) )
@ ( bit_and_not_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ~ ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y3
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
thf(fact_6689_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_6690_and__num_Opelims,axiom,
! [X3: num,Xa: num,Y3: option @ num] :
( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ X3 @ Xa ) )
=> ( ( ( X3 = one2 )
=> ( ( Xa = one2 )
=> ( ( Y3
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y3
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y3
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ( ( Xa = one2 )
=> ( ( Y3
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ one2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y3
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y3
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ( ( Xa = one2 )
=> ( ( Y3
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ one2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y3
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ~ ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y3
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N10: num] : ( some @ num @ ( bit1 @ N10 ) )
@ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
thf(fact_6691_xor__num_Opelims,axiom,
! [X3: num,Xa: num,Y3: option @ num] :
( ( ( bit_un2480387367778600638or_num @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ X3 @ Xa ) )
=> ( ( ( X3 = one2 )
=> ( ( Xa = one2 )
=> ( ( Y3
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y3
= ( some @ num @ ( bit1 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y3
= ( some @ num @ ( bit0 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ( ( Xa = one2 )
=> ( ( Y3
= ( some @ num @ ( bit1 @ M4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ one2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y3
= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y3
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ( ( Xa = one2 )
=> ( ( Y3
= ( some @ num @ ( bit0 @ M4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ one2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y3
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ~ ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y3
= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.pelims
thf(fact_6692_or__not__num__neg_Opelims,axiom,
! [X3: num,Xa: num,Y3: num] :
( ( ( bit_or_not_num_neg @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ X3 @ Xa ) )
=> ( ( ( X3 = one2 )
=> ( ( Xa = one2 )
=> ( ( Y3 = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( ( Y3
= ( bit1 @ M4 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ M4 ) ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( ( Y3
= ( bit1 @ M4 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ M4 ) ) ) ) ) )
=> ( ! [N2: num] :
( ( X3
= ( bit0 @ N2 ) )
=> ( ( Xa = one2 )
=> ( ( Y3
= ( bit0 @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N2 ) @ one2 ) ) ) ) )
=> ( ! [N2: num] :
( ( X3
= ( bit0 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( ( Y3
= ( 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] :
( ( X3
= ( bit0 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( ( Y3
= ( 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] :
( ( X3
= ( bit1 @ N2 ) )
=> ( ( Xa = one2 )
=> ( ( Y3 = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N2 ) @ one2 ) ) ) ) )
=> ( ! [N2: num] :
( ( X3
= ( bit1 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( ( Y3
= ( 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] :
( ( X3
= ( bit1 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( ( Y3
= ( 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_6693_xor__num__rel__dict,axiom,
bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).
% xor_num_rel_dict
thf(fact_6694_and__num__rel__dict,axiom,
bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).
% and_num_rel_dict
thf(fact_6695_sum__list__update,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [K: nat,Xs2: list @ A,X3: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs2 @ K @ X3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs2 ) @ X3 ) @ ( nth @ A @ Xs2 @ K ) ) ) ) ) ).
% sum_list_update
thf(fact_6696_sum__list__eq__0__iff,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [Ns: list @ A] :
( ( ( groups8242544230860333062m_list @ A @ Ns )
= ( zero_zero @ A ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Ns ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_6697_member__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X3: A,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ X3 @ ( groups8242544230860333062m_list @ A @ Xs2 ) ) ) ) ).
% member_le_sum_list
thf(fact_6698_sum__list__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs2: list @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ X4 @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs2 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_nonpos
thf(fact_6699_sum__list__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs2: list @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 ) )
=> ( ( ( groups8242544230860333062m_list @ A @ Xs2 )
= ( zero_zero @ A ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_6700_Groups__List_Osum__list__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs2: list @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs2 ) ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_6701_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_6702_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_6703_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_6704_sum__list__sum__nth,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ( ( groups8242544230860333062m_list @ B )
= ( ^ [Xs: list @ B] : ( groups7311177749621191930dd_sum @ nat @ B @ ( nth @ B @ Xs ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ) ) ).
% sum_list_sum_nth
thf(fact_6705_Code__Target__Nat_ONat_Oabs__eq,axiom,
! [X3: int] :
( ( code_Target_Nat @ ( code_integer_of_int @ X3 ) )
= ( nat2 @ X3 ) ) ).
% Code_Target_Nat.Nat.abs_eq
thf(fact_6706_Code__Target__Nat_ONat_Orep__eq,axiom,
( code_Target_Nat
= ( ^ [X: code_integer] : ( nat2 @ ( code_int_of_integer @ X ) ) ) ) ).
% Code_Target_Nat.Nat.rep_eq
thf(fact_6707_power_Opower__eq__if,axiom,
! [A: $tType] :
( ( power2 @ A )
= ( ^ [One: A,Times: A > A > A,P5: A,M3: nat] :
( if @ A
@ ( M3
= ( zero_zero @ nat ) )
@ One
@ ( Times @ P5 @ ( power2 @ A @ One @ Times @ P5 @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% power.power_eq_if
thf(fact_6708_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X3: A,Xs2: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nth @ A @ ( cons @ A @ X3 @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_6709_nth__Cons__0,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( nth @ A @ ( cons @ A @ X3 @ Xs2 ) @ ( zero_zero @ nat ) )
= X3 ) ).
% nth_Cons_0
thf(fact_6710_nth__Cons__numeral,axiom,
! [A: $tType,X3: A,Xs2: list @ A,V2: num] :
( ( nth @ A @ ( cons @ A @ X3 @ Xs2 ) @ ( numeral_numeral @ nat @ V2 ) )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) ) ) ).
% nth_Cons_numeral
thf(fact_6711_take__Cons__numeral,axiom,
! [A: $tType,V2: num,X3: A,Xs2: list @ A] :
( ( take @ A @ ( numeral_numeral @ nat @ V2 ) @ ( cons @ A @ X3 @ Xs2 ) )
= ( cons @ A @ X3 @ ( take @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) @ Xs2 ) ) ) ).
% take_Cons_numeral
thf(fact_6712_impossible__Cons,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,X3: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs2
!= ( cons @ A @ X3 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_6713_power_Opower_Ocong,axiom,
! [A: $tType] :
( ( power2 @ A )
= ( power2 @ A ) ) ).
% power.power.cong
thf(fact_6714_list__update__code_I2_J,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Y3: A] :
( ( list_update @ A @ ( cons @ A @ X3 @ Xs2 ) @ ( zero_zero @ nat ) @ Y3 )
= ( cons @ A @ Y3 @ Xs2 ) ) ).
% list_update_code(2)
thf(fact_6715_power_Opower_Opower__Suc,axiom,
! [A: $tType,One2: A,Times2: A > A > A,A2: A,N: nat] :
( ( power2 @ A @ One2 @ Times2 @ A2 @ ( suc @ N ) )
= ( Times2 @ A2 @ ( power2 @ A @ One2 @ Times2 @ A2 @ N ) ) ) ).
% power.power.power_Suc
thf(fact_6716_power_Opower_Opower__0,axiom,
! [A: $tType,One2: A,Times2: A > A > A,A2: A] :
( ( power2 @ A @ One2 @ Times2 @ A2 @ ( zero_zero @ nat ) )
= One2 ) ).
% power.power.power_0
thf(fact_6717_find_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X3: A,Xs2: list @ A] :
( ( ( P @ X3 )
=> ( ( find @ A @ P @ ( cons @ A @ X3 @ Xs2 ) )
= ( some @ A @ X3 ) ) )
& ( ~ ( P @ X3 )
=> ( ( find @ A @ P @ ( cons @ A @ X3 @ Xs2 ) )
= ( find @ A @ P @ Xs2 ) ) ) ) ).
% find.simps(2)
thf(fact_6718_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= ( ? [X: A,Ys3: list @ A] :
( ( Xs2
= ( cons @ A @ X @ Ys3 ) )
& ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_6719_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_6720_nth__Cons_H,axiom,
! [A: $tType,N: nat,X3: A,Xs2: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X3 @ Xs2 ) @ N )
= X3 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X3 @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_6721_list_Osize__gen_I2_J,axiom,
! [A: $tType,X3: A > nat,X21: A,X222: list @ A] :
( ( size_list @ A @ X3 @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X3 @ X21 ) @ ( size_list @ A @ X3 @ X222 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_6722_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_6723_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_6724_nth__equal__first__eq,axiom,
! [A: $tType,X3: A,Xs2: list @ A,N: nat] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( nth @ A @ ( cons @ A @ X3 @ Xs2 ) @ N )
= X3 )
= ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_6725_nth__non__equal__first__eq,axiom,
! [A: $tType,X3: A,Y3: A,Xs2: list @ A,N: nat] :
( ( X3 != Y3 )
=> ( ( ( nth @ A @ ( cons @ A @ X3 @ Xs2 ) @ N )
= Y3 )
= ( ( ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= Y3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_6726_Cons__replicate__eq,axiom,
! [A: $tType,X3: A,Xs2: list @ A,N: nat,Y3: A] :
( ( ( cons @ A @ X3 @ Xs2 )
= ( replicate @ A @ N @ Y3 ) )
= ( ( X3 = Y3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( Xs2
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X3 ) ) ) ) ).
% Cons_replicate_eq
thf(fact_6727_Cons__lenlex__iff,axiom,
! [A: $tType,M: A,Ms: list @ A,N: A,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M @ Ms ) @ ( cons @ A @ N @ Ns ) ) @ ( lenlex @ A @ R2 ) )
= ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
| ( ( ( size_size @ ( list @ A ) @ Ms )
= ( size_size @ ( list @ A ) @ Ns ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_6728_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_6729_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_6730_length__0__conv,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( zero_zero @ nat ) )
= ( Xs2
= ( nil @ A ) ) ) ).
% length_0_conv
thf(fact_6731_sum__list_ONil,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A @ ( nil @ A ) )
= ( zero_zero @ A ) ) ) ).
% sum_list.Nil
thf(fact_6732_take__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( nil @ A )
= ( take @ A @ N @ Xs2 ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( Xs2
= ( nil @ A ) ) ) ) ).
% take_eq_Nil2
thf(fact_6733_take__eq__Nil,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( take @ A @ N @ Xs2 )
= ( nil @ A ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( Xs2
= ( nil @ A ) ) ) ) ).
% take_eq_Nil
thf(fact_6734_take0,axiom,
! [A: $tType] :
( ( take @ A @ ( zero_zero @ nat ) )
= ( ^ [Xs: list @ A] : ( nil @ A ) ) ) ).
% take0
thf(fact_6735_empty__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( ( nil @ A )
= ( replicate @ A @ N @ X3 ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_6736_replicate__empty,axiom,
! [A: $tType,N: nat,X3: A] :
( ( ( replicate @ A @ N @ X3 )
= ( nil @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_6737_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_6738_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_6739_length__greater__0__conv,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= ( Xs2
!= ( nil @ A ) ) ) ).
% length_greater_0_conv
thf(fact_6740_nths__singleton,axiom,
! [A: $tType,A4: set @ nat,X3: A] :
( ( ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ( nths @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ A4 )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) )
& ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ( nths @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ A4 )
= ( nil @ A ) ) ) ) ).
% nths_singleton
thf(fact_6741_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs2: list @ A] :
( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs2 )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% n_lists.simps(1)
thf(fact_6742_take__0,axiom,
! [A: $tType,Xs2: list @ A] :
( ( take @ A @ ( zero_zero @ nat ) @ Xs2 )
= ( nil @ A ) ) ).
% take_0
thf(fact_6743_replicate__0,axiom,
! [A: $tType,X3: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X3 )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_6744_list_Osize__gen_I1_J,axiom,
! [A: $tType,X3: A > nat] :
( ( size_list @ A @ X3 @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size_gen(1)
thf(fact_6745_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_6746_lexordp_Omono,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P5: ( list @ A ) > ( list @ A ) > $o,X17: list @ A,X23: list @ A] :
( ? [Y: A,Ys3: list @ A] :
( ( X17
= ( nil @ A ) )
& ( X23
= ( cons @ A @ Y @ Ys3 ) ) )
| ? [X: A,Y: A,Xs: list @ A,Ys3: list @ A] :
( ( X17
= ( cons @ A @ X @ Xs ) )
& ( X23
= ( cons @ A @ Y @ Ys3 ) )
& ( ord_less @ A @ X @ Y ) )
| ? [X: A,Y: A,Xs: list @ A,Ys3: list @ A] :
( ( X17
= ( cons @ A @ X @ Xs ) )
& ( X23
= ( cons @ A @ Y @ Ys3 ) )
& ~ ( ord_less @ A @ X @ Y )
& ~ ( ord_less @ A @ Y @ X )
& ( P5 @ Xs @ Ys3 ) ) ) ) ) ).
% lexordp.mono
thf(fact_6747_remdups__adj__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N @ X3 ) )
= ( nil @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N @ X3 ) )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_replicate
thf(fact_6748_take__Cons_H,axiom,
! [A: $tType,N: nat,X3: A,Xs2: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X3 @ Xs2 ) )
= ( nil @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X3 @ Xs2 ) )
= ( cons @ A @ X3 @ ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs2 ) ) ) ) ) ).
% take_Cons'
thf(fact_6749_remdups__adj__length__ge1,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs2 ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_6750_concat__inth,axiom,
! [A: $tType,Xs2: list @ A,X3: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs2 @ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= X3 ) ).
% concat_inth
thf(fact_6751_take__Suc__conv__app__nth,axiom,
! [A: $tType,I: nat,Xs2: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( take @ A @ ( suc @ I ) @ Xs2 )
= ( append @ A @ ( take @ A @ I @ Xs2 ) @ ( cons @ A @ ( nth @ A @ Xs2 @ I ) @ ( nil @ A ) ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_6752_mono__compose,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ C ) )
=> ! [Q: A > B > C,F2: D > B] :
( ( order_mono @ A @ ( B > C ) @ Q )
=> ( order_mono @ A @ ( D > C )
@ ^ [I4: A,X: D] : ( Q @ I4 @ ( F2 @ X ) ) ) ) ) ).
% mono_compose
thf(fact_6753_list__update__append1,axiom,
! [A: $tType,I: nat,Xs2: list @ A,Ys: list @ A,X3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( list_update @ A @ ( append @ A @ Xs2 @ Ys ) @ I @ X3 )
= ( append @ A @ ( list_update @ A @ Xs2 @ I @ X3 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_6754_nth__append,axiom,
! [A: $tType,N: nat,Xs2: list @ A,Ys: list @ A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( append @ A @ Xs2 @ Ys ) @ N )
= ( nth @ A @ Xs2 @ N ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( append @ A @ Xs2 @ Ys ) @ N )
= ( nth @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% nth_append
thf(fact_6755_list__update__append,axiom,
! [A: $tType,N: nat,Xs2: list @ A,Ys: list @ A,X3: A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( list_update @ A @ ( append @ A @ Xs2 @ Ys ) @ N @ X3 )
= ( append @ A @ ( list_update @ A @ Xs2 @ N @ X3 ) @ Ys ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( list_update @ A @ ( append @ A @ Xs2 @ Ys ) @ N @ X3 )
= ( append @ A @ Xs2 @ ( list_update @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ X3 ) ) ) ) ) ).
% list_update_append
thf(fact_6756_comm__append__is__replicate,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ A ) )
=> ( ( ( append @ A @ Xs2 @ Ys )
= ( append @ A @ Ys @ Xs2 ) )
=> ? [N2: nat,Zs: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N2 @ Zs ) )
= ( append @ A @ Xs2 @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_6757_lexord__sufI,axiom,
! [A: $tType,U: list @ A,W: list @ A,R2: set @ ( product_prod @ A @ A ),V2: list @ A,Z: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W ) @ ( lexord @ A @ R2 ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W ) @ ( size_size @ ( list @ A ) @ U ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V2 ) @ ( append @ A @ W @ Z ) ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_sufI
thf(fact_6758_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F2: B > A,A2: A,Xs2: list @ B,Ys: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ ( append @ B @ Xs2 @ Ys ) )
= ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ Xs2 ) @ ( times_times @ A @ ( power_power @ A @ A2 @ ( size_size @ ( list @ B ) @ Xs2 ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ Ys ) ) ) ) ) ).
% horner_sum_append
thf(fact_6759_nths__Cons,axiom,
! [A: $tType,X3: A,L: list @ A,A4: set @ nat] :
( ( nths @ A @ ( cons @ A @ X3 @ L ) @ A4 )
= ( append @ A @ ( if @ ( list @ A ) @ ( member @ nat @ ( zero_zero @ nat ) @ A4 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( nil @ A ) )
@ ( nths @ A @ L
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( suc @ J3 ) @ A4 ) ) ) ) ) ).
% nths_Cons
thf(fact_6760_nth__repl,axiom,
! [A: $tType,M: nat,Xs2: list @ A,N: nat,X3: A] :
( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( M != N )
=> ( ( nth @ A @ ( append @ A @ ( take @ A @ N @ Xs2 ) @ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( drop @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ Xs2 ) ) ) @ M )
= ( nth @ A @ Xs2 @ M ) ) ) ) ) ).
% nth_repl
thf(fact_6761_pos__n__replace,axiom,
! [A: $tType,N: nat,Xs2: list @ A,Y3: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ ( append @ A @ ( take @ A @ N @ Xs2 ) @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ ( drop @ A @ ( suc @ N ) @ Xs2 ) ) ) ) ) ) ).
% pos_n_replace
thf(fact_6762_drop0,axiom,
! [A: $tType] :
( ( drop @ A @ ( zero_zero @ nat ) )
= ( ^ [X: list @ A] : X ) ) ).
% drop0
thf(fact_6763_drop__update__cancel,axiom,
! [A: $tType,N: nat,M: nat,Xs2: list @ A,X3: A] :
( ( ord_less @ nat @ N @ M )
=> ( ( drop @ A @ M @ ( list_update @ A @ Xs2 @ N @ X3 ) )
= ( drop @ A @ M @ Xs2 ) ) ) ).
% drop_update_cancel
thf(fact_6764_drop__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( nil @ A )
= ( drop @ A @ N @ Xs2 ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_6765_drop__eq__Nil,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( drop @ A @ N @ Xs2 )
= ( nil @ A ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ).
% drop_eq_Nil
thf(fact_6766_drop__all,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N )
=> ( ( drop @ A @ N @ Xs2 )
= ( nil @ A ) ) ) ).
% drop_all
thf(fact_6767_drop__Cons__numeral,axiom,
! [A: $tType,V2: num,X3: A,Xs2: list @ A] :
( ( drop @ A @ ( numeral_numeral @ nat @ V2 ) @ ( cons @ A @ X3 @ Xs2 ) )
= ( drop @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) @ Xs2 ) ) ).
% drop_Cons_numeral
thf(fact_6768_nth__drop,axiom,
! [A: $tType,N: nat,Xs2: list @ A,I: nat] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( drop @ A @ N @ Xs2 ) @ I )
= ( nth @ A @ Xs2 @ ( plus_plus @ nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_6769_drop__eq__nths,axiom,
! [A: $tType] :
( ( drop @ A )
= ( ^ [N3: nat,Xs: list @ A] : ( nths @ A @ Xs @ ( collect @ nat @ ( ord_less_eq @ nat @ N3 ) ) ) ) ) ).
% drop_eq_nths
thf(fact_6770_drop__0,axiom,
! [A: $tType,Xs2: list @ A] :
( ( drop @ A @ ( zero_zero @ nat ) @ Xs2 )
= Xs2 ) ).
% drop_0
thf(fact_6771_set__drop__subset__set__drop,axiom,
! [A: $tType,N: nat,M: nat,Xs2: list @ A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ M @ Xs2 ) ) @ ( set2 @ A @ ( drop @ A @ N @ Xs2 ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_6772_drop__update__swap,axiom,
! [A: $tType,M: nat,N: nat,Xs2: list @ A,X3: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( drop @ A @ M @ ( list_update @ A @ Xs2 @ N @ X3 ) )
= ( list_update @ A @ ( drop @ A @ M @ Xs2 ) @ ( minus_minus @ nat @ N @ M ) @ X3 ) ) ) ).
% drop_update_swap
thf(fact_6773_drop__Cons_H,axiom,
! [A: $tType,N: nat,X3: A,Xs2: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X3 @ Xs2 ) )
= ( cons @ A @ X3 @ Xs2 ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X3 @ Xs2 ) )
= ( drop @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs2 ) ) ) ) ).
% drop_Cons'
thf(fact_6774_append__eq__append__conv__if,axiom,
! [A: $tType,Xs_1: list @ A,Xs_2: list @ A,Ys_1: list @ A,Ys_2: list @ A] :
( ( ( append @ A @ Xs_1 @ Xs_2 )
= ( append @ A @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs_1 ) @ ( size_size @ ( list @ A ) @ Ys_1 ) )
=> ( ( Xs_1
= ( take @ A @ ( size_size @ ( list @ A ) @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append @ A @ ( drop @ A @ ( size_size @ ( list @ A ) @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs_1 ) @ ( size_size @ ( list @ A ) @ Ys_1 ) )
=> ( ( ( take @ A @ ( size_size @ ( list @ A ) @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append @ A @ ( drop @ A @ ( size_size @ ( list @ A ) @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_6775_Cons__nth__drop__Suc,axiom,
! [A: $tType,I: nat,Xs2: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( cons @ A @ ( nth @ A @ Xs2 @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs2 ) )
= ( drop @ A @ I @ Xs2 ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_6776_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Vs )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ I @ Vs ) ) @ ( set2 @ A @ ( drop @ A @ J @ Vs ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_take_disj_set_drop_if_distinct
thf(fact_6777_id__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs2: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( Xs2
= ( append @ A @ ( take @ A @ I @ Xs2 ) @ ( cons @ A @ ( nth @ A @ Xs2 @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs2 ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_6778_upd__conv__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs2: list @ A,A2: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( list_update @ A @ Xs2 @ I @ A2 )
= ( append @ A @ ( take @ A @ I @ Xs2 ) @ ( cons @ A @ A2 @ ( drop @ A @ ( suc @ I ) @ Xs2 ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_6779_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_6780_upto_Opelims,axiom,
! [X3: int,Xa: int,Y3: list @ int] :
( ( ( upto @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X3 @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X3 @ Xa )
=> ( Y3
= ( cons @ int @ X3 @ ( upto @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X3 @ Xa )
=> ( Y3
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X3 @ Xa ) ) ) ) ) ).
% upto.pelims
thf(fact_6781_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= ( nil @ int ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil
thf(fact_6782_upto__Nil2,axiom,
! [I: int,J: int] :
( ( ( nil @ int )
= ( upto @ I @ J ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil2
thf(fact_6783_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less @ int @ J @ I )
=> ( ( upto @ I @ J )
= ( nil @ int ) ) ) ).
% upto_empty
thf(fact_6784_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_6785_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_6786_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_6787_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_6788_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_6789_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_6790_upto__split2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_6791_upto__split1,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_6792_upto_Oelims,axiom,
! [X3: int,Xa: int,Y3: list @ int] :
( ( ( upto @ X3 @ Xa )
= Y3 )
=> ( ( ( ord_less_eq @ int @ X3 @ Xa )
=> ( Y3
= ( cons @ int @ X3 @ ( upto @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X3 @ Xa )
=> ( Y3
= ( nil @ int ) ) ) ) ) ).
% upto.elims
thf(fact_6793_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_6794_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_6795_upto__rec2,axiom,
! [I: int,J: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( nil @ int ) ) ) ) ) ).
% upto_rec2
thf(fact_6796_upto__split3,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_6797_take__hd__drop,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( append @ A @ ( take @ A @ N @ Xs2 ) @ ( cons @ A @ ( hd @ A @ ( drop @ A @ N @ Xs2 ) ) @ ( nil @ A ) ) )
= ( take @ A @ ( suc @ N ) @ Xs2 ) ) ) ).
% take_hd_drop
thf(fact_6798_remdups__adj__singleton__iff,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs2 ) )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( Xs2
!= ( nil @ A ) )
& ( Xs2
= ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( hd @ A @ Xs2 ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_6799_hd__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( hd @ A @ ( replicate @ A @ N @ X3 ) )
= X3 ) ) ).
% hd_replicate
thf(fact_6800_hd__take,axiom,
! [A: $tType,J: nat,Xs2: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ J )
=> ( ( hd @ A @ ( take @ A @ J @ Xs2 ) )
= ( hd @ A @ Xs2 ) ) ) ).
% hd_take
thf(fact_6801_hd__conv__nth,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( hd @ A @ Xs2 )
= ( nth @ A @ Xs2 @ ( zero_zero @ nat ) ) ) ) ).
% hd_conv_nth
thf(fact_6802_hd__drop__conv__nth,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( hd @ A @ ( drop @ A @ N @ Xs2 ) )
= ( nth @ A @ Xs2 @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_6803_Nitpick_Osize__list__simp_I1_J,axiom,
! [A: $tType] :
( ( size_list @ A )
= ( ^ [F3: A > nat,Xs: list @ A] :
( if @ nat
@ ( Xs
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( plus_plus @ nat @ ( F3 @ ( hd @ A @ Xs ) ) @ ( size_list @ A @ F3 @ ( tl @ A @ Xs ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_6804_finite__enumerate__initial__segment,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat,S: A] :
( ( finite_finite @ 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_6805_enumerate__mono__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,M: nat,N: nat] :
( ~ ( finite_finite @ A @ S2 )
=> ( ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ M ) @ ( infini527867602293511546merate @ A @ S2 @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ) ).
% enumerate_mono_iff
thf(fact_6806_finite__enumerate__mono__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,M: nat,N: nat] :
( ( finite_finite @ 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_6807_le__enumerate,axiom,
! [S2: set @ nat,N: nat] :
( ~ ( finite_finite @ nat @ S2 )
=> ( ord_less_eq @ nat @ N @ ( infini527867602293511546merate @ nat @ S2 @ N ) ) ) ).
% le_enumerate
thf(fact_6808_enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ~ ( finite_finite @ A @ S2 )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ N ) @ ( infini527867602293511546merate @ A @ S2 @ ( suc @ N ) ) ) ) ) ).
% enumerate_step
thf(fact_6809_enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M: nat,N: nat,S2: set @ A] :
( ( ord_less @ nat @ M @ N )
=> ( ~ ( finite_finite @ A @ S2 )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ M ) @ ( infini527867602293511546merate @ A @ S2 @ N ) ) ) ) ) ).
% enumerate_mono
thf(fact_6810_finite__enum__ext,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X8: set @ A,Y7: set @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( finite_card @ A @ X8 ) )
=> ( ( infini527867602293511546merate @ A @ X8 @ I3 )
= ( infini527867602293511546merate @ A @ Y7 @ I3 ) ) )
=> ( ( finite_finite @ A @ X8 )
=> ( ( finite_finite @ A @ Y7 )
=> ( ( ( finite_card @ A @ X8 )
= ( finite_card @ A @ Y7 ) )
=> ( X8 = Y7 ) ) ) ) ) ) ).
% finite_enum_ext
thf(fact_6811_finite__enumerate__Ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,S: A] :
( ( finite_finite @ 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_6812_finite__enumerate__in__set,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ( finite_finite @ A @ S2 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S2 ) )
=> ( member @ A @ ( infini527867602293511546merate @ A @ S2 @ N ) @ S2 ) ) ) ) ).
% finite_enumerate_in_set
thf(fact_6813_Nitpick_Osize__list__simp_I2_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( ^ [Xs: list @ A] :
( if @ nat
@ ( Xs
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_6814_nth__tl,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs2 ) ) )
=> ( ( nth @ A @ ( tl @ A @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_6815_finite__enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M: nat,N: nat,S2: set @ A] :
( ( ord_less @ nat @ M @ N )
=> ( ( finite_finite @ 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_6816_finite__le__enumerate,axiom,
! [S2: set @ nat,N: nat] :
( ( finite_finite @ nat @ S2 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ nat @ S2 ) )
=> ( ord_less_eq @ nat @ N @ ( infini527867602293511546merate @ nat @ S2 @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_6817_finite__enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ( finite_finite @ 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_6818_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_6819_finite__enum__subset,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X8: set @ A,Y7: set @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( finite_card @ A @ X8 ) )
=> ( ( infini527867602293511546merate @ A @ X8 @ I3 )
= ( infini527867602293511546merate @ A @ Y7 @ I3 ) ) )
=> ( ( finite_finite @ A @ X8 )
=> ( ( finite_finite @ A @ Y7 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ X8 ) @ ( finite_card @ A @ Y7 ) )
=> ( ord_less_eq @ ( set @ A ) @ X8 @ Y7 ) ) ) ) ) ) ).
% finite_enum_subset
thf(fact_6820_finite__enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ( finite_finite @ A @ S2 )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( finite_card @ A @ S2 ) )
=> ( ( infini527867602293511546merate @ A @ S2 @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S6: A] :
( ( member @ A @ S6 @ S2 )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ N ) @ S6 ) ) ) ) ) ) ) ).
% finite_enumerate_Suc''
thf(fact_6821_DeMoivre2,axiom,
! [R2: real,A2: real,N: nat] :
( ( power_power @ complex @ ( rcis @ R2 @ A2 ) @ N )
= ( rcis @ ( power_power @ real @ R2 @ N ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ A2 ) ) ) ).
% DeMoivre2
thf(fact_6822_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( zero_zero @ nat ) ) ) ).
% Least_eq_0
thf(fact_6823_rcis__zero__arg,axiom,
! [R2: real] :
( ( rcis @ R2 @ ( zero_zero @ real ) )
= ( real_Vector_of_real @ complex @ R2 ) ) ).
% rcis_zero_arg
thf(fact_6824_rcis__zero__mod,axiom,
! [A2: real] :
( ( rcis @ ( zero_zero @ real ) @ A2 )
= ( zero_zero @ complex ) ) ).
% rcis_zero_mod
thf(fact_6825_rcis__eq__zero__iff,axiom,
! [R2: real,A2: real] :
( ( ( rcis @ R2 @ A2 )
= ( zero_zero @ complex ) )
= ( R2
= ( zero_zero @ real ) ) ) ).
% rcis_eq_zero_iff
thf(fact_6826_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [A6: A] :
( ( P @ A6 )
=> ( ! [B8: A] :
( ( P @ B8 )
=> ( ord_less_eq @ A @ A6 @ B8 ) )
=> ( Q @ A6 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_6827_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [A6: A] :
( ( P @ A6 )
=> ( ! [B8: A] :
( ( P @ B8 )
=> ( ord_less_eq @ A @ A6 @ B8 ) )
=> ( Q @ A6 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_6828_Least__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ X3 @ Y4 ) )
=> ( ( ord_Least @ A @ P )
= X3 ) ) ) ) ).
% Least_equality
thf(fact_6829_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A,Q: A > $o] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ X3 @ Y4 ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ X4 @ Y5 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_6830_Least1__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,Z: A] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ X6 @ Y4 ) )
& ! [Y4: A] :
( ( ( P @ Y4 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y4 @ Ya2 ) ) )
=> ( Y4 = X6 ) ) )
=> ( ( P @ Z )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ Z ) ) ) ) ).
% Least1_le
thf(fact_6831_Least1I,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ X6 @ Y4 ) )
& ! [Y4: A] :
( ( ( P @ Y4 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y4 @ Ya2 ) ) )
=> ( Y4 = X6 ) ) )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% Least1I
thf(fact_6832_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_6833_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_6834_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI
thf(fact_6835_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2
thf(fact_6836_LeastI__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI_ex
thf(fact_6837_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_ex
thf(fact_6838_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_6839_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_6840_Inf__nat__def,axiom,
( ( complete_Inf_Inf @ nat )
= ( ^ [X9: set @ nat] :
( ord_Least @ nat
@ ^ [N3: nat] : ( member @ nat @ N3 @ X9 ) ) ) ) ).
% Inf_nat_def
thf(fact_6841_Least__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( finite_finite @ A @ ( collect @ A @ P ) )
=> ( ? [X_12: A] : ( P @ X_12 )
=> ( ( ord_Least @ A @ P )
= ( lattic643756798350308766er_Min @ A @ ( collect @ A @ P ) ) ) ) ) ) ).
% Least_Min
thf(fact_6842_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_6843_Sup__real__def,axiom,
( ( complete_Sup_Sup @ real )
= ( ^ [X9: set @ real] :
( ord_Least @ real
@ ^ [Z6: real] :
! [X: real] :
( ( member @ real @ X @ X9 )
=> ( ord_less_eq @ real @ X @ Z6 ) ) ) ) ) ).
% Sup_real_def
thf(fact_6844_Least__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,S2: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ? [X6: A] :
( ( member @ A @ X6 @ S2 )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ S2 )
=> ( ord_less_eq @ A @ X6 @ Xa3 ) ) )
=> ( ( ord_Least @ B
@ ^ [Y: B] : ( member @ B @ Y @ ( image @ A @ B @ F2 @ S2 ) ) )
= ( F2
@ ( ord_Least @ A
@ ^ [X: A] : ( member @ A @ X @ S2 ) ) ) ) ) ) ) ).
% Least_mono
thf(fact_6845_Least__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_Least @ A )
= ( ^ [P3: A > $o] :
( the @ A
@ ^ [X: A] :
( ( P3 @ X )
& ! [Y: A] :
( ( P3 @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ).
% Least_def
thf(fact_6846_enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ~ ( finite_finite @ A @ S2 )
=> ( ( infini527867602293511546merate @ A @ S2 @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S6: A] :
( ( member @ A @ S6 @ S2 )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ N ) @ S6 ) ) ) ) ) ) ).
% enumerate_Suc''
thf(fact_6847_butlast__take,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( butlast @ A @ ( take @ A @ N @ Xs2 ) )
= ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs2 ) ) ) ).
% butlast_take
thf(fact_6848_Gcd__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( ^ [A9: set @ A] : ( if @ A @ ( finite_finite @ A @ A9 ) @ ( finite_fold @ A @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ A9 ) @ ( one_one @ A ) ) ) ) ) ).
% Gcd_fin.eq_fold
thf(fact_6849_Gcd__fin_Oempty,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_fin.empty
thf(fact_6850_nth__butlast,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs2 ) ) )
=> ( ( nth @ A @ ( butlast @ A @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ N ) ) ) ).
% nth_butlast
thf(fact_6851_take__butlast,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( take @ A @ N @ ( butlast @ A @ Xs2 ) )
= ( take @ A @ N @ Xs2 ) ) ) ).
% take_butlast
thf(fact_6852_Gcd__fin__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( ( semiring_gcd_Gcd_fin @ A @ A4 )
= ( zero_zero @ A ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) )
& ( finite_finite @ A @ A4 ) ) ) ) ).
% Gcd_fin_0_iff
thf(fact_6853_independent__explicit__finite__subsets,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A4: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ A4 ) )
= ( ! [S8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S8 @ A4 )
=> ( ( finite_finite @ A @ S8 )
=> ! [U2: A > real] :
( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V5 ) @ V5 )
@ S8 )
= ( zero_zero @ A ) )
=> ! [X: A] :
( ( member @ A @ X @ S8 )
=> ( ( U2 @ X )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% independent_explicit_finite_subsets
thf(fact_6854_independent__explicit__module,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ S ) )
= ( ! [T3: set @ A,U2: A > real,V5: A] :
( ( finite_finite @ A @ T3 )
=> ( ( ord_less_eq @ ( set @ A ) @ T3 @ S )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [W3: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ W3 ) @ W3 )
@ T3 )
= ( zero_zero @ A ) )
=> ( ( member @ A @ V5 @ T3 )
=> ( ( U2 @ V5 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% independent_explicit_module
thf(fact_6855_dependent__single,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A] :
( ( real_V358717886546972837endent @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( X3
= ( zero_zero @ A ) ) ) ) ).
% dependent_single
thf(fact_6856_dependent__zero,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A4: set @ A] :
( ( member @ A @ ( zero_zero @ A ) @ A4 )
=> ( real_V358717886546972837endent @ A @ A4 ) ) ) ).
% dependent_zero
thf(fact_6857_unique__representation,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,F2: A > real,G: A > real] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ! [V4: A] :
( ( ( F2 @ V4 )
!= ( zero_zero @ real ) )
=> ( member @ A @ V4 @ Basis ) )
=> ( ! [V4: A] :
( ( ( G @ V4 )
!= ( zero_zero @ real ) )
=> ( member @ A @ V4 @ Basis ) )
=> ( ( finite_finite @ A
@ ( collect @ A
@ ^ [V5: A] :
( ( F2 @ V5 )
!= ( zero_zero @ real ) ) ) )
=> ( ( finite_finite @ A
@ ( collect @ A
@ ^ [V5: A] :
( ( G @ V5 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( F2 @ V5 ) @ V5 )
@ ( collect @ A
@ ^ [V5: A] :
( ( F2 @ V5 )
!= ( zero_zero @ real ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( G @ V5 ) @ V5 )
@ ( collect @ A
@ ^ [V5: A] :
( ( G @ V5 )
!= ( zero_zero @ real ) ) ) ) )
=> ( F2 = G ) ) ) ) ) ) ) ) ).
% unique_representation
thf(fact_6858_independent__if__scalars__zero,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ! [F5: A > real,X4: A] :
( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [Y: A] : ( real_V8093663219630862766scaleR @ A @ ( F5 @ Y ) @ Y )
@ A4 )
= ( zero_zero @ A ) )
=> ( ( member @ A @ X4 @ A4 )
=> ( ( F5 @ X4 )
= ( zero_zero @ real ) ) ) )
=> ~ ( real_V358717886546972837endent @ A @ A4 ) ) ) ) ).
% independent_if_scalars_zero
thf(fact_6859_dependent__finite,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A] :
( ( finite_finite @ A @ S2 )
=> ( ( real_V358717886546972837endent @ A @ S2 )
= ( ? [U2: A > real] :
( ? [X: A] :
( ( member @ A @ X @ S2 )
& ( ( U2 @ X )
!= ( zero_zero @ real ) ) )
& ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V5 ) @ V5 )
@ S2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% dependent_finite
thf(fact_6860_independentD__unique,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B6: set @ A,X8: A > real,Y7: A > real] :
( ~ ( real_V358717886546972837endent @ A @ B6 )
=> ( ( finite_finite @ A
@ ( collect @ A
@ ^ [X: A] :
( ( X8 @ X )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( X8 @ X )
!= ( zero_zero @ real ) ) )
@ B6 )
=> ( ( finite_finite @ A
@ ( collect @ A
@ ^ [X: A] :
( ( Y7 @ X )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( Y7 @ X )
!= ( zero_zero @ real ) ) )
@ B6 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X: A] : ( real_V8093663219630862766scaleR @ A @ ( X8 @ X ) @ X )
@ ( collect @ A
@ ^ [X: A] :
( ( X8 @ X )
!= ( zero_zero @ real ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X: A] : ( real_V8093663219630862766scaleR @ A @ ( Y7 @ X ) @ X )
@ ( collect @ A
@ ^ [X: A] :
( ( Y7 @ X )
!= ( zero_zero @ real ) ) ) ) )
=> ( X8 = Y7 ) ) ) ) ) ) ) ) ).
% independentD_unique
thf(fact_6861_independentD,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A,T2: set @ A,U: A > real,V2: A] :
( ~ ( real_V358717886546972837endent @ A @ S )
=> ( ( finite_finite @ A @ T2 )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( U @ V5 ) @ V5 )
@ T2 )
= ( zero_zero @ A ) )
=> ( ( member @ A @ V2 @ T2 )
=> ( ( U @ V2 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% independentD
thf(fact_6862_dependent__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V358717886546972837endent @ A )
= ( ^ [B9: set @ A] :
? [X9: A > real] :
( ( finite_finite @ A
@ ( collect @ A
@ ^ [X: A] :
( ( X9 @ X )
!= ( zero_zero @ real ) ) ) )
& ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( X9 @ X )
!= ( zero_zero @ real ) ) )
@ B9 )
& ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X: A] : ( real_V8093663219630862766scaleR @ A @ ( X9 @ X ) @ X )
@ ( collect @ A
@ ^ [X: A] :
( ( X9 @ X )
!= ( zero_zero @ real ) ) ) )
= ( zero_zero @ A ) )
& ? [X: A] :
( ( X9 @ X )
!= ( zero_zero @ real ) ) ) ) ) ) ).
% dependent_alt
thf(fact_6863_independent__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B6: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ B6 ) )
= ( ! [X9: A > real] :
( ( finite_finite @ A
@ ( collect @ A
@ ^ [X: A] :
( ( X9 @ X )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( X9 @ X )
!= ( zero_zero @ real ) ) )
@ B6 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X: A] : ( real_V8093663219630862766scaleR @ A @ ( X9 @ X ) @ X )
@ ( collect @ A
@ ^ [X: A] :
( ( X9 @ X )
!= ( zero_zero @ real ) ) ) )
= ( zero_zero @ A ) )
=> ! [X: A] :
( ( X9 @ X )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% independent_alt
thf(fact_6864_independentD__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B6: set @ A,X8: A > real,X3: A] :
( ~ ( real_V358717886546972837endent @ A @ B6 )
=> ( ( finite_finite @ A
@ ( collect @ A
@ ^ [X: A] :
( ( X8 @ X )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( X8 @ X )
!= ( zero_zero @ real ) ) )
@ B6 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X: A] : ( real_V8093663219630862766scaleR @ A @ ( X8 @ X ) @ X )
@ ( collect @ A
@ ^ [X: A] :
( ( X8 @ X )
!= ( zero_zero @ real ) ) ) )
= ( zero_zero @ A ) )
=> ( ( X8 @ X3 )
= ( zero_zero @ real ) ) ) ) ) ) ) ).
% independentD_alt
thf(fact_6865_dependent__explicit,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V358717886546972837endent @ A )
= ( ^ [S6: set @ A] :
? [T3: set @ A] :
( ( finite_finite @ A @ T3 )
& ( ord_less_eq @ ( set @ A ) @ T3 @ S6 )
& ? [U2: A > real] :
( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V5 ) @ V5 )
@ T3 )
= ( zero_zero @ A ) )
& ? [X: A] :
( ( member @ A @ X @ T3 )
& ( ( U2 @ X )
!= ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% dependent_explicit
thf(fact_6866_isUCont__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F2: A > B] :
( ( topolo6026614971017936543ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S6 )
& ! [X: A,Y: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ S6 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R5 ) ) ) ) ) ) ) ).
% isUCont_def
thf(fact_6867_possible__bit__def,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se6407376104438227557le_bit @ A )
= ( ^ [Tyrep: itself @ A,N3: nat] :
( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
!= ( zero_zero @ A ) ) ) ) ) ).
% possible_bit_def
thf(fact_6868_possible__bit__min,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [Tyrep2: itself @ A,I: nat,J: nat] :
( ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ ( ord_min @ nat @ I @ J ) )
= ( ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ I )
| ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ J ) ) ) ) ).
% possible_bit_min
thf(fact_6869_possible__bit__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [Ty: itself @ A] : ( bit_se6407376104438227557le_bit @ A @ Ty @ ( zero_zero @ nat ) ) ) ).
% possible_bit_0
thf(fact_6870_possible__bit__less__imp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [Tyrep2: itself @ A,I: nat,J: nat] :
( ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ I )
=> ( ( ord_less_eq @ nat @ J @ I )
=> ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ J ) ) ) ) ).
% possible_bit_less_imp
thf(fact_6871_uniformly__continuous__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ( ( topolo6026614971017936543ous_on @ A @ B )
= ( ^ [S6: set @ A,F3: A > B] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X: A] :
( ( member @ A @ X @ S6 )
=> ! [Y: A] :
( ( member @ A @ Y @ S6 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X ) @ D4 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ Y ) @ ( F3 @ X ) ) @ E3 ) ) ) ) ) ) ) ) ) ).
% uniformly_continuous_on_def
thf(fact_6872_drop__bit__exp__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% drop_bit_exp_eq
thf(fact_6873_bit__minus__2__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% bit_minus_2_iff
thf(fact_6874_CHAR__eq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) ) ) ).
% CHAR_eq_0
thf(fact_6875_of__nat__CHAR,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% of_nat_CHAR
thf(fact_6876_bit__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ).
% bit_minus_1_iff
thf(fact_6877_bit__imp__possible__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
=> ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ).
% bit_imp_possible_bit
thf(fact_6878_impossible__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat,A2: A] :
( ~ ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
=> ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ).
% impossible_bit
thf(fact_6879_bit__eq__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [A5: A,B4: A] :
! [N3: nat] :
( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A5 @ N3 )
= ( bit_se5641148757651400278ts_bit @ A @ B4 @ N3 ) ) ) ) ) ) ).
% bit_eq_iff
thf(fact_6880_bit__eqI,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,B2: A] :
( ! [N2: nat] :
( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 )
= ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 ) ) )
=> ( A2 = B2 ) ) ) ).
% bit_eqI
thf(fact_6881_bit__not__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ N )
= ( ~ ~ ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% bit_not_iff
thf(fact_6882_bit__set__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
| ( ( M = N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ) ).
% bit_set_bit_iff
thf(fact_6883_bit__flip__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) @ N )
= ( ( ( M = N )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% bit_flip_bit_iff
thf(fact_6884_bit__mask__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2239418461657761734s_mask @ A @ M ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ N @ M ) ) ) ) ).
% bit_mask_iff
thf(fact_6885_CHAR__eqI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C2: nat] :
( ( ( semiring_1_of_nat @ A @ C2 )
= ( zero_zero @ A ) )
=> ( ! [X4: nat] :
( ( ( semiring_1_of_nat @ A @ X4 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ nat @ C2 @ X4 ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C2 ) ) ) ) ).
% CHAR_eqI
thf(fact_6886_of__nat__eq__0__iff__char__dvd,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( semiring_1_of_nat @ A @ N )
= ( zero_zero @ A ) )
= ( dvd_dvd @ nat @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) @ N ) ) ) ).
% of_nat_eq_0_iff_char_dvd
thf(fact_6887_bit__of__int__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( ring_1_of_int @ A @ K ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ) ).
% bit_of_int_iff
thf(fact_6888_bit__of__nat__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( semiring_1_of_nat @ A @ M ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ nat @ M @ N ) ) ) ) ).
% bit_of_nat_iff
thf(fact_6889_bit__signed__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4674362597316999326ke_bit @ A @ M @ A2 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( ord_min @ nat @ M @ N ) ) ) ) ) ).
% bit_signed_take_bit_iff
thf(fact_6890_bit__minus__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% bit_minus_iff
thf(fact_6891_CHAR__eq0__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) )
= ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( semiring_1_of_nat @ A @ N3 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_eq0_iff
thf(fact_6892_CHAR__eq__posI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( ( ( semiring_1_of_nat @ A @ C2 )
= ( zero_zero @ A ) )
=> ( ! [X4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ X4 )
=> ( ( ord_less @ nat @ X4 @ C2 )
=> ( ( semiring_1_of_nat @ A @ X4 )
!= ( zero_zero @ A ) ) ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C2 ) ) ) ) ) ).
% CHAR_eq_posI
thf(fact_6893_CHAR__pos__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( ( semiring_1_of_nat @ A @ N3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_pos_iff
thf(fact_6894_bit__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4730199178511100633sh_bit @ A @ M @ A2 ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% bit_push_bit_iff
thf(fact_6895_fold__possible__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
= ( ~ ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% fold_possible_bit
thf(fact_6896_bit__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( M = N ) ) ) ) ).
% bit_exp_iff
thf(fact_6897_bit__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ) ).
% bit_2_iff
thf(fact_6898_bit__not__exp__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( N != M ) ) ) ) ).
% bit_not_exp_iff
thf(fact_6899_bit__minus__exp__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% bit_minus_exp_iff
thf(fact_6900_bit__mask__sub__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ N @ M ) ) ) ) ).
% bit_mask_sub_iff
thf(fact_6901_bit__double__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
& ( N
!= ( zero_zero @ nat ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% bit_double_iff
thf(fact_6902_Code__Target__Nat_ONat__def,axiom,
( code_Target_Nat
= ( map_fun @ code_integer @ int @ nat @ nat @ code_int_of_integer @ ( id @ nat ) @ nat2 ) ) ).
% Code_Target_Nat.Nat_def
thf(fact_6903_sum__list__upt,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups8242544230860333062m_list @ nat @ ( upt @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum_list_upt
thf(fact_6904_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( hd @ nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_6905_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( upt @ I @ J )
= ( nil @ nat ) ) ) ).
% upt_conv_Nil
thf(fact_6906_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_6907_take__upt,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ M ) @ N )
=> ( ( take @ nat @ M @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus @ nat @ I @ M ) ) ) ) ).
% take_upt
thf(fact_6908_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_6909_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_6910_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus @ nat @ J @ K ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_6911_atLeast__upt,axiom,
( ( set_ord_lessThan @ nat )
= ( ^ [N3: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% atLeast_upt
thf(fact_6912_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% upt_0
thf(fact_6913_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N3: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) ) ) ) ).
% atMost_upto
thf(fact_6914_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_6915_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_6916_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( nil @ nat ) ) ) ) ).
% upt_Suc
thf(fact_6917_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) ) ).
% upt_Suc_append
thf(fact_6918_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X3: nat,Xs2: list @ nat] :
( ( ( upt @ I @ J )
= ( cons @ nat @ X3 @ Xs2 ) )
= ( ( ord_less @ nat @ I @ J )
& ( I = X3 )
& ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
= Xs2 ) ) ) ).
% upt_eq_Cons_conv
thf(fact_6919_nat__of__integer__def,axiom,
( code_nat_of_integer
= ( map_fun @ code_integer @ int @ nat @ nat @ code_int_of_integer @ ( id @ nat ) @ nat2 ) ) ).
% nat_of_integer_def
thf(fact_6920_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_6921_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys ) @ ( listrel @ A @ B @ R2 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ N3 ) @ ( nth @ B @ Ys @ N3 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_6922_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add @ A )
=> ! [Xs2: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X: B] : ( zero_zero @ A )
@ Xs2 ) )
= ( zero_zero @ A ) ) ) ).
% sum_list_0
thf(fact_6923_nth__map,axiom,
! [B: $tType,A: $tType,N: nat,Xs2: list @ A,F2: A > B] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ B @ ( map @ A @ B @ F2 @ Xs2 ) @ N )
= ( F2 @ ( nth @ A @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_6924_sum__list__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [Xs2: list @ A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( groups8242544230860333062m_list @ A @ Xs2 ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ A @ A @ ( abs_abs @ A ) @ Xs2 ) ) ) ) ).
% sum_list_abs
thf(fact_6925_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( ordere6658533253407199908up_add @ B ) )
=> ! [Xs2: list @ A,F2: A > B,G: A > B] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F2 @ Xs2 ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs2 ) ) ) ) ) ).
% sum_list_mono
thf(fact_6926_map__replicate__trivial,axiom,
! [A: $tType,X3: A,I: nat] :
( ( map @ nat @ A
@ ^ [I4: nat] : X3
@ ( upt @ ( zero_zero @ nat ) @ I ) )
= ( replicate @ A @ I @ X3 ) ) ).
% map_replicate_trivial
thf(fact_6927_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( strict9044650504122735259up_add @ B ) )
=> ! [Xs2: list @ A,F2: A > B,G: A > B] :
( ( Xs2
!= ( nil @ A ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F2 @ Xs2 ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs2 ) ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_6928_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_6929_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [R2: B,Xs2: list @ C] :
( ( groups8242544230860333062m_list @ B
@ ( map @ C @ B
@ ^ [X: C] : R2
@ Xs2 ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs2 ) ) @ R2 ) ) ) ).
% sum_list_triv
thf(fact_6930_map__nth,axiom,
! [A: $tType,Xs2: list @ A] :
( ( map @ nat @ A @ ( nth @ A @ Xs2 ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) )
= Xs2 ) ).
% map_nth
thf(fact_6931_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_6932_map__upt__eqI,axiom,
! [A: $tType,Xs2: list @ A,N: nat,M: nat,F2: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( minus_minus @ nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I3 )
= ( F2 @ ( plus_plus @ nat @ M @ I3 ) ) ) )
=> ( ( map @ nat @ A @ F2 @ ( upt @ M @ N ) )
= Xs2 ) ) ) ).
% map_upt_eqI
thf(fact_6933_transpose__rectangle,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),N: nat] :
( ( ( Xs2
= ( nil @ ( list @ A ) ) )
=> ( N
= ( zero_zero @ nat ) ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs2 @ I3 ) )
= N ) )
=> ( ( transpose @ A @ Xs2 )
= ( map @ nat @ ( list @ A )
@ ^ [I4: nat] :
( map @ nat @ A
@ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs2 @ J3 ) @ I4 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_6934_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [F2: nat > B,Ns: list @ nat] :
( ! [X4: nat,Y4: nat] :
( ( ord_less_eq @ nat @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ nat @ B @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ nat ) @ Ns ) ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F2 @ Ns ) ) ) ) ) ) ).
% sorted_wrt_less_sum_mono_lowerbound
thf(fact_6935_sorted__distinct__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ( distinct @ A @ Xs2 )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
=> ( ( distinct @ A @ Ys )
=> ( ( ( set2 @ A @ Xs2 )
= ( set2 @ A @ Ys ) )
=> ( Xs2 = Ys ) ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_6936_strict__sorted__simps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less @ A ) @ ( nil @ A ) ) ) ).
% strict_sorted_simps(1)
thf(fact_6937_sorted0,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ).
% sorted0
thf(fact_6938_sorted1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% sorted1
thf(fact_6939_sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X3 @ Ys ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X3 @ X ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys ) ) ) ) ).
% sorted_simps(2)
thf(fact_6940_strict__sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ ( cons @ A @ X3 @ Ys ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Ys ) )
=> ( ord_less @ A @ X3 @ X ) )
& ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys ) ) ) ) ).
% strict_sorted_simps(2)
thf(fact_6941_sorted__take,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,N: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( take @ A @ N @ Xs2 ) ) ) ) ).
% sorted_take
thf(fact_6942_sorted__map,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs2: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs2 ) )
= ( sorted_wrt @ B
@ ^ [X: B,Y: B] : ( ord_less_eq @ A @ ( F2 @ X ) @ ( F2 @ Y ) )
@ Xs2 ) ) ) ).
% sorted_map
thf(fact_6943_sorted__enumerate,axiom,
! [A: $tType,N: nat,Xs2: list @ A] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs2 ) ) ) ).
% sorted_enumerate
thf(fact_6944_strict__sorted__equal,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs2 )
=> ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys )
=> ( ( ( set2 @ A @ Ys )
= ( set2 @ A @ Xs2 ) )
=> ( Ys = Xs2 ) ) ) ) ) ).
% strict_sorted_equal
thf(fact_6945_sorted__append,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs2 @ Ys ) )
= ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
& ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ! [Y: A] :
( ( member @ A @ Y @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ).
% sorted_append
thf(fact_6946_sorted__wrt__nth__less,axiom,
! [A: $tType,P: A > A > $o,Xs2: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ P @ Xs2 )
=> ( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ I ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_6947_sorted__wrt__iff__nth__less,axiom,
! [A: $tType] :
( ( sorted_wrt @ A )
= ( ^ [P3: A > A > $o,Xs: list @ A] :
! [I4: nat,J3: nat] :
( ( ord_less @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P3 @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_6948_sorted__wrt01,axiom,
! [A: $tType,Xs2: list @ A,P: A > A > $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ P @ Xs2 ) ) ).
% sorted_wrt01
thf(fact_6949_sorted__replicate,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N: nat,X3: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( replicate @ A @ N @ X3 ) ) ) ).
% sorted_replicate
thf(fact_6950_sorted__drop,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,N: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( drop @ A @ N @ Xs2 ) ) ) ) ).
% sorted_drop
thf(fact_6951_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_6952_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linord4507533701916653071of_set @ A @ A4 ) ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_6953_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] : ( sorted_wrt @ A @ ( ord_less @ A ) @ ( linord4507533701916653071of_set @ A @ A4 ) ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_6954_strict__sorted__imp__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 ) ) ) ).
% strict_sorted_imp_sorted
thf(fact_6955_sorted__remdups__adj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remdups_adj @ A @ Xs2 ) ) ) ) ).
% sorted_remdups_adj
thf(fact_6956_sorted__nths,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,I6: set @ nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nths @ A @ Xs2 @ I6 ) ) ) ) ).
% sorted_nths
thf(fact_6957_sorted2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A,Zs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs2 ) ) )
= ( ( ord_less_eq @ A @ X3 @ Y3 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) ) ).
% sorted2
thf(fact_6958_sorted__tl,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( tl @ A @ Xs2 ) ) ) ) ).
% sorted_tl
thf(fact_6959_sorted__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( upt @ M @ N ) ) ).
% sorted_upt
thf(fact_6960_sorted__wrt__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less @ nat ) @ ( upt @ M @ N ) ) ).
% sorted_wrt_upt
thf(fact_6961_sorted__butlast,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( butlast @ A @ Xs2 ) ) ) ) ) ).
% sorted_butlast
thf(fact_6962_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_6963_sorted01,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 ) ) ) ).
% sorted01
thf(fact_6964_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
= ( ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I4 ) @ ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_6965_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ? [X4: list @ A] :
( ( ( set2 @ A @ X4 )
= A4 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ X4 )
& ( distinct @ A @ X4 )
& ! [Y5: list @ A] :
( ( ( ( set2 @ A @ Y5 )
= A4 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Y5 )
& ( distinct @ A @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_6966_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ( distinct @ A @ Xs2 )
=> ( ( linord4507533701916653071of_set @ A @ ( set2 @ A @ Xs2 ) )
= Xs2 ) ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_6967_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_6968_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
= ( ! [I4: nat] :
( ( ord_less @ nat @ ( suc @ I4 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I4 ) @ ( nth @ A @ Xs2 @ ( suc @ I4 ) ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_6969_sorted__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_6970_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_eq @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I4 ) @ ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_6971_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ~ ! [L4: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ L4 )
=> ( ( ( set2 @ A @ L4 )
= A4 )
=> ( ( size_size @ ( list @ A ) @ L4 )
!= ( finite_card @ A @ A4 ) ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_6972_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_6973_sorted__find__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ? [X6: A] :
( ( member @ A @ X6 @ ( set2 @ A @ Xs2 ) )
& ( P @ X6 ) )
=> ( ( find @ A @ P @ Xs2 )
= ( some @ A
@ ( lattic643756798350308766er_Min @ A
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
& ( P @ X ) ) ) ) ) ) ) ) ) ).
% sorted_find_Min
thf(fact_6974_map__sorted__distinct__set__unique,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs2: list @ B,Ys: list @ B] :
( ( inj_on @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs2 ) @ ( set2 @ B @ Ys ) ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs2 ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F2 @ Xs2 ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Ys ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F2 @ Ys ) )
=> ( ( ( set2 @ B @ Xs2 )
= ( set2 @ B @ Ys ) )
=> ( Xs2 = Ys ) ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_6975_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,L: list @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
& ( ( set2 @ A @ L )
= A4 )
& ( ( size_size @ ( list @ A ) @ L )
= ( finite_card @ A @ A4 ) ) )
= ( ( linord4507533701916653071of_set @ A @ A4 )
= L ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_6976_length__transpose__sorted,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs2 ) ) )
=> ( ( ( Xs2
= ( nil @ ( list @ A ) ) )
=> ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) )
= ( zero_zero @ nat ) ) )
& ( ( Xs2
!= ( nil @ ( list @ A ) ) )
=> ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs2 @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% length_transpose_sorted
thf(fact_6977_length__transpose,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) )
= ( foldr @ ( list @ A ) @ nat
@ ^ [Xs: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs ) )
@ Xs2
@ ( zero_zero @ nat ) ) ) ).
% length_transpose
thf(fact_6978_sorted__upto,axiom,
! [M: int,N: int] : ( sorted_wrt @ int @ ( ord_less_eq @ int ) @ ( upto @ M @ N ) ) ).
% sorted_upto
thf(fact_6979_sorted__wrt__upto,axiom,
! [I: int,J: int] : ( sorted_wrt @ int @ ( ord_less @ int ) @ ( upto @ I @ J ) ) ).
% sorted_wrt_upto
thf(fact_6980_foldr__max__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,Y3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs2 ) )
=> ( ( ( Xs2
= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs2 @ Y3 )
= Y3 ) )
& ( ( Xs2
!= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs2 @ Y3 )
= ( ord_max @ A @ ( nth @ A @ Xs2 @ ( zero_zero @ nat ) ) @ Y3 ) ) ) ) ) ) ).
% foldr_max_sorted
thf(fact_6981_sum__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( ^ [Xs: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs @ ( zero_zero @ A ) ) ) ) ) ).
% sum_list.eq_foldr
thf(fact_6982_sorted__transpose,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) ) ) ) ).
% sorted_transpose
thf(fact_6983_rev__nth,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( rev @ A @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_6984_rev__update,axiom,
! [A: $tType,K: nat,Xs2: list @ A,Y3: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( rev @ A @ ( list_update @ A @ Xs2 @ K @ Y3 ) )
= ( list_update @ A @ ( rev @ A @ Xs2 ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ K ) @ ( one_one @ nat ) ) @ Y3 ) ) ) ).
% rev_update
thf(fact_6985_sorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs2 ) )
= ( ! [I4: nat] :
( ( ord_less @ nat @ ( suc @ I4 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ ( suc @ I4 ) ) @ ( nth @ A @ Xs2 @ I4 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_6986_sorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs2 ) )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_eq @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ J3 ) @ ( nth @ A @ Xs2 @ I4 ) ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_6987_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs2 ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ J ) @ ( nth @ A @ Xs2 @ I ) ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_6988_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F3: B > A,A5: A,Xs: list @ B] :
( foldr @ B @ A
@ ^ [X: B,B4: A] : ( plus_plus @ A @ ( F3 @ X ) @ ( times_times @ A @ A5 @ B4 ) )
@ Xs
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_6989_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),I: nat,J: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs2 ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) ) )
=> ( ( ord_less @ nat @ J
@ ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs2 ) ) )
=> ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs2 ) @ I ) @ J )
= ( nth @ A @ ( nth @ ( list @ A ) @ Xs2 @ J ) @ I ) ) ) ) ) ).
% nth_nth_transpose_sorted
thf(fact_6990_transpose__column,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs2 ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) )
=> ( ( map @ ( list @ A ) @ A
@ ^ [Ys3: list @ A] : ( nth @ A @ Ys3 @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs2 ) ) )
= ( nth @ ( list @ A ) @ Xs2 @ I ) ) ) ) ).
% transpose_column
thf(fact_6991_sorted__filter,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs2: list @ B,P: B > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs2 ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs2 ) ) ) ) ) ).
% sorted_filter
thf(fact_6992_sorted__same,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [G: ( list @ A ) > A,Xs2: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( filter2 @ A
@ ^ [X: A] :
( X
= ( G @ Xs2 ) )
@ Xs2 ) ) ) ).
% sorted_same
thf(fact_6993_length__filter__less,axiom,
! [A: $tType,X3: A,Xs2: list @ A,P: A > $o] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ~ ( P @ X3 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% length_filter_less
thf(fact_6994_length__filter__le,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_filter_le
thf(fact_6995_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,G: ( list @ B ) > A,Xs2: list @ B] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( map @ B @ A @ F2
@ ( filter2 @ B
@ ^ [X: B] :
( ( F2 @ X )
= ( G @ Xs2 ) )
@ Xs2 ) ) ) ) ).
% sorted_map_same
thf(fact_6996_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [F2: B > A,P: B > $o,Xs2: list @ B] :
( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs2 ) ) )
= ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( F2 @ X ) @ ( zero_zero @ A ) )
@ Xs2 ) ) ) ) ).
% sum_list_map_filter'
thf(fact_6997_sum__list__filter__le__nat,axiom,
! [A: $tType,F2: A > nat,P: A > $o,Xs2: list @ A] : ( ord_less_eq @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ ( filter2 @ A @ P @ Xs2 ) ) ) @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs2 ) ) ) ).
% sum_list_filter_le_nat
thf(fact_6998_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [Xs2: list @ B,P: B > $o,F2: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Xs2 ) )
=> ( ~ ( P @ X4 )
=> ( ( F2 @ X4 )
= ( zero_zero @ A ) ) ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs2 ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs2 ) ) ) ) ) ).
% sum_list_map_filter
thf(fact_6999_filter__eq__nths,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P3: A > $o,Xs: list @ A] :
( nths @ A @ Xs
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P3 @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_7000_length__filter__conv__card,axiom,
! [A: $tType,P6: A > $o,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P6 @ Xs2 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P6 @ ( nth @ A @ Xs2 @ I4 ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_7001_transpose__aux__max,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Xss: list @ ( list @ B )] :
( ( ord_max @ nat @ ( suc @ ( size_size @ ( list @ A ) @ Xs2 ) )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [Xs: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs ) )
@ Xss
@ ( zero_zero @ nat ) ) )
= ( suc
@ ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs2 )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [X: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( filter2 @ ( list @ B )
@ ^ [Ys3: list @ B] :
( Ys3
!= ( nil @ B ) )
@ Xss )
@ ( zero_zero @ nat ) ) ) ) ) ).
% transpose_aux_max
thf(fact_7002_nth__transpose,axiom,
! [A: $tType,I: nat,Xs2: list @ ( list @ A )] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) ) )
=> ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs2 ) @ I )
= ( map @ ( list @ A ) @ A
@ ^ [Xs: list @ A] : ( nth @ A @ Xs @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs2 ) ) ) ) ).
% nth_transpose
thf(fact_7003_transpose__max__length,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( foldr @ ( list @ A ) @ nat
@ ^ [Xs: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs ) )
@ ( transpose @ A @ Xs2 )
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [X: list @ A] :
( X
!= ( nil @ A ) )
@ Xs2 ) ) ) ).
% transpose_max_length
thf(fact_7004_transpose__column__length,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs2 ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) )
=> ( ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs2 ) ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs2 @ I ) ) ) ) ) ).
% transpose_column_length
thf(fact_7005_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F2: B > A,P: B > $o,Xs2: list @ B] :
( ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs2 ) )
= ( map_filter @ B @ A
@ ^ [X: B] : ( if @ ( option @ A ) @ ( P @ X ) @ ( some @ A @ ( F2 @ X ) ) @ ( none @ A ) )
@ Xs2 ) ) ).
% map_filter_map_filter
thf(fact_7006_transpose__transpose,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs2 ) ) )
=> ( ( transpose @ A @ ( transpose @ A @ Xs2 ) )
= ( takeWhile @ ( list @ A )
@ ^ [X: list @ A] :
( X
!= ( nil @ A ) )
@ Xs2 ) ) ) ).
% transpose_transpose
thf(fact_7007_sorted__takeWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( takeWhile @ A @ P @ Xs2 ) ) ) ) ).
% sorted_takeWhile
thf(fact_7008_length__takeWhile__le,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_takeWhile_le
thf(fact_7009_nth__length__takeWhile,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ~ ( P @ ( nth @ A @ Xs2 @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs2 ) ) ) ) ) ).
% nth_length_takeWhile
thf(fact_7010_takeWhile__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs2: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs2 ) ) )
=> ( ( nth @ A @ ( takeWhile @ A @ P @ Xs2 ) @ J )
= ( nth @ A @ Xs2 @ J ) ) ) ).
% takeWhile_nth
thf(fact_7011_length__takeWhile__less__P__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs2: list @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J )
=> ( P @ ( nth @ A @ Xs2 @ I3 ) ) )
=> ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs2 ) ) ) ) ) ).
% length_takeWhile_less_P_nth
thf(fact_7012_takeWhile__eq__take__P__nth,axiom,
! [A: $tType,N: nat,Xs2: list @ A,P: A > $o] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ I3 ) ) ) )
=> ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ~ ( P @ ( nth @ A @ Xs2 @ N ) ) )
=> ( ( takeWhile @ A @ P @ Xs2 )
= ( take @ A @ N @ Xs2 ) ) ) ) ).
% takeWhile_eq_take_P_nth
thf(fact_7013_nths__shift__lemma,axiom,
! [A: $tType,A4: set @ nat,Xs2: list @ A,I: nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P5: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P5 ) @ A4 )
@ ( zip @ A @ nat @ Xs2 @ ( upt @ I @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P5: product_prod @ A @ nat] : ( member @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P5 ) @ I ) @ A4 )
@ ( zip @ A @ nat @ Xs2 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% nths_shift_lemma
thf(fact_7014_nths__def,axiom,
! [A: $tType] :
( ( nths @ A )
= ( ^ [Xs: list @ A,A9: set @ nat] :
( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P5: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P5 ) @ A9 )
@ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ) ).
% nths_def
thf(fact_7015_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs2: list @ B,T2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F2 @ Xs2 ) ) )
=> ( ( filter2 @ B
@ ^ [X: B] : ( ord_less @ A @ T2 @ ( F2 @ X ) )
@ Xs2 )
= ( takeWhile @ B
@ ^ [X: B] : ( ord_less @ A @ T2 @ ( F2 @ X ) )
@ Xs2 ) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
thf(fact_7016_sorted__insort__is__snoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,A2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ X4 @ A2 ) )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ A2
@ Xs2 )
= ( append @ A @ Xs2 @ ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_insort_is_snoc
thf(fact_7017_count__notin,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( count_list @ A @ Xs2 @ X3 )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_7018_insort__key_Osimps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X3: B,Y3: B,Ys: list @ B] :
( ( ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
=> ( ( linorder_insort_key @ B @ A @ F2 @ X3 @ ( cons @ B @ Y3 @ Ys ) )
= ( cons @ B @ X3 @ ( cons @ B @ Y3 @ Ys ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
=> ( ( linorder_insort_key @ B @ A @ F2 @ X3 @ ( cons @ B @ Y3 @ Ys ) )
= ( cons @ B @ Y3 @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Ys ) ) ) ) ) ) ).
% insort_key.simps(2)
thf(fact_7019_sorted__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X3
@ Xs2 ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 ) ) ) ).
% sorted_insort
thf(fact_7020_insort__is__Cons,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ B,F2: B > A,A2: B] :
( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Xs2 ) )
=> ( ord_less_eq @ A @ ( F2 @ A2 ) @ ( F2 @ X4 ) ) )
=> ( ( linorder_insort_key @ B @ A @ F2 @ A2 @ Xs2 )
= ( cons @ B @ A2 @ Xs2 ) ) ) ) ).
% insort_is_Cons
thf(fact_7021_sorted__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X3: B,Xs2: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs2 ) ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs2 ) ) ) ) ).
% sorted_insort_key
thf(fact_7022_count__list_Osimps_I1_J,axiom,
! [A: $tType,Y3: A] :
( ( count_list @ A @ ( nil @ A ) @ Y3 )
= ( zero_zero @ nat ) ) ).
% count_list.simps(1)
thf(fact_7023_count__le__length,axiom,
! [A: $tType,Xs2: list @ A,X3: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs2 @ X3 ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% count_le_length
thf(fact_7024_filter__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs2: list @ B,P: B > $o,X3: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs2 ) )
=> ( ( P @ X3 )
=> ( ( filter2 @ B @ P @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs2 ) )
= ( linorder_insort_key @ B @ A @ F2 @ X3 @ ( filter2 @ B @ P @ Xs2 ) ) ) ) ) ) ).
% filter_insort
thf(fact_7025_insort__key__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [A2: B,Xs2: list @ B,F2: B > A] :
( ( member @ B @ A2 @ ( set2 @ B @ Xs2 ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs2 ) )
=> ( ( ( hd @ B
@ ( filter2 @ B
@ ^ [X: B] :
( ( F2 @ A2 )
= ( F2 @ X ) )
@ Xs2 ) )
= A2 )
=> ( ( linorder_insort_key @ B @ A @ F2 @ A2 @ ( remove1 @ B @ A2 @ Xs2 ) )
= Xs2 ) ) ) ) ) ).
% insort_key_remove1
thf(fact_7026_num__of__integer__def,axiom,
( code_num_of_integer
= ( map_fun @ code_integer @ int @ num @ num @ code_int_of_integer @ ( id @ num ) @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 ) ) ) ).
% num_of_integer_def
thf(fact_7027_sorted__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,A2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remove1 @ A @ A2 @ Xs2 ) ) ) ) ).
% sorted_remove1
thf(fact_7028_sorted__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs2: list @ B,X3: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs2 ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ ( remove1 @ B @ X3 @ Xs2 ) ) ) ) ) ).
% sorted_map_remove1
thf(fact_7029_insort__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,Xs2: list @ A] :
( ( member @ A @ A2 @ ( set2 @ A @ Xs2 ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ A2
@ ( remove1 @ A @ A2 @ Xs2 ) )
= Xs2 ) ) ) ) ).
% insort_remove1
thf(fact_7030_last__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( last @ nat @ ( upt @ I @ J ) )
= ( minus_minus @ nat @ J @ ( one_one @ nat ) ) ) ) ).
% last_upt
thf(fact_7031_arg__min__inj__eq,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F2: A > B,P: A > $o,A2: A] :
( ( inj_on @ A @ B @ F2 @ ( collect @ A @ P ) )
=> ( ( P @ A2 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ A2 ) @ ( F2 @ Y4 ) ) )
=> ( ( lattices_ord_arg_min @ A @ B @ F2 @ P )
= A2 ) ) ) ) ) ).
% arg_min_inj_eq
thf(fact_7032_last__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( replicate @ A @ N @ X3 ) )
= X3 ) ) ).
% last_replicate
thf(fact_7033_last__drop,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( last @ A @ ( drop @ A @ N @ Xs2 ) )
= ( last @ A @ Xs2 ) ) ) ).
% last_drop
thf(fact_7034_arg__minI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [P: A > $o,X3: A,F2: A > B,Q: A > $o] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ~ ( ord_less @ B @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ~ ( ord_less @ B @ ( F2 @ Y5 ) @ ( F2 @ X4 ) ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( lattices_ord_arg_min @ A @ B @ F2 @ P ) ) ) ) ) ) ).
% arg_minI
thf(fact_7035_arg__min__nat__le,axiom,
! [A: $tType,P: A > $o,X3: A,M: A > nat] :
( ( P @ X3 )
=> ( ord_less_eq @ nat @ ( M @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) @ ( M @ X3 ) ) ) ).
% arg_min_nat_le
thf(fact_7036_arg__min__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ( ( P @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) @ ( M @ Y5 ) ) ) ) ) ).
% arg_min_nat_lemma
thf(fact_7037_arg__min__equality,axiom,
! [A: $tType,C: $tType] :
( ( order @ A )
=> ! [P: C > $o,K: C,F2: C > A] :
( ( P @ K )
=> ( ! [X4: C] :
( ( P @ X4 )
=> ( ord_less_eq @ A @ ( F2 @ K ) @ ( F2 @ X4 ) ) )
=> ( ( F2 @ ( lattices_ord_arg_min @ C @ A @ F2 @ P ) )
= ( F2 @ K ) ) ) ) ) ).
% arg_min_equality
thf(fact_7038_arg__min__on__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic7623131987881927897min_on @ B @ A )
= ( ^ [F3: B > A,S8: set @ B] :
( lattices_ord_arg_min @ B @ A @ F3
@ ^ [X: B] : ( member @ B @ X @ S8 ) ) ) ) ) ).
% arg_min_on_def
thf(fact_7039_arg__min__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattices_ord_arg_min @ B @ A )
= ( ^ [F3: B > A,P3: B > $o] : ( fChoice @ B @ ( lattic501386751177426532rg_min @ B @ A @ F3 @ P3 ) ) ) ) ) ).
% arg_min_def
thf(fact_7040_in__measures_I2_J,axiom,
! [A: $tType,X3: A,Y3: A,F2: A > nat,Fs: list @ ( A > nat )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F2 @ Fs ) ) )
= ( ( ord_less @ nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
| ( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_7041_is__arg__min__arg__min__nat,axiom,
! [A: $tType,P: A > $o,X3: A,M: A > nat] :
( ( P @ X3 )
=> ( lattic501386751177426532rg_min @ A @ nat @ M @ P @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) ) ).
% is_arg_min_arg_min_nat
thf(fact_7042_arg__min__natI,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ( P @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) ) ).
% arg_min_natI
thf(fact_7043_is__arg__min__antimono,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F2: A > B,P: A > $o,X3: A,Y3: A] :
( ( lattic501386751177426532rg_min @ A @ B @ F2 @ P @ X3 )
=> ( ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) )
=> ( ( P @ Y3 )
=> ( lattic501386751177426532rg_min @ A @ B @ F2 @ P @ Y3 ) ) ) ) ) ).
% is_arg_min_antimono
thf(fact_7044_is__arg__min__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ( ( lattic501386751177426532rg_min @ A @ B )
= ( ^ [F3: A > B,P3: A > $o,X: A] :
( ( P3 @ X )
& ! [Y: A] :
( ( P3 @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ) ) ).
% is_arg_min_linorder
thf(fact_7045_is__arg__min__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic501386751177426532rg_min @ B @ A )
= ( ^ [F3: B > A,P3: B > $o,X: B] :
( ( P3 @ X )
& ~ ? [Y: B] :
( ( P3 @ Y )
& ( ord_less @ A @ ( F3 @ Y ) @ ( F3 @ X ) ) ) ) ) ) ) ).
% is_arg_min_def
thf(fact_7046_measures__less,axiom,
! [A: $tType,F2: A > nat,X3: A,Y3: A,Fs: list @ ( A > nat )] :
( ( ord_less @ nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F2 @ Fs ) ) ) ) ).
% measures_less
thf(fact_7047_measures__lesseq,axiom,
! [A: $tType,F2: A > nat,X3: A,Y3: A,Fs: list @ ( A > nat )] :
( ( ord_less_eq @ nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( measures @ A @ Fs ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F2 @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_7048_ex__is__arg__min__if__finite,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S2: set @ A,F2: A > B] :
( ( finite_finite @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_1: A] :
( lattic501386751177426532rg_min @ A @ B @ F2
@ ^ [X: A] : ( member @ A @ X @ S2 )
@ X_1 ) ) ) ) ).
% ex_is_arg_min_if_finite
thf(fact_7049_nhds__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo7230453075368039082e_nhds @ A )
= ( ^ [X: A] :
( complete_Inf_Inf @ ( filter @ A )
@ ( image @ real @ ( filter @ A )
@ ^ [E3: real] :
( principal @ A
@ ( collect @ A
@ ^ [Y: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X ) @ E3 ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ).
% nhds_metric
thf(fact_7050_map__of__eq__Some__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),X3: A,Y3: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( map_of @ A @ B @ Xys @ X3 )
= ( some @ B @ Y3 ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% map_of_eq_Some_iff
thf(fact_7051_map__of__is__SomeI,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ A @ B ),X3: A,Y3: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) )
=> ( ( map_of @ A @ B @ Xys @ X3 )
= ( some @ B @ Y3 ) ) ) ) ).
% map_of_is_SomeI
thf(fact_7052_Some__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),Y3: B,X3: A] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( some @ B @ Y3 )
= ( map_of @ A @ B @ Xys @ X3 ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% Some_eq_map_of_iff
thf(fact_7053_map__of__Cons__code_I2_J,axiom,
! [C: $tType,B: $tType,L: B,K: B,V2: C,Ps: list @ ( product_prod @ B @ C )] :
( ( ( L = K )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V2 ) @ Ps ) @ K )
= ( some @ C @ V2 ) ) )
& ( ( L != K )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V2 ) @ Ps ) @ K )
= ( map_of @ B @ C @ Ps @ K ) ) ) ) ).
% map_of_Cons_code(2)
thf(fact_7054_map__of__SomeD,axiom,
! [A: $tType,B: $tType,Xs2: list @ ( product_prod @ B @ A ),K: B,Y3: A] :
( ( ( map_of @ B @ A @ Xs2 @ K )
= ( some @ A @ Y3 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ Y3 ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs2 ) ) ) ).
% map_of_SomeD
thf(fact_7055_weak__map__of__SomeI,axiom,
! [A: $tType,B: $tType,K: A,X3: B,L: list @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ X3 ) @ ( set2 @ ( product_prod @ A @ B ) @ L ) )
=> ? [X4: B] :
( ( map_of @ A @ B @ L @ K )
= ( some @ B @ X4 ) ) ) ).
% weak_map_of_SomeI
thf(fact_7056_map__of__filter__in,axiom,
! [B: $tType,A: $tType,Xs2: list @ ( product_prod @ B @ A ),K: B,Z: A,P: B > A > $o] :
( ( ( map_of @ B @ A @ Xs2 @ K )
= ( some @ A @ Z ) )
=> ( ( P @ K @ Z )
=> ( ( map_of @ B @ A @ ( filter2 @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) @ Xs2 ) @ K )
= ( some @ A @ Z ) ) ) ) ).
% map_of_filter_in
thf(fact_7057_map__of__zip__is__Some,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,X3: A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
= ( ? [Y: B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys ) @ X3 )
= ( some @ B @ Y ) ) ) ) ) ).
% map_of_zip_is_Some
thf(fact_7058_map__of__zip__map,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,F2: A > B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ ( map @ A @ B @ F2 @ Xs2 ) ) )
= ( ^ [X: A] : ( if @ ( option @ B ) @ ( member @ A @ X @ ( set2 @ A @ Xs2 ) ) @ ( some @ B @ ( F2 @ X ) ) @ ( none @ B ) ) ) ) ).
% map_of_zip_map
thf(fact_7059_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,T2: list @ ( product_prod @ A @ C ),K: A,X3: C] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map_of @ A @ C @ T2 @ K )
= ( some @ C @ X3 ) )
=> ( ( map_of @ B @ C
@ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C )
@ ( product_case_prod @ A @ C @ ( product_prod @ B @ C )
@ ^ [K3: A] : ( product_Pair @ B @ C @ ( F2 @ K3 ) ) )
@ T2 )
@ ( F2 @ K ) )
= ( some @ C @ X3 ) ) ) ) ).
% map_of_mapk_SomeI
thf(fact_7060_at__left__eq,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less @ A @ Y3 @ X3 )
=> ( ( topolo174197925503356063within @ A @ X3 @ ( set_ord_lessThan @ A @ X3 ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( set_or5935395276787703475ssThan @ A @ A5 @ X3 ) )
@ ( set_ord_lessThan @ A @ X3 ) ) ) ) ) ) ).
% at_left_eq
thf(fact_7061_at__right__eq,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( topolo174197925503356063within @ A @ X3 @ ( set_ord_greaterThan @ A @ X3 ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( set_or5935395276787703475ssThan @ A @ X3 @ A5 ) )
@ ( set_ord_greaterThan @ A @ X3 ) ) ) ) ) ) ).
% at_right_eq
thf(fact_7062_map__of__zip__nth,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,I: nat] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( distinct @ A @ Xs2 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys ) @ ( nth @ A @ Xs2 @ I ) )
= ( some @ B @ ( nth @ B @ Ys @ I ) ) ) ) ) ) ).
% map_of_zip_nth
thf(fact_7063_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
@ ^ [X: A] : ( ord_less_eq @ real @ R5 @ ( real_V7770717601297561774m_norm @ A @ X ) ) ) )
@ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% at_infinity_def
thf(fact_7064_set__map__of__compr,axiom,
! [B: $tType,A: $tType,Xs2: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs2 ) )
=> ( ( set2 @ ( product_prod @ A @ B ) @ Xs2 )
= ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K3: A,V5: B] :
( ( map_of @ A @ B @ Xs2 @ K3 )
= ( some @ B @ V5 ) ) ) ) ) ) ).
% set_map_of_compr
thf(fact_7065_uniformity__dist,axiom,
! [A: $tType] :
( ( real_V768167426530841204y_dist @ A )
=> ( ( topolo7806501430040627800ormity @ A )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ A @ A ) )
@ ( image @ real @ ( filter @ ( product_prod @ A @ A ) )
@ ^ [E3: real] :
( principal @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X: A,Y: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E3 ) ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ).
% uniformity_dist
thf(fact_7066_map__of__map__restrict,axiom,
! [B: $tType,A: $tType,F2: A > B,Ks: list @ A] :
( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K3: A] : ( product_Pair @ A @ B @ K3 @ ( F2 @ K3 ) )
@ Ks ) )
= ( restrict_map @ A @ B @ ( comp @ B @ ( option @ B ) @ A @ ( some @ B ) @ F2 ) @ ( set2 @ A @ Ks ) ) ) ).
% map_of_map_restrict
thf(fact_7067_Cauchy__uniform__iff,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X9: nat > A] :
! [P3: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ P3 @ ( topolo7806501430040627800ormity @ A ) )
=> ? [N6: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ! [M3: nat] :
( ( ord_less_eq @ nat @ N6 @ M3 )
=> ( P3 @ ( product_Pair @ A @ A @ ( X9 @ N3 ) @ ( X9 @ M3 ) ) ) ) ) ) ) ) ) ).
% Cauchy_uniform_iff
thf(fact_7068_uniformity__real__def,axiom,
( ( topolo7806501430040627800ormity @ real )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ real @ real ) )
@ ( image @ real @ ( filter @ ( product_prod @ real @ real ) )
@ ^ [E3: real] :
( principal @ ( product_prod @ real @ real )
@ ( collect @ ( product_prod @ real @ real )
@ ( product_case_prod @ real @ real @ $o
@ ^ [X: real,Y: real] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ real @ X @ Y ) @ E3 ) ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% uniformity_real_def
thf(fact_7069_uniformity__complex__def,axiom,
( ( topolo7806501430040627800ormity @ complex )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ complex @ complex ) )
@ ( image @ real @ ( filter @ ( product_prod @ complex @ complex ) )
@ ^ [E3: real] :
( principal @ ( product_prod @ complex @ complex )
@ ( collect @ ( product_prod @ complex @ complex )
@ ( product_case_prod @ complex @ complex @ $o
@ ^ [X: complex,Y: complex] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ complex @ X @ Y ) @ E3 ) ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% uniformity_complex_def
thf(fact_7070_eventually__uniformity__metric,axiom,
! [A: $tType] :
( ( real_V768167426530841204y_dist @ A )
=> ! [P: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ P @ ( topolo7806501430040627800ormity @ A ) )
= ( ? [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
& ! [X: A,Y: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E3 )
=> ( P @ ( product_Pair @ A @ A @ X @ Y ) ) ) ) ) ) ) ).
% eventually_uniformity_metric
thf(fact_7071_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),X3: A,Y3: B] :
( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X3 @ ( some @ B @ Y3 ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( restrict_map @ A @ B @ M @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% restrict_upd_same
thf(fact_7072_zero__rat__def,axiom,
( ( zero_zero @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ) ).
% zero_rat_def
thf(fact_7073_image__map__upd,axiom,
! [B: $tType,A: $tType,X3: A,A4: set @ A,M: A > ( option @ B ),Y3: B] :
( ~ ( member @ A @ X3 @ A4 )
=> ( ( image @ A @ ( option @ B ) @ ( fun_upd @ A @ ( option @ B ) @ M @ X3 @ ( some @ B @ Y3 ) ) @ A4 )
= ( image @ A @ ( option @ B ) @ M @ A4 ) ) ) ).
% image_map_upd
thf(fact_7074_map__option__o__map__upd,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > B,M: A > ( option @ C ),A2: A,B2: C] :
( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F2 ) @ ( fun_upd @ A @ ( option @ C ) @ M @ A2 @ ( some @ C @ B2 ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F2 ) @ M ) @ A2 @ ( some @ B @ ( F2 @ B2 ) ) ) ) ).
% map_option_o_map_upd
thf(fact_7075_map__upd__eqD1,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),A2: A,X3: B,N: A > ( option @ B ),Y3: B] :
( ( ( fun_upd @ A @ ( option @ B ) @ M @ A2 @ ( some @ B @ X3 ) )
= ( fun_upd @ A @ ( option @ B ) @ N @ A2 @ ( some @ B @ Y3 ) ) )
=> ( X3 = Y3 ) ) ).
% map_upd_eqD1
thf(fact_7076_map__upd__triv,axiom,
! [A: $tType,B: $tType,T2: B > ( option @ A ),K: B,X3: A] :
( ( ( T2 @ K )
= ( some @ A @ X3 ) )
=> ( ( fun_upd @ B @ ( option @ A ) @ T2 @ K @ ( some @ A @ X3 ) )
= T2 ) ) ).
% map_upd_triv
thf(fact_7077_map__upd__Some__unfold,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),A2: B,B2: A,X3: B,Y3: A] :
( ( ( fun_upd @ B @ ( option @ A ) @ M @ A2 @ ( some @ A @ B2 ) @ X3 )
= ( some @ A @ Y3 ) )
= ( ( ( X3 = A2 )
& ( B2 = Y3 ) )
| ( ( X3 != A2 )
& ( ( M @ X3 )
= ( some @ A @ Y3 ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_7078_map__upd__nonempty,axiom,
! [B: $tType,A: $tType,T2: A > ( option @ B ),K: A,X3: B] :
( ( fun_upd @ A @ ( option @ B ) @ T2 @ K @ ( some @ B @ X3 ) )
!= ( ^ [X: A] : ( none @ B ) ) ) ).
% map_upd_nonempty
thf(fact_7079_finite__range__updI,axiom,
! [A: $tType,B: $tType,F2: B > ( option @ A ),A2: B,B2: A] :
( ( finite_finite @ ( option @ A ) @ ( image @ B @ ( option @ A ) @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite @ ( option @ A ) @ ( image @ B @ ( option @ A ) @ ( fun_upd @ B @ ( option @ A ) @ F2 @ A2 @ ( some @ A @ B2 ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_updI
thf(fact_7080_map__of__zip__upd,axiom,
! [A: $tType,B: $tType,Ys: list @ B,Xs2: list @ A,Zs2: list @ B,X3: A,Y3: B,Z: B] :
( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Zs2 )
= ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys ) ) @ X3 @ ( some @ B @ Y3 ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Zs2 ) ) @ X3 @ ( some @ B @ Z ) ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys ) )
= ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Zs2 ) ) ) ) ) ) ) ).
% map_of_zip_upd
thf(fact_7081_map__of_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,P6: product_prod @ A @ B,Ps: list @ ( product_prod @ A @ B )] :
( ( map_of @ A @ B @ ( cons @ ( product_prod @ A @ B ) @ P6 @ Ps ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Ps ) @ ( product_fst @ A @ B @ P6 ) @ ( some @ B @ ( product_snd @ A @ B @ P6 ) ) ) ) ).
% map_of.simps(2)
thf(fact_7082_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ B,M: A > ( option @ B ),X3: A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ Ys )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M @ Xs2 @ Ys ) @ X3 @ ( some @ B @ ( nth @ B @ Ys @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% map_upds_append1
thf(fact_7083_inverse__rat_Oabs__eq,axiom,
! [X3: product_prod @ int @ int] :
( ( ratrel @ X3 @ X3 )
=> ( ( inverse_inverse @ rat @ ( abs_Rat @ X3 ) )
= ( abs_Rat
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X3 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) ) ) ) ) ).
% inverse_rat.abs_eq
thf(fact_7084_map__upds__list__update2__drop,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,I: nat,M: A > ( option @ B ),Ys: list @ B,Y3: B] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ I )
=> ( ( map_upds @ A @ B @ M @ Xs2 @ ( list_update @ B @ Ys @ I @ Y3 ) )
= ( map_upds @ A @ B @ M @ Xs2 @ Ys ) ) ) ).
% map_upds_list_update2_drop
thf(fact_7085_map__upds__Cons,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),A2: A,As: list @ A,B2: B,Bs: list @ B] :
( ( map_upds @ A @ B @ M @ ( cons @ A @ A2 @ As ) @ ( cons @ B @ B2 @ Bs ) )
= ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ A2 @ ( some @ B @ B2 ) ) @ As @ Bs ) ) ).
% map_upds_Cons
thf(fact_7086_map__upds__twist,axiom,
! [A: $tType,B: $tType,A2: A,As: list @ A,M: A > ( option @ B ),B2: B,Bs: list @ B] :
( ~ ( member @ A @ A2 @ ( set2 @ A @ As ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ A2 @ ( some @ B @ B2 ) ) @ As @ Bs )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M @ As @ Bs ) @ A2 @ ( some @ B @ B2 ) ) ) ) ).
% map_upds_twist
thf(fact_7087_ratrel__iff,axiom,
( ratrel
= ( ^ [X: product_prod @ int @ int,Y: product_prod @ int @ int] :
( ( ( product_snd @ int @ int @ X )
!= ( zero_zero @ int ) )
& ( ( product_snd @ int @ int @ Y )
!= ( zero_zero @ int ) )
& ( ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ Y ) )
= ( times_times @ int @ ( product_fst @ int @ int @ Y ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ).
% ratrel_iff
thf(fact_7088_zero__rat_Orsp,axiom,
ratrel @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ).
% zero_rat.rsp
thf(fact_7089_ratrel__def,axiom,
( ratrel
= ( ^ [X: product_prod @ int @ int,Y: product_prod @ int @ int] :
( ( ( product_snd @ int @ int @ X )
!= ( zero_zero @ int ) )
& ( ( product_snd @ int @ int @ Y )
!= ( zero_zero @ int ) )
& ( ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ Y ) )
= ( times_times @ int @ ( product_fst @ int @ int @ Y ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ).
% ratrel_def
thf(fact_7090_map__upd__upds__conv__if,axiom,
! [A: $tType,B: $tType,X3: A,Ys: list @ B,Xs2: list @ A,F2: A > ( option @ B ),Y3: B] :
( ( ( member @ A @ X3 @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs2 ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X3 @ ( some @ B @ Y3 ) ) @ Xs2 @ Ys )
= ( map_upds @ A @ B @ F2 @ Xs2 @ Ys ) ) )
& ( ~ ( member @ A @ X3 @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs2 ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X3 @ ( some @ B @ Y3 ) ) @ Xs2 @ Ys )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ F2 @ Xs2 @ Ys ) @ X3 @ ( some @ B @ Y3 ) ) ) ) ) ).
% map_upd_upds_conv_if
thf(fact_7091_Rat_Opositive_Oabs__eq,axiom,
! [X3: product_prod @ int @ int] :
( ( ratrel @ X3 @ X3 )
=> ( ( positive @ ( abs_Rat @ X3 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ X3 ) ) ) ) ) ).
% Rat.positive.abs_eq
thf(fact_7092_ran__map__upd,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A2: B,B2: A] :
( ( ( M @ A2 )
= ( none @ A ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M @ A2 @ ( some @ A @ B2 ) ) )
= ( insert @ A @ B2 @ ( ran @ B @ A @ M ) ) ) ) ).
% ran_map_upd
thf(fact_7093_Rat_Opositive__zero,axiom,
~ ( positive @ ( zero_zero @ rat ) ) ).
% Rat.positive_zero
thf(fact_7094_ranI,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A2: B,B2: A] :
( ( ( M @ A2 )
= ( some @ A @ B2 ) )
=> ( member @ A @ B2 @ ( ran @ B @ A @ M ) ) ) ).
% ranI
thf(fact_7095_ran__restrictD,axiom,
! [B: $tType,A: $tType,Y3: A,M: B > ( option @ A ),A4: set @ B] :
( ( member @ A @ Y3 @ ( ran @ B @ A @ ( restrict_map @ B @ A @ M @ A4 ) ) )
=> ? [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( ( M @ X4 )
= ( some @ A @ Y3 ) ) ) ) ).
% ran_restrictD
thf(fact_7096_ran__def,axiom,
! [B: $tType,A: $tType] :
( ( ran @ A @ B )
= ( ^ [M3: A > ( option @ B )] :
( collect @ B
@ ^ [B4: B] :
? [A5: A] :
( ( M3 @ A5 )
= ( some @ B @ B4 ) ) ) ) ) ).
% ran_def
thf(fact_7097_Rat_Opositive__minus,axiom,
! [X3: rat] :
( ~ ( positive @ X3 )
=> ( ( X3
!= ( zero_zero @ rat ) )
=> ( positive @ ( uminus_uminus @ rat @ X3 ) ) ) ) ).
% Rat.positive_minus
thf(fact_7098_Rat_Opositive_Orep__eq,axiom,
( positive
= ( ^ [X: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ ( rep_Rat @ X ) ) @ ( product_snd @ int @ int @ ( rep_Rat @ X ) ) ) ) ) ) ).
% Rat.positive.rep_eq
thf(fact_7099_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),X3: B,Y3: A,Z: A] :
( ( ( M @ X3 )
= ( some @ A @ Y3 ) )
=> ( ( inj_on @ B @ ( option @ A ) @ M @ ( dom @ B @ A @ M ) )
=> ( ~ ( member @ A @ Z @ ( ran @ B @ A @ M ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M @ X3 @ ( some @ A @ Z ) ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M ) @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Z @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_7100_dom__const,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( dom @ A @ B
@ ^ [X: A] : ( some @ B @ ( F2 @ X ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% dom_const
thf(fact_7101_domI,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A2: B,B2: A] :
( ( ( M @ A2 )
= ( some @ A @ B2 ) )
=> ( member @ B @ A2 @ ( dom @ B @ A @ M ) ) ) ).
% domI
thf(fact_7102_domD,axiom,
! [A: $tType,B: $tType,A2: A,M: A > ( option @ B )] :
( ( member @ A @ A2 @ ( dom @ A @ B @ M ) )
=> ? [B5: B] :
( ( M @ A2 )
= ( some @ B @ B5 ) ) ) ).
% domD
thf(fact_7103_insert__dom,axiom,
! [A: $tType,B: $tType,F2: B > ( option @ A ),X3: B,Y3: A] :
( ( ( F2 @ X3 )
= ( some @ A @ Y3 ) )
=> ( ( insert @ B @ X3 @ ( dom @ B @ A @ F2 ) )
= ( dom @ B @ A @ F2 ) ) ) ).
% insert_dom
thf(fact_7104_finite__Map__induct,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),P: ( A > ( option @ B ) ) > $o] :
( ( finite_finite @ A @ ( dom @ A @ B @ M ) )
=> ( ( P
@ ^ [X: A] : ( none @ B ) )
=> ( ! [K2: A,V4: B,M4: A > ( option @ B )] :
( ( finite_finite @ A @ ( dom @ A @ B @ M4 ) )
=> ( ~ ( member @ A @ K2 @ ( dom @ A @ B @ M4 ) )
=> ( ( P @ M4 )
=> ( P @ ( fun_upd @ A @ ( option @ B ) @ M4 @ K2 @ ( some @ B @ V4 ) ) ) ) ) )
=> ( P @ M ) ) ) ) ).
% finite_Map_induct
thf(fact_7105_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F2: A > ( option @ B ),X3: A] :
( ( ( dom @ A @ B @ F2 )
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ? [V5: B] :
( F2
= ( fun_upd @ A @ ( option @ B )
@ ^ [X: A] : ( none @ B )
@ X3
@ ( some @ B @ V5 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_7106_Rat_Opositive__def,axiom,
( positive
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ $o @ $o @ rep_Rat @ ( id @ $o )
@ ^ [X: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% Rat.positive_def
thf(fact_7107_inverse__rat__def,axiom,
( ( inverse_inverse @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
@ ^ [X: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X ) @ ( product_fst @ int @ int @ X ) ) ) ) ) ).
% inverse_rat_def
thf(fact_7108_of__rat__def,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ A @ A @ rep_Rat @ ( id @ A )
@ ^ [X: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ).
% of_rat_def
thf(fact_7109_map__upds__fold__map__upd,axiom,
! [B: $tType,A: $tType] :
( ( map_upds @ A @ B )
= ( ^ [M3: A > ( option @ B ),Ks2: list @ A,Vs2: list @ B] :
( foldl @ ( A > ( option @ B ) ) @ ( product_prod @ A @ B )
@ ^ [N3: A > ( option @ B )] :
( product_case_prod @ A @ B @ ( A > ( option @ B ) )
@ ^ [K3: A,V5: B] : ( fun_upd @ A @ ( option @ B ) @ N3 @ K3 @ ( some @ B @ V5 ) ) )
@ M3
@ ( zip @ A @ B @ Ks2 @ Vs2 ) ) ) ) ).
% map_upds_fold_map_upd
thf(fact_7110_of__rat__of__nat__eq,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat] :
( ( field_char_0_of_rat @ A @ ( semiring_1_of_nat @ rat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ).
% of_rat_of_nat_eq
thf(fact_7111_of__rat__of__int__eq,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z: int] :
( ( field_char_0_of_rat @ A @ ( ring_1_of_int @ rat @ Z ) )
= ( ring_1_of_int @ A @ Z ) ) ) ).
% of_rat_of_int_eq
thf(fact_7112_of__rat__0,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A @ ( zero_zero @ rat ) )
= ( zero_zero @ A ) ) ) ).
% of_rat_0
thf(fact_7113_of__rat__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: rat] :
( ( ( field_char_0_of_rat @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ rat ) ) ) ) ).
% of_rat_eq_0_iff
thf(fact_7114_zero__eq__of__rat__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: rat] :
( ( ( zero_zero @ A )
= ( field_char_0_of_rat @ A @ A2 ) )
= ( ( zero_zero @ rat )
= A2 ) ) ) ).
% zero_eq_of_rat_iff
thf(fact_7115_of__rat__numeral__eq,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [W: num] :
( ( field_char_0_of_rat @ A @ ( numeral_numeral @ rat @ W ) )
= ( numeral_numeral @ A @ W ) ) ) ).
% of_rat_numeral_eq
thf(fact_7116_of__rat__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R2 ) @ ( zero_zero @ A ) )
= ( ord_less @ rat @ R2 @ ( zero_zero @ rat ) ) ) ) ).
% of_rat_less_0_iff
thf(fact_7117_zero__less__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( field_char_0_of_rat @ A @ R2 ) )
= ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 ) ) ) ).
% zero_less_of_rat_iff
thf(fact_7118_of__rat__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R2 ) @ ( one_one @ A ) )
= ( ord_less @ rat @ R2 @ ( one_one @ rat ) ) ) ) ).
% of_rat_less_1_iff
thf(fact_7119_one__less__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R2 ) )
= ( ord_less @ rat @ ( one_one @ rat ) @ R2 ) ) ) ).
% one_less_of_rat_iff
thf(fact_7120_zero__le__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( field_char_0_of_rat @ A @ R2 ) )
= ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ R2 ) ) ) ).
% zero_le_of_rat_iff
thf(fact_7121_of__rat__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ rat @ R2 @ ( zero_zero @ rat ) ) ) ) ).
% of_rat_le_0_iff
thf(fact_7122_of__rat__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ rat @ R2 @ ( one_one @ rat ) ) ) ) ).
% of_rat_le_1_iff
thf(fact_7123_one__le__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R2 ) )
= ( ord_less_eq @ rat @ ( one_one @ rat ) @ R2 ) ) ) ).
% one_le_of_rat_iff
thf(fact_7124_of__rat__neg__numeral__eq,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [W: num] :
( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ W ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ).
% of_rat_neg_numeral_eq
thf(fact_7125_of__rat__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat,S: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R2 ) @ ( field_char_0_of_rat @ A @ S ) )
= ( ord_less_eq @ rat @ R2 @ S ) ) ) ).
% of_rat_less_eq
thf(fact_7126_of__rat__dense,axiom,
! [X3: real,Y3: real] :
( ( ord_less @ real @ X3 @ Y3 )
=> ? [Q4: rat] :
( ( ord_less @ real @ X3 @ ( field_char_0_of_rat @ real @ Q4 ) )
& ( ord_less @ real @ ( field_char_0_of_rat @ real @ Q4 ) @ Y3 ) ) ) ).
% of_rat_dense
thf(fact_7127_of__rat__power,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: rat,N: nat] :
( ( field_char_0_of_rat @ A @ ( power_power @ rat @ A2 @ N ) )
= ( power_power @ A @ ( field_char_0_of_rat @ A @ A2 ) @ N ) ) ) ).
% of_rat_power
thf(fact_7128_of__rat__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat,S: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R2 ) @ ( field_char_0_of_rat @ A @ S ) )
= ( ord_less @ rat @ R2 @ S ) ) ) ).
% of_rat_less
thf(fact_7129_nonzero__of__rat__inverse,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: rat] :
( ( A2
!= ( zero_zero @ rat ) )
=> ( ( field_char_0_of_rat @ A @ ( inverse_inverse @ rat @ A2 ) )
= ( inverse_inverse @ A @ ( field_char_0_of_rat @ A @ A2 ) ) ) ) ) ).
% nonzero_of_rat_inverse
thf(fact_7130_nonzero__of__rat__divide,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [B2: rat,A2: rat] :
( ( B2
!= ( zero_zero @ rat ) )
=> ( ( field_char_0_of_rat @ A @ ( divide_divide @ rat @ A2 @ B2 ) )
= ( divide_divide @ A @ ( field_char_0_of_rat @ A @ A2 ) @ ( field_char_0_of_rat @ A @ B2 ) ) ) ) ) ).
% nonzero_of_rat_divide
thf(fact_7131_of__rat_Orep__eq,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A )
= ( ^ [X: rat] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ ( rep_Rat @ X ) ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ ( rep_Rat @ X ) ) ) ) ) ) ) ).
% of_rat.rep_eq
thf(fact_7132_of__rat_Oabs__eq,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [X3: product_prod @ int @ int] :
( ( ratrel @ X3 @ X3 )
=> ( ( field_char_0_of_rat @ A @ ( abs_Rat @ X3 ) )
= ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X3 ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X3 ) ) ) ) ) ) ).
% of_rat.abs_eq
thf(fact_7133_graph__map__upd,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),K: A,V2: B] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( some @ B @ V2 ) ) )
= ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( none @ B ) ) ) ) ) ).
% graph_map_upd
thf(fact_7134_Rat_Opositive_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ $o @ $o @ ratrel
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ^ [X: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ X ) ) )
@ ^ [X: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ X ) ) ) ) ).
% Rat.positive.rsp
thf(fact_7135_transfer__rule__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( semiring_numeral @ B )
& ( monoid_add @ A )
& ( semiring_numeral @ A ) )
=> ! [R3: A > B > $o] :
( ( R3 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R3 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R3 @ ( bNF_rel_fun @ A @ B @ A @ B @ R3 @ R3 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ R3
@ ( numeral_numeral @ A )
@ ( numeral_numeral @ B ) ) ) ) ) ) ).
% transfer_rule_numeral
thf(fact_7136_power__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( power @ B )
& ( power @ A ) )
=> ! [R3: A > B > $o] :
( ( R3 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R3 @ ( bNF_rel_fun @ A @ B @ A @ B @ R3 @ R3 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ A @ B @ ( nat > A ) @ ( nat > B ) @ R3
@ ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ R3 )
@ ( power_power @ A )
@ ( power_power @ B ) ) ) ) ) ).
% power_transfer
thf(fact_7137_in__graphD,axiom,
! [A: $tType,B: $tType,K: A,V2: B,M: A > ( option @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( graph @ A @ B @ M ) )
=> ( ( M @ K )
= ( some @ B @ V2 ) ) ) ).
% in_graphD
thf(fact_7138_in__graphI,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),K: B,V2: A] :
( ( ( M @ K )
= ( some @ A @ V2 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V2 ) @ ( graph @ B @ A @ M ) ) ) ).
% in_graphI
thf(fact_7139_transfer__rule__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ( ring_1 @ B )
& ( ring_1 @ A ) )
=> ! [R3: A > B > $o] :
( ( R3 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R3 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R3 @ ( bNF_rel_fun @ A @ B @ A @ B @ R3 @ R3 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ A @ B @ R3 @ R3 @ ( uminus_uminus @ A ) @ ( uminus_uminus @ B ) )
=> ( bNF_rel_fun @ int @ int @ A @ B
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ R3
@ ( ring_1_of_int @ A )
@ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% transfer_rule_of_int
thf(fact_7140_graph__restrictD_I2_J,axiom,
! [A: $tType,B: $tType,K: A,V2: B,M: A > ( option @ B ),A4: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M @ A4 ) ) )
=> ( ( M @ K )
= ( some @ B @ V2 ) ) ) ).
% graph_restrictD(2)
thf(fact_7141_graph__def,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M3: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [A5: A,B4: B] :
( ( Uu3
= ( product_Pair @ A @ B @ A5 @ B4 ) )
& ( ( M3 @ A5 )
= ( some @ B @ B4 ) ) ) ) ) ) ).
% graph_def
thf(fact_7142_of__rat_Orsp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ A @ A @ ratrel
@ ^ [Y6: A,Z2: A] : Y6 = Z2
@ ^ [X: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X ) ) )
@ ^ [X: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% of_rat.rsp
thf(fact_7143_transfer__rule__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( ( semiring_1 @ B )
& ( semiring_1 @ A ) )
=> ! [R3: A > B > $o] :
( ( R3 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R3 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R3 @ ( bNF_rel_fun @ A @ B @ A @ B @ R3 @ R3 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ R3
@ ( semiring_1_of_nat @ A )
@ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% transfer_rule_of_nat
thf(fact_7144_inverse__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
@ ^ [X: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X ) @ ( product_fst @ int @ int @ X ) ) )
@ ^ [X: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X ) @ ( product_fst @ int @ int @ X ) ) ) ) ).
% inverse_rat.rsp
thf(fact_7145_Fract_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > ( product_prod @ int @ int ) )
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int )
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ratrel )
@ ^ [A5: int,B4: int] :
( if @ ( product_prod @ int @ int )
@ ( B4
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A5 @ B4 ) )
@ ^ [A5: int,B4: int] :
( if @ ( product_prod @ int @ int )
@ ( B4
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A5 @ B4 ) ) ) ).
% Fract.rsp
thf(fact_7146_num_Ocase__transfer,axiom,
! [A: $tType,B: $tType,S2: A > B > $o] :
( bNF_rel_fun @ A @ B @ ( ( num > A ) > ( num > A ) > num > A ) @ ( ( num > B ) > ( num > B ) > num > B ) @ S2
@ ( bNF_rel_fun @ ( num > A ) @ ( num > B ) @ ( ( num > A ) > num > A ) @ ( ( num > B ) > num > B )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ S2 )
@ ( bNF_rel_fun @ ( num > A ) @ ( num > B ) @ ( num > A ) @ ( num > B )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ S2 )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ S2 ) ) )
@ ( case_num @ A )
@ ( case_num @ B ) ) ).
% num.case_transfer
thf(fact_7147_transfer__rule__of__bool,axiom,
! [A: $tType,B: $tType] :
( ( ( zero_neq_one @ B )
& ( zero_neq_one @ A ) )
=> ! [R3: A > B > $o] :
( ( R3 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R3 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( bNF_rel_fun @ $o @ $o @ A @ B
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ R3
@ ( zero_neq_one_of_bool @ A )
@ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% transfer_rule_of_bool
thf(fact_7148_integer__of__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ int @ int
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ( semiring_1_of_nat @ int )
@ ( semiring_1_of_nat @ int ) ) ).
% integer_of_natural.rsp
thf(fact_7149_natural__of__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ nat @ nat
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ nat2
@ nat2 ) ).
% natural_of_integer.rsp
thf(fact_7150_sub_Orsp,axiom,
( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > int )
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ ( bNF_rel_fun @ num @ num @ int @ int
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ ^ [Y6: int,Z2: int] : Y6 = Z2 )
@ ^ [M3: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M3 ) @ ( numeral_numeral @ int @ N3 ) )
@ ^ [M3: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M3 ) @ ( numeral_numeral @ int @ N3 ) ) ) ).
% sub.rsp
thf(fact_7151_less__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( ord_less @ int )
@ ( ord_less @ int ) ) ).
% less_integer.rsp
thf(fact_7152_less__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( ord_less @ nat )
@ ( ord_less @ nat ) ) ).
% less_natural.rsp
thf(fact_7153_less__eq__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( ord_less_eq @ int )
@ ( ord_less_eq @ int ) ) ).
% less_eq_integer.rsp
thf(fact_7154_less__eq__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( ord_less_eq @ nat )
@ ( ord_less_eq @ nat ) ) ).
% less_eq_natural.rsp
thf(fact_7155_num__of__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ num @ num
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
@ ( comp @ nat @ num @ int @ num_of_nat @ nat2 ) ) ).
% num_of_integer.rsp
thf(fact_7156_inverse__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
@ ^ [X: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X ) @ ( product_fst @ int @ int @ X ) ) )
@ ( inverse_inverse @ rat ) ) ).
% inverse_rat.transfer
thf(fact_7157_Rat_Opositive_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ $o @ $o @ pcr_rat
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ^ [X: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ X ) ) )
@ positive ) ).
% Rat.positive.transfer
thf(fact_7158_zero__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( zero_zero @ rat ) ).
% zero_rat.transfer
thf(fact_7159_of__rat_Otransfer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ A @ A @ pcr_rat
@ ^ [Y6: A,Z2: A] : Y6 = Z2
@ ^ [X: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X ) ) )
@ ( field_char_0_of_rat @ A ) ) ) ).
% of_rat.transfer
thf(fact_7160_times__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ U2 ) @ ( times_times @ nat @ Y @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X @ V5 ) @ ( times_times @ nat @ Y @ U2 ) ) ) ) )
@ ( times_times @ int ) ) ).
% times_int.transfer
thf(fact_7161_num_Orec__transfer,axiom,
! [A: $tType,B: $tType,S2: A > B > $o] :
( bNF_rel_fun @ A @ B @ ( ( num > A > A ) > ( num > A > A ) > num > A ) @ ( ( num > B > B ) > ( num > B > B ) > num > B ) @ S2
@ ( bNF_rel_fun @ ( num > A > A ) @ ( num > B > B ) @ ( ( num > A > A ) > num > A ) @ ( ( num > B > B ) > num > B )
@ ( bNF_rel_fun @ num @ num @ ( A > A ) @ ( B > B )
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ ( bNF_rel_fun @ A @ B @ A @ B @ S2 @ S2 ) )
@ ( bNF_rel_fun @ ( num > A > A ) @ ( num > B > B ) @ ( num > A ) @ ( num > B )
@ ( bNF_rel_fun @ num @ num @ ( A > A ) @ ( B > B )
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ ( bNF_rel_fun @ A @ B @ A @ B @ S2 @ S2 ) )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ S2 ) ) )
@ ( rec_num @ A )
@ ( rec_num @ B ) ) ).
% num.rec_transfer
thf(fact_7162_verit__eq__simplify_I20_J,axiom,
! [A: $tType,F1: A,F22: num > A > A,F32: num > A > A,X2: num] :
( ( rec_num @ A @ F1 @ F22 @ F32 @ ( bit0 @ X2 ) )
= ( F22 @ X2 @ ( rec_num @ A @ F1 @ F22 @ F32 @ X2 ) ) ) ).
% verit_eq_simplify(20)
thf(fact_7163_verit__eq__simplify_I19_J,axiom,
! [A: $tType,F1: A,F22: num > A > A,F32: num > A > A] :
( ( rec_num @ A @ F1 @ F22 @ F32 @ one2 )
= F1 ) ).
% verit_eq_simplify(19)
thf(fact_7164_verit__eq__simplify_I21_J,axiom,
! [A: $tType,F1: A,F22: num > A > A,F32: num > A > A,X32: num] :
( ( rec_num @ A @ F1 @ F22 @ F32 @ ( bit1 @ X32 ) )
= ( F32 @ X32 @ ( rec_num @ A @ F1 @ F22 @ F32 @ X32 ) ) ) ).
% verit_eq_simplify(21)
thf(fact_7165_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( zero_zero @ int ) ).
% zero_int.transfer
thf(fact_7166_int__transfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( product_prod @ nat @ nat ) @ int
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ pcr_int
@ ^ [N3: nat] : ( product_Pair @ nat @ nat @ N3 @ ( zero_zero @ nat ) )
@ ( semiring_1_of_nat @ int ) ) ).
% int_transfer
thf(fact_7167_uminus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X: nat,Y: nat] : ( product_Pair @ nat @ nat @ Y @ X ) )
@ ( uminus_uminus @ int ) ) ).
% uminus_int.transfer
thf(fact_7168_nat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ nat @ nat @ pcr_int
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
@ nat2 ) ).
% nat.transfer
thf(fact_7169_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( one_one @ int ) ).
% one_int.transfer
thf(fact_7170_of__int_Otransfer,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ A @ A @ pcr_int
@ ^ [Y6: A,Z2: A] : Y6 = Z2
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I4: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J3 ) ) )
@ ( ring_1_of_int @ A ) ) ) ).
% of_int.transfer
thf(fact_7171_less__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y ) ) ) )
@ ( ord_less @ int ) ) ).
% less_int.transfer
thf(fact_7172_less__eq__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y ) ) ) )
@ ( ord_less_eq @ int ) ) ).
% less_eq_int.transfer
thf(fact_7173_plus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ U2 ) @ ( plus_plus @ nat @ Y @ V5 ) ) ) )
@ ( plus_plus @ int ) ) ).
% plus_int.transfer
thf(fact_7174_minus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ Y @ U2 ) ) ) )
@ ( minus_minus @ int ) ) ).
% minus_int.transfer
thf(fact_7175_times__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ U2 ) @ ( times_times @ nat @ Y @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X @ V5 ) @ ( times_times @ nat @ Y @ U2 ) ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ U2 ) @ ( times_times @ nat @ Y @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X @ V5 ) @ ( times_times @ nat @ Y @ U2 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_7176_Real_Opositive_Orsp,axiom,
( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ $o @ $o @ realrel
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ^ [X9: nat > rat] :
? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less @ rat @ R5 @ ( X9 @ N3 ) ) ) )
@ ^ [X9: nat > rat] :
? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less @ rat @ R5 @ ( X9 @ N3 ) ) ) ) ) ).
% Real.positive.rsp
thf(fact_7177_intrel__iff,axiom,
! [X3: nat,Y3: nat,U: nat,V2: nat] :
( ( intrel @ ( product_Pair @ nat @ nat @ X3 @ Y3 ) @ ( product_Pair @ nat @ nat @ U @ V2 ) )
= ( ( plus_plus @ nat @ X3 @ V2 )
= ( plus_plus @ nat @ U @ Y3 ) ) ) ).
% intrel_iff
thf(fact_7178_plus__real_Orsp,axiom,
( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ ( ( nat > rat ) > nat > rat ) @ ( ( nat > rat ) > nat > rat ) @ realrel @ ( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ ( nat > rat ) @ ( nat > rat ) @ realrel @ realrel )
@ ^ [X9: nat > rat,Y8: nat > rat,N3: nat] : ( plus_plus @ rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) )
@ ^ [X9: nat > rat,Y8: nat > rat,N3: nat] : ( plus_plus @ rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ).
% plus_real.rsp
thf(fact_7179_int_Orel__eq__transfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ intrel
@ ^ [Y6: int,Z2: int] : Y6 = Z2 ) ).
% int.rel_eq_transfer
thf(fact_7180_times__real_Orsp,axiom,
( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ ( ( nat > rat ) > nat > rat ) @ ( ( nat > rat ) > nat > rat ) @ realrel @ ( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ ( nat > rat ) @ ( nat > rat ) @ realrel @ realrel )
@ ^ [X9: nat > rat,Y8: nat > rat,N3: nat] : ( times_times @ rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) )
@ ^ [X9: nat > rat,Y8: nat > rat,N3: nat] : ( times_times @ rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ).
% times_real.rsp
thf(fact_7181_uminus__real_Orsp,axiom,
( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ ( nat > rat ) @ ( nat > rat ) @ realrel @ realrel
@ ^ [X9: nat > rat,N3: nat] : ( uminus_uminus @ rat @ ( X9 @ N3 ) )
@ ^ [X9: nat > rat,N3: nat] : ( uminus_uminus @ rat @ ( X9 @ N3 ) ) ) ).
% uminus_real.rsp
thf(fact_7182_one__real_Orsp,axiom,
( realrel
@ ^ [N3: nat] : ( one_one @ rat )
@ ^ [N3: nat] : ( one_one @ rat ) ) ).
% one_real.rsp
thf(fact_7183_int_Oabs__eq__iff,axiom,
! [X3: product_prod @ nat @ nat,Y3: product_prod @ nat @ nat] :
( ( ( abs_Integ @ X3 )
= ( abs_Integ @ Y3 ) )
= ( intrel @ X3 @ Y3 ) ) ).
% int.abs_eq_iff
thf(fact_7184_zero__real_Orsp,axiom,
( realrel
@ ^ [N3: nat] : ( zero_zero @ rat )
@ ^ [N3: nat] : ( zero_zero @ rat ) ) ).
% zero_real.rsp
thf(fact_7185_zero__int_Orsp,axiom,
intrel @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ).
% zero_int.rsp
thf(fact_7186_uminus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X: nat,Y: nat] : ( product_Pair @ nat @ nat @ Y @ X ) )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X: nat,Y: nat] : ( product_Pair @ nat @ nat @ Y @ X ) ) ) ).
% uminus_int.rsp
thf(fact_7187_nat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ nat @ nat @ intrel
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ).
% nat.rsp
thf(fact_7188_one__int_Orsp,axiom,
intrel @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ).
% one_int.rsp
thf(fact_7189_of__int_Orsp,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ A @ A @ intrel
@ ^ [Y6: A,Z2: A] : Y6 = Z2
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I4: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J3 ) ) )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I4: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ).
% of_int.rsp
thf(fact_7190_intrel__def,axiom,
( intrel
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] :
( ( plus_plus @ nat @ X @ V5 )
= ( plus_plus @ nat @ U2 @ Y ) ) ) ) ) ).
% intrel_def
thf(fact_7191_less__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y ) ) ) ) ) ).
% less_int.rsp
thf(fact_7192_less__eq__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_7193_plus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ U2 ) @ ( plus_plus @ nat @ Y @ V5 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ U2 ) @ ( plus_plus @ nat @ Y @ V5 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_7194_minus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ Y @ U2 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ Y @ U2 ) ) ) ) ) ).
% minus_int.rsp
thf(fact_7195_mono__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ D )
& ( order @ C )
& ( order @ A ) )
=> ! [A4: A > B > $o,B6: C > D > $o] :
( ( bi_total @ A @ B @ A4 )
=> ( ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A4
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( ord_less_eq @ A )
@ ( ord_less_eq @ B ) )
=> ( ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B6
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ B6
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( ord_less_eq @ C )
@ ( ord_less_eq @ D ) )
=> ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B6 )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ( order_mono @ A @ C )
@ ( order_mono @ B @ D ) ) ) ) ) ) ).
% mono_transfer
thf(fact_7196_sorted__insort__insert__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs2: list @ B,X3: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs2 ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ ( linord329482645794927042rt_key @ B @ A @ F2 @ X3 @ Xs2 ) ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_7197_int_Obi__total,axiom,
bi_total @ ( product_prod @ nat @ nat ) @ int @ pcr_int ).
% int.bi_total
thf(fact_7198_sorted__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,X3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X: A] : X
@ X3
@ Xs2 ) ) ) ) ).
% sorted_insort_insert
thf(fact_7199_inverse__real_Orsp,axiom,
( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ ( nat > rat ) @ ( nat > rat ) @ realrel @ realrel
@ ^ [X9: nat > rat] :
( if @ ( nat > rat ) @ ( vanishes @ X9 )
@ ^ [N3: nat] : ( zero_zero @ rat )
@ ^ [N3: nat] : ( inverse_inverse @ rat @ ( X9 @ N3 ) ) )
@ ^ [X9: nat > rat] :
( if @ ( nat > rat ) @ ( vanishes @ X9 )
@ ^ [N3: nat] : ( zero_zero @ rat )
@ ^ [N3: nat] : ( inverse_inverse @ rat @ ( X9 @ N3 ) ) ) ) ).
% inverse_real.rsp
thf(fact_7200_vanishes__mult__bounded,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ? [A8: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ A8 )
& ! [N2: nat] : ( ord_less @ rat @ ( abs_abs @ rat @ ( X8 @ N2 ) ) @ A8 ) )
=> ( ( vanishes @ Y7 )
=> ( vanishes
@ ^ [N3: nat] : ( times_times @ rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ) ).
% vanishes_mult_bounded
thf(fact_7201_vanishes__const,axiom,
! [C2: rat] :
( ( vanishes
@ ^ [N3: nat] : C2 )
= ( C2
= ( zero_zero @ rat ) ) ) ).
% vanishes_const
thf(fact_7202_vanishes__add,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( vanishes @ X8 )
=> ( ( vanishes @ Y7 )
=> ( vanishes
@ ^ [N3: nat] : ( plus_plus @ rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ) ).
% vanishes_add
thf(fact_7203_vanishes__diff,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( vanishes @ X8 )
=> ( ( vanishes @ Y7 )
=> ( vanishes
@ ^ [N3: nat] : ( minus_minus @ rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ) ).
% vanishes_diff
thf(fact_7204_vanishes__minus,axiom,
! [X8: nat > rat] :
( ( vanishes @ X8 )
=> ( vanishes
@ ^ [N3: nat] : ( uminus_uminus @ rat @ ( X8 @ N3 ) ) ) ) ).
% vanishes_minus
thf(fact_7205_vanishes__def,axiom,
( vanishes
= ( ^ [X9: nat > rat] :
! [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
=> ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( X9 @ N3 ) ) @ R5 ) ) ) ) ) ).
% vanishes_def
thf(fact_7206_vanishesI,axiom,
! [X8: nat > rat] :
( ! [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
=> ? [K4: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K4 @ N2 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( X8 @ N2 ) ) @ R ) ) )
=> ( vanishes @ X8 ) ) ).
% vanishesI
thf(fact_7207_vanishesD,axiom,
! [X8: nat > rat,R2: rat] :
( ( vanishes @ X8 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 )
=> ? [K2: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ K2 @ N9 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( X8 @ N9 ) ) @ R2 ) ) ) ) ).
% vanishesD
thf(fact_7208_inverse__real_Otransfer,axiom,
( bNF_rel_fun @ ( nat > rat ) @ real @ ( nat > rat ) @ real @ pcr_real @ pcr_real
@ ^ [X9: nat > rat] :
( if @ ( nat > rat ) @ ( vanishes @ X9 )
@ ^ [N3: nat] : ( zero_zero @ rat )
@ ^ [N3: nat] : ( inverse_inverse @ rat @ ( X9 @ N3 ) ) )
@ ( inverse_inverse @ real ) ) ).
% inverse_real.transfer
thf(fact_7209_inverse__real_Oabs__eq,axiom,
! [X3: nat > rat] :
( ( realrel @ X3 @ X3 )
=> ( ( inverse_inverse @ real @ ( real2 @ X3 ) )
= ( real2
@ ( if @ ( nat > rat ) @ ( vanishes @ X3 )
@ ^ [N3: nat] : ( zero_zero @ rat )
@ ^ [N3: nat] : ( inverse_inverse @ rat @ ( X3 @ N3 ) ) ) ) ) ) ).
% inverse_real.abs_eq
thf(fact_7210_real_Oabs__induct,axiom,
! [P: real > $o,X3: real] :
( ! [Y4: nat > rat] :
( ( realrel @ Y4 @ Y4 )
=> ( P @ ( real2 @ Y4 ) ) )
=> ( P @ X3 ) ) ).
% real.abs_induct
thf(fact_7211_of__rat__Real,axiom,
( ( field_char_0_of_rat @ real )
= ( ^ [X: rat] :
( real2
@ ^ [N3: nat] : X ) ) ) ).
% of_rat_Real
thf(fact_7212_zero__real__def,axiom,
( ( zero_zero @ real )
= ( real2
@ ^ [N3: nat] : ( zero_zero @ rat ) ) ) ).
% zero_real_def
thf(fact_7213_one__real__def,axiom,
( ( one_one @ real )
= ( real2
@ ^ [N3: nat] : ( one_one @ rat ) ) ) ).
% one_real_def
thf(fact_7214_real_Orel__eq__transfer,axiom,
( bNF_rel_fun @ ( nat > rat ) @ real @ ( ( nat > rat ) > $o ) @ ( real > $o ) @ pcr_real
@ ( bNF_rel_fun @ ( nat > rat ) @ real @ $o @ $o @ pcr_real
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ realrel
@ ^ [Y6: real,Z2: real] : Y6 = Z2 ) ).
% real.rel_eq_transfer
thf(fact_7215_of__int__Real,axiom,
( ( ring_1_of_int @ real )
= ( ^ [X: int] :
( real2
@ ^ [N3: nat] : ( ring_1_of_int @ rat @ X ) ) ) ) ).
% of_int_Real
thf(fact_7216_zero__real_Otransfer,axiom,
( pcr_real
@ ^ [N3: nat] : ( zero_zero @ rat )
@ ( zero_zero @ real ) ) ).
% zero_real.transfer
thf(fact_7217_one__real_Otransfer,axiom,
( pcr_real
@ ^ [N3: nat] : ( one_one @ rat )
@ ( one_one @ real ) ) ).
% one_real.transfer
thf(fact_7218_of__nat__Real,axiom,
( ( semiring_1_of_nat @ real )
= ( ^ [X: nat] :
( real2
@ ^ [N3: nat] : ( semiring_1_of_nat @ rat @ X ) ) ) ) ).
% of_nat_Real
thf(fact_7219_uminus__real_Otransfer,axiom,
( bNF_rel_fun @ ( nat > rat ) @ real @ ( nat > rat ) @ real @ pcr_real @ pcr_real
@ ^ [X9: nat > rat,N3: nat] : ( uminus_uminus @ rat @ ( X9 @ N3 ) )
@ ( uminus_uminus @ real ) ) ).
% uminus_real.transfer
thf(fact_7220_plus__real_Otransfer,axiom,
( bNF_rel_fun @ ( nat > rat ) @ real @ ( ( nat > rat ) > nat > rat ) @ ( real > real ) @ pcr_real @ ( bNF_rel_fun @ ( nat > rat ) @ real @ ( nat > rat ) @ real @ pcr_real @ pcr_real )
@ ^ [X9: nat > rat,Y8: nat > rat,N3: nat] : ( plus_plus @ rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) )
@ ( plus_plus @ real ) ) ).
% plus_real.transfer
thf(fact_7221_uminus__real_Oabs__eq,axiom,
! [X3: nat > rat] :
( ( realrel @ X3 @ X3 )
=> ( ( uminus_uminus @ real @ ( real2 @ X3 ) )
= ( real2
@ ^ [N3: nat] : ( uminus_uminus @ rat @ ( X3 @ N3 ) ) ) ) ) ).
% uminus_real.abs_eq
thf(fact_7222_times__real_Otransfer,axiom,
( bNF_rel_fun @ ( nat > rat ) @ real @ ( ( nat > rat ) > nat > rat ) @ ( real > real ) @ pcr_real @ ( bNF_rel_fun @ ( nat > rat ) @ real @ ( nat > rat ) @ real @ pcr_real @ pcr_real )
@ ^ [X9: nat > rat,Y8: nat > rat,N3: nat] : ( times_times @ rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) )
@ ( times_times @ real ) ) ).
% times_real.transfer
thf(fact_7223_plus__real_Oabs__eq,axiom,
! [Xa: nat > rat,X3: nat > rat] :
( ( realrel @ Xa @ Xa )
=> ( ( realrel @ X3 @ X3 )
=> ( ( plus_plus @ real @ ( real2 @ Xa ) @ ( real2 @ X3 ) )
= ( real2
@ ^ [N3: nat] : ( plus_plus @ rat @ ( Xa @ N3 ) @ ( X3 @ N3 ) ) ) ) ) ) ).
% plus_real.abs_eq
thf(fact_7224_times__real_Oabs__eq,axiom,
! [Xa: nat > rat,X3: nat > rat] :
( ( realrel @ Xa @ Xa )
=> ( ( realrel @ X3 @ X3 )
=> ( ( times_times @ real @ ( real2 @ Xa ) @ ( real2 @ X3 ) )
= ( real2
@ ^ [N3: nat] : ( times_times @ rat @ ( Xa @ N3 ) @ ( X3 @ N3 ) ) ) ) ) ) ).
% times_real.abs_eq
thf(fact_7225_Real_Opositive_Otransfer,axiom,
( bNF_rel_fun @ ( nat > rat ) @ real @ $o @ $o @ pcr_real
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ^ [X9: nat > rat] :
? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less @ rat @ R5 @ ( X9 @ N3 ) ) ) )
@ positive2 ) ).
% Real.positive.transfer
thf(fact_7226_Real_Opositive_Oabs__eq,axiom,
! [X3: nat > rat] :
( ( realrel @ X3 @ X3 )
=> ( ( positive2 @ ( real2 @ X3 ) )
= ( ? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less @ rat @ R5 @ ( X3 @ N3 ) ) ) ) ) ) ) ).
% Real.positive.abs_eq
thf(fact_7227_Real_Opositive__mult,axiom,
! [X3: real,Y3: real] :
( ( positive2 @ X3 )
=> ( ( positive2 @ Y3 )
=> ( positive2 @ ( times_times @ real @ X3 @ Y3 ) ) ) ) ).
% Real.positive_mult
thf(fact_7228_Real_Opositive__add,axiom,
! [X3: real,Y3: real] :
( ( positive2 @ X3 )
=> ( ( positive2 @ Y3 )
=> ( positive2 @ ( plus_plus @ real @ X3 @ Y3 ) ) ) ) ).
% Real.positive_add
thf(fact_7229_Real_Opositive__zero,axiom,
~ ( positive2 @ ( zero_zero @ real ) ) ).
% Real.positive_zero
thf(fact_7230_Real_Opositive__minus,axiom,
! [X3: real] :
( ~ ( positive2 @ X3 )
=> ( ( X3
!= ( zero_zero @ real ) )
=> ( positive2 @ ( uminus_uminus @ real @ X3 ) ) ) ) ).
% Real.positive_minus
thf(fact_7231_less__real__def,axiom,
( ( ord_less @ real )
= ( ^ [X: real,Y: real] : ( positive2 @ ( minus_minus @ real @ Y @ X ) ) ) ) ).
% less_real_def
thf(fact_7232_le__Real,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( ( ord_less_eq @ real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
= ( ! [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
=> ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less_eq @ rat @ ( X8 @ N3 ) @ ( plus_plus @ rat @ ( Y7 @ N3 ) @ R5 ) ) ) ) ) ) ) ) ).
% le_Real
thf(fact_7233_Real_Opositive_Orep__eq,axiom,
( positive2
= ( ^ [X: real] :
? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less @ rat @ R5 @ ( rep_real @ X @ N3 ) ) ) ) ) ) ).
% Real.positive.rep_eq
thf(fact_7234_realrel__refl,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( realrel @ X8 @ X8 ) ) ).
% realrel_refl
thf(fact_7235_cauchy__minus,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( cauchy
@ ^ [N3: nat] : ( uminus_uminus @ rat @ ( X8 @ N3 ) ) ) ) ).
% cauchy_minus
thf(fact_7236_cauchy__const,axiom,
! [X3: rat] :
( cauchy
@ ^ [N3: nat] : X3 ) ).
% cauchy_const
thf(fact_7237_cauchy__add,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( cauchy
@ ^ [N3: nat] : ( plus_plus @ rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ) ).
% cauchy_add
thf(fact_7238_cauchy__mult,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( cauchy
@ ^ [N3: nat] : ( times_times @ rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ) ).
% cauchy_mult
thf(fact_7239_Real__induct,axiom,
! [P: real > $o,X3: real] :
( ! [X18: nat > rat] :
( ( cauchy @ X18 )
=> ( P @ ( real2 @ X18 ) ) )
=> ( P @ X3 ) ) ).
% Real_induct
thf(fact_7240_cauchy__diff,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( cauchy
@ ^ [N3: nat] : ( minus_minus @ rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ) ).
% cauchy_diff
thf(fact_7241_cr__real__eq,axiom,
( pcr_real
= ( ^ [X: nat > rat,Y: real] :
( ( cauchy @ X )
& ( ( real2 @ X )
= Y ) ) ) ) ).
% cr_real_eq
thf(fact_7242_cauchy__inverse,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ~ ( vanishes @ X8 )
=> ( cauchy
@ ^ [N3: nat] : ( inverse_inverse @ rat @ ( X8 @ N3 ) ) ) ) ) ).
% cauchy_inverse
thf(fact_7243_cauchy__imp__bounded,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ? [B5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ B5 )
& ! [N9: nat] : ( ord_less @ rat @ ( abs_abs @ rat @ ( X8 @ N9 ) ) @ B5 ) ) ) ).
% cauchy_imp_bounded
thf(fact_7244_less__RealD,axiom,
! [Y7: nat > rat,X3: real] :
( ( cauchy @ Y7 )
=> ( ( ord_less @ real @ X3 @ ( real2 @ Y7 ) )
=> ? [N2: nat] : ( ord_less @ real @ X3 @ ( field_char_0_of_rat @ real @ ( Y7 @ N2 ) ) ) ) ) ).
% less_RealD
thf(fact_7245_le__RealI,axiom,
! [Y7: nat > rat,X3: real] :
( ( cauchy @ Y7 )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ X3 @ ( field_char_0_of_rat @ real @ ( Y7 @ N2 ) ) )
=> ( ord_less_eq @ real @ X3 @ ( real2 @ Y7 ) ) ) ) ).
% le_RealI
thf(fact_7246_Real__leI,axiom,
! [X8: nat > rat,Y3: real] :
( ( cauchy @ X8 )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( field_char_0_of_rat @ real @ ( X8 @ N2 ) ) @ Y3 )
=> ( ord_less_eq @ real @ ( real2 @ X8 ) @ Y3 ) ) ) ).
% Real_leI
thf(fact_7247_minus__Real,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ( uminus_uminus @ real @ ( real2 @ X8 ) )
= ( real2
@ ^ [N3: nat] : ( uminus_uminus @ rat @ ( X8 @ N3 ) ) ) ) ) ).
% minus_Real
thf(fact_7248_add__Real,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( ( plus_plus @ real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
= ( real2
@ ^ [N3: nat] : ( plus_plus @ rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ) ) ).
% add_Real
thf(fact_7249_mult__Real,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( ( times_times @ real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
= ( real2
@ ^ [N3: nat] : ( times_times @ rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ) ) ).
% mult_Real
thf(fact_7250_diff__Real,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( ( minus_minus @ real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
= ( real2
@ ^ [N3: nat] : ( minus_minus @ rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ) ) ).
% diff_Real
thf(fact_7251_realrelI,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( ( vanishes
@ ^ [N3: nat] : ( minus_minus @ rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) )
=> ( realrel @ X8 @ Y7 ) ) ) ) ).
% realrelI
thf(fact_7252_eq__Real,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( ( ( real2 @ X8 )
= ( real2 @ Y7 ) )
= ( vanishes
@ ^ [N3: nat] : ( minus_minus @ rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) ) ) ) ).
% eq_Real
thf(fact_7253_vanishes__diff__inverse,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ~ ( vanishes @ X8 )
=> ( ( cauchy @ Y7 )
=> ( ~ ( vanishes @ Y7 )
=> ( ( vanishes
@ ^ [N3: nat] : ( minus_minus @ rat @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) )
=> ( vanishes
@ ^ [N3: nat] : ( minus_minus @ rat @ ( inverse_inverse @ rat @ ( X8 @ N3 ) ) @ ( inverse_inverse @ rat @ ( Y7 @ N3 ) ) ) ) ) ) ) ) ) ).
% vanishes_diff_inverse
thf(fact_7254_realrel__def,axiom,
( realrel
= ( ^ [X9: nat > rat,Y8: nat > rat] :
( ( cauchy @ X9 )
& ( cauchy @ Y8 )
& ( vanishes
@ ^ [N3: nat] : ( minus_minus @ rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ) ) ).
% realrel_def
thf(fact_7255_cauchy__not__vanishes__cases,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ~ ( vanishes @ X8 )
=> ? [B5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ B5 )
& ? [K2: nat] :
( ! [N9: nat] :
( ( ord_less_eq @ nat @ K2 @ N9 )
=> ( ord_less @ rat @ B5 @ ( uminus_uminus @ rat @ ( X8 @ N9 ) ) ) )
| ! [N9: nat] :
( ( ord_less_eq @ nat @ K2 @ N9 )
=> ( ord_less @ rat @ B5 @ ( X8 @ N9 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes_cases
thf(fact_7256_positive__Real,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ( positive2 @ ( real2 @ X8 ) )
= ( ? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less @ rat @ R5 @ ( X8 @ N3 ) ) ) ) ) ) ) ).
% positive_Real
thf(fact_7257_cauchy__not__vanishes,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ~ ( vanishes @ X8 )
=> ? [B5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ B5 )
& ? [K2: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ K2 @ N9 )
=> ( ord_less @ rat @ B5 @ ( abs_abs @ rat @ ( X8 @ N9 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes
thf(fact_7258_cauchy__def,axiom,
( cauchy
= ( ^ [X9: nat > rat] :
! [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
=> ? [K3: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ K3 @ M3 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( minus_minus @ rat @ ( X9 @ M3 ) @ ( X9 @ N3 ) ) ) @ R5 ) ) ) ) ) ) ).
% cauchy_def
thf(fact_7259_cauchyI,axiom,
! [X8: nat > rat] :
( ! [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
=> ? [K4: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ K4 @ M4 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ K4 @ N2 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( minus_minus @ rat @ ( X8 @ M4 ) @ ( X8 @ N2 ) ) ) @ R ) ) ) )
=> ( cauchy @ X8 ) ) ).
% cauchyI
thf(fact_7260_cauchyD,axiom,
! [X8: nat > rat,R2: rat] :
( ( cauchy @ X8 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 )
=> ? [K2: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ K2 @ M2 )
=> ! [N9: nat] :
( ( ord_less_eq @ nat @ K2 @ N9 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( minus_minus @ rat @ ( X8 @ M2 ) @ ( X8 @ N9 ) ) ) @ R2 ) ) ) ) ) ).
% cauchyD
thf(fact_7261_inverse__Real,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ( ( vanishes @ X8 )
=> ( ( inverse_inverse @ real @ ( real2 @ X8 ) )
= ( zero_zero @ real ) ) )
& ( ~ ( vanishes @ X8 )
=> ( ( inverse_inverse @ real @ ( real2 @ X8 ) )
= ( real2
@ ^ [N3: nat] : ( inverse_inverse @ rat @ ( X8 @ N3 ) ) ) ) ) ) ) ).
% inverse_Real
thf(fact_7262_not__positive__Real,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ( ~ ( positive2 @ ( real2 @ X8 ) ) )
= ( ! [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
=> ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less_eq @ rat @ ( X8 @ N3 ) @ R5 ) ) ) ) ) ) ).
% not_positive_Real
thf(fact_7263_Real_Opositive__def,axiom,
( positive2
= ( map_fun @ real @ ( nat > rat ) @ $o @ $o @ rep_real @ ( id @ $o )
@ ^ [X9: nat > rat] :
? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K3 @ N3 )
=> ( ord_less @ rat @ R5 @ ( X9 @ N3 ) ) ) ) ) ) ).
% Real.positive_def
thf(fact_7264_inverse__real__def,axiom,
( ( inverse_inverse @ real )
= ( map_fun @ real @ ( nat > rat ) @ ( nat > rat ) @ real @ rep_real @ real2
@ ^ [X9: nat > rat] :
( if @ ( nat > rat ) @ ( vanishes @ X9 )
@ ^ [N3: nat] : ( zero_zero @ rat )
@ ^ [N3: nat] : ( inverse_inverse @ rat @ ( X9 @ N3 ) ) ) ) ) ).
% inverse_real_def
thf(fact_7265_uminus__real__def,axiom,
( ( uminus_uminus @ real )
= ( map_fun @ real @ ( nat > rat ) @ ( nat > rat ) @ real @ rep_real @ real2
@ ^ [X9: nat > rat,N3: nat] : ( uminus_uminus @ rat @ ( X9 @ N3 ) ) ) ) ).
% uminus_real_def
thf(fact_7266_times__real__def,axiom,
( ( times_times @ real )
= ( map_fun @ real @ ( nat > rat ) @ ( ( nat > rat ) > nat > rat ) @ ( real > real ) @ rep_real @ ( map_fun @ real @ ( nat > rat ) @ ( nat > rat ) @ real @ rep_real @ real2 )
@ ^ [X9: nat > rat,Y8: nat > rat,N3: nat] : ( times_times @ rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ).
% times_real_def
thf(fact_7267_plus__real__def,axiom,
( ( plus_plus @ real )
= ( map_fun @ real @ ( nat > rat ) @ ( ( nat > rat ) > nat > rat ) @ ( real > real ) @ rep_real @ ( map_fun @ real @ ( nat > rat ) @ ( nat > rat ) @ real @ rep_real @ real2 )
@ ^ [X9: nat > rat,Y8: nat > rat,N3: nat] : ( plus_plus @ rat @ ( X9 @ N3 ) @ ( Y8 @ N3 ) ) ) ) ).
% plus_real_def
thf(fact_7268_cr__real__def,axiom,
( cr_real
= ( ^ [X: nat > rat,Y: real] :
( ( realrel @ X @ X )
& ( ( real2 @ X )
= Y ) ) ) ) ).
% cr_real_def
thf(fact_7269_Bseq__monoseq__convergent_H__dec,axiom,
! [F2: nat > real,M7: nat] :
( ( bfun @ nat @ real
@ ^ [N3: nat] : ( F2 @ ( plus_plus @ nat @ N3 @ M7 ) )
@ ( at_top @ nat ) )
=> ( ! [M4: nat,N2: nat] :
( ( ord_less_eq @ nat @ M7 @ M4 )
=> ( ( ord_less_eq @ nat @ M4 @ N2 )
=> ( ord_less_eq @ real @ ( F2 @ N2 ) @ ( F2 @ M4 ) ) ) )
=> ( topolo6863149650580417670ergent @ real @ F2 ) ) ) ).
% Bseq_monoseq_convergent'_dec
thf(fact_7270_convergent__mult__const__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( topolo6863149650580417670ergent @ A
@ ^ [N3: nat] : ( times_times @ A @ C2 @ ( F2 @ N3 ) ) )
= ( topolo6863149650580417670ergent @ A @ F2 ) ) ) ) ).
% convergent_mult_const_iff
thf(fact_7271_convergent__mult__const__right__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( topolo6863149650580417670ergent @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ C2 ) )
= ( topolo6863149650580417670ergent @ A @ F2 ) ) ) ) ).
% convergent_mult_const_right_iff
thf(fact_7272_lim__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: nat > A,X3: A] :
( ( topolo6863149650580417670ergent @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( F2 @ N2 ) @ X3 )
=> ( ord_less_eq @ A @ ( topolo3827282254853284352ce_Lim @ nat @ A @ ( at_top @ nat ) @ F2 ) @ X3 ) ) ) ) ).
% lim_le
thf(fact_7273_real_Opcr__cr__eq,axiom,
pcr_real = cr_real ).
% real.pcr_cr_eq
thf(fact_7274_Bseq__mono__convergent,axiom,
! [X8: nat > real] :
( ( bfun @ nat @ real @ X8 @ ( at_top @ nat ) )
=> ( ! [M4: nat,N2: nat] :
( ( ord_less_eq @ nat @ M4 @ N2 )
=> ( ord_less_eq @ real @ ( X8 @ M4 ) @ ( X8 @ N2 ) ) )
=> ( topolo6863149650580417670ergent @ real @ X8 ) ) ) ).
% Bseq_mono_convergent
thf(fact_7275_convergent__realpow,axiom,
! [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) )
=> ( topolo6863149650580417670ergent @ real @ ( power_power @ real @ X3 ) ) ) ) ).
% convergent_realpow
thf(fact_7276_Bseq__monoseq__convergent_H__inc,axiom,
! [F2: nat > real,M7: nat] :
( ( bfun @ nat @ real
@ ^ [N3: nat] : ( F2 @ ( plus_plus @ nat @ N3 @ M7 ) )
@ ( at_top @ nat ) )
=> ( ! [M4: nat,N2: nat] :
( ( ord_less_eq @ nat @ M7 @ M4 )
=> ( ( ord_less_eq @ nat @ M4 @ N2 )
=> ( ord_less_eq @ real @ ( F2 @ M4 ) @ ( F2 @ N2 ) ) ) )
=> ( topolo6863149650580417670ergent @ real @ F2 ) ) ) ).
% Bseq_monoseq_convergent'_inc
thf(fact_7277_cauchy__filter__metric__filtermap,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V768167426530841204y_dist @ B )
& ( topolo7287701948861334536_space @ B ) )
=> ! [F2: A > B,F4: filter @ A] :
( ( topolo6773858410816713723filter @ B @ ( filtermap @ A @ B @ F2 @ F4 ) )
= ( ! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [P3: A > $o] :
( ( eventually @ A @ P3 @ F4 )
& ! [X: A,Y: A] :
( ( ( P3 @ X )
& ( P3 @ Y ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) @ E3 ) ) ) ) ) ) ) ).
% cauchy_filter_metric_filtermap
thf(fact_7278_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( ( comm_semiring_0 @ B )
& ( comm_semiring_0 @ A ) )
=> ! [A4: A > B > $o,B6: C > D > $o] :
( ( A4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( A > ( list @ C ) > A ) @ ( B > ( list @ D ) > B ) @ ( bNF_rel_fun @ C @ D @ A @ B @ B6 @ A4 ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ C ) > A ) @ ( ( list @ D ) > B ) @ A4 @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ A @ B @ ( list_all2 @ C @ D @ B6 ) @ A4 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% horner_sum_transfer
thf(fact_7279_filtermap__nhds__times,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,A2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filtermap @ A @ A @ ( times_times @ A @ C2 ) @ ( topolo7230453075368039082e_nhds @ A @ A2 ) )
= ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ A2 ) ) ) ) ) ).
% filtermap_nhds_times
thf(fact_7280_list__all2__conv__all__nth,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P3: A > B > $o,Xs: list @ A,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P3 @ ( nth @ A @ Xs @ I4 ) @ ( nth @ B @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_7281_list__all2__all__nthI,axiom,
! [A: $tType,B: $tType,A2: list @ A,B2: list @ B,P: A > B > $o] :
( ( ( size_size @ ( list @ A ) @ A2 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ A2 ) )
=> ( P @ ( nth @ A @ A2 @ N2 ) @ ( nth @ B @ B2 @ N2 ) ) )
=> ( list_all2 @ A @ B @ P @ A2 @ B2 ) ) ) ).
% list_all2_all_nthI
thf(fact_7282_list__all2__nthD2,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs2: list @ A,Ys: list @ B,P6: nat] :
( ( list_all2 @ A @ B @ P @ Xs2 @ Ys )
=> ( ( ord_less @ nat @ P6 @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( P @ ( nth @ A @ Xs2 @ P6 ) @ ( nth @ B @ Ys @ P6 ) ) ) ) ).
% list_all2_nthD2
thf(fact_7283_list__all2__nthD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs2: list @ A,Ys: list @ B,P6: nat] :
( ( list_all2 @ A @ B @ P @ Xs2 @ Ys )
=> ( ( ord_less @ nat @ P6 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ P6 ) @ ( nth @ B @ Ys @ P6 ) ) ) ) ).
% list_all2_nthD
thf(fact_7284_at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A] :
( ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
= ( filtermap @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ X @ A2 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_to_0
thf(fact_7285_at__right__to__0,axiom,
! [A2: real] :
( ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) )
= ( filtermap @ real @ real
@ ^ [X: real] : ( plus_plus @ real @ X @ A2 )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% at_right_to_0
thf(fact_7286_sum__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( monoid_add @ A ) )
=> ! [A4: A > B > $o] :
( ( A4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A4 ) @ A4 @ ( groups8242544230860333062m_list @ A ) @ ( groups8242544230860333062m_list @ B ) ) ) ) ) ).
% sum_list_transfer
thf(fact_7287_filtermap__times__pos__at__right,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [C2: A,P6: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( filtermap @ A @ A @ ( times_times @ A @ C2 ) @ ( topolo174197925503356063within @ A @ P6 @ ( set_ord_greaterThan @ A @ P6 ) ) )
= ( topolo174197925503356063within @ A @ ( times_times @ A @ C2 @ P6 ) @ ( set_ord_greaterThan @ A @ ( times_times @ A @ C2 @ P6 ) ) ) ) ) ) ).
% filtermap_times_pos_at_right
thf(fact_7288_at__to__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) )
= ( filtermap @ A @ A @ ( inverse_inverse @ A ) @ ( at_infinity @ A ) ) ) ) ).
% at_to_infinity
thf(fact_7289_at__right__to__top,axiom,
( ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) )
= ( filtermap @ real @ real @ ( inverse_inverse @ real ) @ ( at_top @ real ) ) ) ).
% at_right_to_top
thf(fact_7290_at__top__to__right,axiom,
( ( at_top @ real )
= ( filtermap @ real @ real @ ( inverse_inverse @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% at_top_to_right
thf(fact_7291_filtermap__ln__at__right,axiom,
( ( filtermap @ real @ real @ ( ln_ln @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
= ( at_bot @ real ) ) ).
% filtermap_ln_at_right
thf(fact_7292_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ( ordering_top @ A
@ ^ [X: A,Y: A] : ( ord_less_eq @ A @ Y @ X )
@ ^ [X: A,Y: A] : ( ord_less @ A @ Y @ X )
@ ( bot_bot @ A ) ) ) ).
% bot.ordering_top_axioms
thf(fact_7293_pair__lessI2,axiom,
! [A2: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( ord_less @ nat @ S @ T2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A2 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_7294_pair__less__iff1,axiom,
! [X3: nat,Y3: nat,Z: nat] :
( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X3 @ Y3 ) @ ( product_Pair @ nat @ nat @ X3 @ Z ) ) @ fun_pair_less )
= ( ord_less @ nat @ Y3 @ Z ) ) ).
% pair_less_iff1
thf(fact_7295_ordering__top_Oextremum__uniqueI,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
=> ( ( Less_eq2 @ Top @ A2 )
=> ( A2 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_7296_ordering__top_Onot__eq__extremum,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
=> ( ( A2 != Top )
= ( Less @ A2 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_7297_ordering__top_Oextremum__unique,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
=> ( ( Less_eq2 @ Top @ A2 )
= ( A2 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_7298_ordering__top_Oextremum__strict,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
=> ~ ( Less @ Top @ A2 ) ) ).
% ordering_top.extremum_strict
thf(fact_7299_ordering__top_Oextremum,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
=> ( Less_eq2 @ A2 @ Top ) ) ).
% ordering_top.extremum
thf(fact_7300_pair__lessI1,axiom,
! [A2: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A2 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_7301_gcd__nat_Oordering__top__axioms,axiom,
( ordering_top @ nat @ ( dvd_dvd @ nat )
@ ^ [M3: nat,N3: nat] :
( ( dvd_dvd @ nat @ M3 @ N3 )
& ( M3 != N3 ) )
@ ( zero_zero @ nat ) ) ).
% gcd_nat.ordering_top_axioms
thf(fact_7302_top_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ( ordering_top @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ A ) ) ) ).
% top.ordering_top_axioms
thf(fact_7303_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top @ nat
@ ^ [X: nat,Y: nat] : ( ord_less_eq @ nat @ Y @ X )
@ ^ [X: nat,Y: nat] : ( ord_less @ nat @ Y @ X )
@ ( zero_zero @ nat ) ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_7304_pair__leqI2,axiom,
! [A2: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( ord_less_eq @ nat @ S @ T2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A2 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_7305_pair__leqI1,axiom,
! [A2: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A2 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_7306_wmax__insertI,axiom,
! [Y3: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat ),X3: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y3 @ YS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y3 ) @ fun_pair_leq )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_max_weak )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X3 @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% wmax_insertI
thf(fact_7307_wmin__insertI,axiom,
! [X3: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y3: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X3 @ XS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y3 ) @ fun_pair_leq )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_min_weak )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ ( insert @ ( product_prod @ nat @ nat ) @ Y3 @ YS ) ) @ fun_min_weak ) ) ) ) ).
% wmin_insertI
thf(fact_7308_wmin__emptyI,axiom,
! [X8: set @ ( product_prod @ nat @ nat )] : ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X8 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_weak ) ).
% wmin_emptyI
thf(fact_7309_wmax__emptyI,axiom,
! [X8: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite @ ( product_prod @ nat @ nat ) @ X8 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ X8 ) @ fun_max_weak ) ) ).
% wmax_emptyI
thf(fact_7310_smax__insertI,axiom,
! [Y3: product_prod @ nat @ nat,Y7: set @ ( product_prod @ nat @ nat ),X3: product_prod @ nat @ nat,X8: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y3 @ Y7 )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y3 ) @ fun_pair_less )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X8 @ Y7 ) @ fun_max_strict )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X3 @ X8 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).
% smax_insertI
thf(fact_7311_smin__insertI,axiom,
! [X3: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y3: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X3 @ XS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y3 ) @ fun_pair_less )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_min_strict )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ ( insert @ ( product_prod @ nat @ nat ) @ Y3 @ YS ) ) @ fun_min_strict ) ) ) ) ).
% smin_insertI
thf(fact_7312_less__by__empty,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A )] :
( ( A4
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B6 ) ) ).
% less_by_empty
thf(fact_7313_smax__emptyI,axiom,
! [Y7: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite @ ( product_prod @ nat @ nat ) @ Y7 )
=> ( ( Y7
!= ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ Y7 ) @ fun_max_strict ) ) ) ).
% smax_emptyI
thf(fact_7314_smin__emptyI,axiom,
! [X8: set @ ( product_prod @ nat @ nat )] :
( ( X8
!= ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X8 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict ) ) ).
% smin_emptyI
thf(fact_7315_min__weak__def,axiom,
( fun_min_weak
= ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% min_weak_def
thf(fact_7316_max__weak__def,axiom,
( fun_max_weak
= ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% max_weak_def
thf(fact_7317_max__strict__def,axiom,
( fun_max_strict
= ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_less ) ) ).
% max_strict_def
thf(fact_7318_min__strict__def,axiom,
( fun_min_strict
= ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_less ) ) ).
% min_strict_def
thf(fact_7319_min__rpair__set,axiom,
fun_reduction_pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict @ fun_min_weak ) ).
% min_rpair_set
thf(fact_7320_max__rpair__set,axiom,
fun_reduction_pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_max_strict @ fun_max_weak ) ).
% max_rpair_set
thf(fact_7321_rp__inv__image__rp,axiom,
! [A: $tType,B: $tType,P: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F2: B > A] :
( ( fun_reduction_pair @ A @ P )
=> ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P @ F2 ) ) ) ).
% rp_inv_image_rp
thf(fact_7322_relpow__finite__bounded1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite @ ( product_prod @ A @ A ) @ R3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R3 )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R3 )
@ ( collect @ nat
@ ^ [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R3 ) ) ) ) ) ) ) ) ) ).
% relpow_finite_bounded1
thf(fact_7323_finite__relpow,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),N: nat] :
( ( finite_finite @ ( product_prod @ A @ A ) @ R3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite @ ( product_prod @ A @ A ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R3 ) ) ) ) ).
% finite_relpow
thf(fact_7324_relpow__0__E,axiom,
! [A: $tType,X3: A,Y3: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R3 ) )
=> ( X3 = Y3 ) ) ).
% relpow_0_E
thf(fact_7325_relpow__0__I,axiom,
! [A: $tType,X3: A,R3: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R3 ) ) ).
% relpow_0_I
thf(fact_7326_relpow__E2,axiom,
! [A: $tType,X3: A,Z: A,N: nat,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R3 ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X3 != Z ) )
=> ~ ! [Y4: A,M4: nat] :
( ( N
= ( suc @ M4 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M4 @ R3 ) ) ) ) ) ) ).
% relpow_E2
thf(fact_7327_relpow__E,axiom,
! [A: $tType,X3: A,Z: A,N: nat,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R3 ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X3 != Z ) )
=> ~ ! [Y4: A,M4: nat] :
( ( N
= ( suc @ M4 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M4 @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ R3 ) ) ) ) ) ).
% relpow_E
thf(fact_7328_relpow__empty,axiom,
! [A: $tType,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% relpow_empty
thf(fact_7329_relpow__finite__bounded,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite @ ( product_prod @ A @ A ) @ R3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R3 )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R3 )
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R3 ) ) ) ) ) ) ) ).
% relpow_finite_bounded
thf(fact_7330_relpow__fun__conv,axiom,
! [A: $tType,A2: A,B2: A,N: nat,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R3 ) )
= ( ? [F3: nat > A] :
( ( ( F3 @ ( zero_zero @ nat ) )
= A2 )
& ( ( F3 @ N )
= B2 )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F3 @ I4 ) @ ( F3 @ ( suc @ I4 ) ) ) @ R3 ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_7331_ntrancl__def,axiom,
! [A: $tType] :
( ( transitive_ntrancl @ A )
= ( ^ [N3: nat,R6: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [I4: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ I4 @ R6 )
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I4 )
& ( ord_less_eq @ nat @ I4 @ ( suc @ N3 ) ) ) ) ) ) ) ) ).
% ntrancl_def
thf(fact_7332_trancl__finite__eq__relpow,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite @ ( product_prod @ A @ A ) @ R3 )
=> ( ( transitive_trancl @ A @ R3 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R3 )
@ ( collect @ nat
@ ^ [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R3 ) ) ) ) ) ) ) ) ).
% trancl_finite_eq_relpow
thf(fact_7333_ntrancl__Zero,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_ntrancl @ A @ ( zero_zero @ nat ) @ R3 )
= R3 ) ).
% ntrancl_Zero
thf(fact_7334_trancl__power,axiom,
! [A: $tType,P6: product_prod @ A @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P6 @ ( transitive_trancl @ A @ R3 ) )
= ( ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( member @ ( product_prod @ A @ A ) @ P6 @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R3 ) ) ) ) ) ).
% trancl_power
thf(fact_7335_rtrancl__finite__eq__relpow,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite @ ( product_prod @ A @ A ) @ R3 )
=> ( ( transitive_rtrancl @ A @ R3 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R3 )
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R3 ) ) ) ) ) ) ) ).
% rtrancl_finite_eq_relpow
thf(fact_7336_rp__inv__image__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_rp_inv_image @ A @ B )
= ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
@ ^ [R6: set @ ( product_prod @ A @ A ),S8: set @ ( product_prod @ A @ A ),F3: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R6 @ F3 ) @ ( inv_image @ A @ B @ S8 @ F3 ) ) ) ) ).
% rp_inv_image_def
thf(fact_7337_pred__nat__trancl__eq__le,axiom,
! [M: nat,N: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M @ N ) @ ( transitive_rtrancl @ nat @ pred_nat ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% pred_nat_trancl_eq_le
thf(fact_7338_less__eq,axiom,
! [M: nat,N: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M @ N ) @ ( transitive_trancl @ nat @ pred_nat ) )
= ( ord_less @ nat @ M @ N ) ) ).
% less_eq
thf(fact_7339_euclidean__size__times__nonunit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ B2 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ) ) ).
% euclidean_size_times_nonunit
thf(fact_7340_dropWhile__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs2: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs2 ) ) )
=> ( ( nth @ A @ ( dropWhile @ A @ P @ Xs2 ) @ J )
= ( nth @ A @ Xs2 @ ( plus_plus @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs2 ) ) ) ) ) ) ).
% dropWhile_nth
thf(fact_7341_euclidean__size__of__nat,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( euclid6346220572633701492n_size @ A @ ( semiring_1_of_nat @ A @ N ) )
= N ) ) ).
% euclidean_size_of_nat
thf(fact_7342_euclidean__size__eq__0__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A] :
( ( ( euclid6346220572633701492n_size @ A @ B2 )
= ( zero_zero @ nat ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% euclidean_size_eq_0_iff
thf(fact_7343_size__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ( ( euclid6346220572633701492n_size @ A @ ( zero_zero @ A ) )
= ( zero_zero @ nat ) ) ) ).
% size_0
thf(fact_7344_euclidean__size__numeral,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [K: num] :
( ( euclid6346220572633701492n_size @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ nat @ K ) ) ) ).
% euclidean_size_numeral
thf(fact_7345_euclidean__size__greater__0__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
= ( B2
!= ( zero_zero @ A ) ) ) ) ).
% euclidean_size_greater_0_iff
thf(fact_7346_dvd__euclidean__size__eq__imp__dvd,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( ( euclid6346220572633701492n_size @ A @ A2 )
= ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ) ).
% dvd_euclidean_size_eq_imp_dvd
thf(fact_7347_sorted__dropWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( dropWhile @ A @ P @ Xs2 ) ) ) ) ).
% sorted_dropWhile
thf(fact_7348_length__dropWhile__le,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_dropWhile_le
thf(fact_7349_unit__iff__euclidean__size,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
= ( ( ( euclid6346220572633701492n_size @ A @ A2 )
= ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) )
& ( A2
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_iff_euclidean_size
thf(fact_7350_size__mult__mono_H,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A2 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ B2 @ A2 ) ) ) ) ) ).
% size_mult_mono'
thf(fact_7351_size__mult__mono,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A2 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% size_mult_mono
thf(fact_7352_dvd__proper__imp__size__less,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ~ ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A2 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ) ) ).
% dvd_proper_imp_size_less
thf(fact_7353_dvd__imp__size__le,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A2 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ) ).
% dvd_imp_size_le
thf(fact_7354_mod__size__less,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% mod_size_less
thf(fact_7355_euclidean__size__int__def,axiom,
( ( euclid6346220572633701492n_size @ int )
= ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) ) ).
% euclidean_size_int_def
thf(fact_7356_divmod__cases,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,A2: A] :
( ( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( modulo_modulo @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( A2
!= ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) ) ) )
=> ( ( ( B2
!= ( zero_zero @ A ) )
=> ! [Q4: A,R: A] :
( ( ( euclid7384307370059645450egment @ A @ R )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( R
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= Q4 )
=> ( ( ( modulo_modulo @ A @ A2 @ B2 )
= R )
=> ( A2
!= ( plus_plus @ A @ ( times_times @ A @ Q4 @ B2 ) @ R ) ) ) ) ) ) ) )
=> ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divmod_cases
thf(fact_7357_mod__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R2: A,Q3: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R2 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q3 @ B2 ) @ R2 )
= A2 )
=> ( ( modulo_modulo @ A @ A2 @ B2 )
= R2 ) ) ) ) ) ) ).
% mod_eqI
thf(fact_7358_division__segment__numeral,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [K: num] :
( ( euclid7384307370059645450egment @ A @ ( numeral_numeral @ A @ K ) )
= ( one_one @ A ) ) ) ).
% division_segment_numeral
thf(fact_7359_division__segment__of__nat,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( euclid7384307370059645450egment @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( one_one @ A ) ) ) ).
% division_segment_of_nat
thf(fact_7360_division__segment__euclidean__size,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A2 ) @ ( semiring_1_of_nat @ A @ ( euclid6346220572633701492n_size @ A @ A2 ) ) )
= A2 ) ) ).
% division_segment_euclidean_size
thf(fact_7361_division__segment__eq__sgn,axiom,
! [K: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ( euclid7384307370059645450egment @ int @ K )
= ( sgn_sgn @ int @ K ) ) ) ).
% division_segment_eq_sgn
thf(fact_7362_division__segment__not__0,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A] :
( ( euclid7384307370059645450egment @ A @ A2 )
!= ( zero_zero @ A ) ) ) ).
% division_segment_not_0
thf(fact_7363_division__segment__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( euclid7384307370059645450egment @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A2 ) @ ( euclid7384307370059645450egment @ A @ B2 ) ) ) ) ) ) ).
% division_segment_mult
thf(fact_7364_division__segment__mod,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( euclid7384307370059645450egment @ A @ ( modulo_modulo @ A @ A2 @ B2 ) )
= ( euclid7384307370059645450egment @ A @ B2 ) ) ) ) ) ).
% division_segment_mod
thf(fact_7365_of__nat__euclidean__size,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ( semiring_1_of_nat @ A @ ( euclid6346220572633701492n_size @ A @ A2 ) )
= ( divide_divide @ A @ A2 @ ( euclid7384307370059645450egment @ A @ A2 ) ) ) ) ).
% of_nat_euclidean_size
thf(fact_7366_division__segment__int__def,axiom,
( ( euclid7384307370059645450egment @ int )
= ( ^ [K3: int] : ( if @ int @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ K3 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% division_segment_int_def
thf(fact_7367_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A,B2: A] :
( ( ( euclid7384307370059645450egment @ A @ A2 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A2 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ) ).
% unique_euclidean_semiring_class.div_eq_0_iff
thf(fact_7368_div__bounded,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R2: A,Q3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R2 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ Q3 @ B2 ) @ R2 ) @ B2 )
= Q3 ) ) ) ) ) ).
% div_bounded
thf(fact_7369_div__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R2: A,Q3: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R2 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q3 @ B2 ) @ R2 )
= A2 )
=> ( ( divide_divide @ A @ A2 @ B2 )
= Q3 ) ) ) ) ) ) ).
% div_eqI
thf(fact_7370_find__dropWhile,axiom,
! [A: $tType] :
( ( find @ A )
= ( ^ [P3: A > $o,Xs: list @ A] :
( case_list @ ( option @ A ) @ A @ ( none @ A )
@ ^ [X: A,Xa4: list @ A] : ( some @ A @ X )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs ) ) ) ) ).
% find_dropWhile
thf(fact_7371_rat__number__expand_I5_J,axiom,
! [K: num] :
( ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) )
= ( fract @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ).
% rat_number_expand(5)
thf(fact_7372_less__rat,axiom,
! [B2: int,D3: int,A2: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( fract @ C2 @ D3 ) )
= ( ord_less @ int @ ( times_times @ int @ ( times_times @ int @ A2 @ D3 ) @ ( times_times @ int @ B2 @ D3 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B2 ) @ ( times_times @ int @ B2 @ D3 ) ) ) ) ) ) ).
% less_rat
thf(fact_7373_add__rat,axiom,
! [B2: int,D3: int,A2: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( plus_plus @ rat @ ( fract @ A2 @ B2 ) @ ( fract @ C2 @ D3 ) )
= ( fract @ ( plus_plus @ int @ ( times_times @ int @ A2 @ D3 ) @ ( times_times @ int @ C2 @ B2 ) ) @ ( times_times @ int @ B2 @ D3 ) ) ) ) ) ).
% add_rat
thf(fact_7374_le__rat,axiom,
! [B2: int,D3: int,A2: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( ord_less_eq @ rat @ ( fract @ A2 @ B2 ) @ ( fract @ C2 @ D3 ) )
= ( ord_less_eq @ int @ ( times_times @ int @ ( times_times @ int @ A2 @ D3 ) @ ( times_times @ int @ B2 @ D3 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B2 ) @ ( times_times @ int @ B2 @ D3 ) ) ) ) ) ) ).
% le_rat
thf(fact_7375_diff__rat,axiom,
! [B2: int,D3: int,A2: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( minus_minus @ rat @ ( fract @ A2 @ B2 ) @ ( fract @ C2 @ D3 ) )
= ( fract @ ( minus_minus @ int @ ( times_times @ int @ A2 @ D3 ) @ ( times_times @ int @ C2 @ B2 ) ) @ ( times_times @ int @ B2 @ D3 ) ) ) ) ) ).
% diff_rat
thf(fact_7376_eq__rat_I1_J,axiom,
! [B2: int,D3: int,A2: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( ( fract @ A2 @ B2 )
= ( fract @ C2 @ D3 ) )
= ( ( times_times @ int @ A2 @ D3 )
= ( times_times @ int @ C2 @ B2 ) ) ) ) ) ).
% eq_rat(1)
thf(fact_7377_mult__rat__cancel,axiom,
! [C2: int,A2: int,B2: int] :
( ( C2
!= ( zero_zero @ int ) )
=> ( ( fract @ ( times_times @ int @ C2 @ A2 ) @ ( times_times @ int @ C2 @ B2 ) )
= ( fract @ A2 @ B2 ) ) ) ).
% mult_rat_cancel
thf(fact_7378_eq__rat_I2_J,axiom,
! [A2: int] :
( ( fract @ A2 @ ( zero_zero @ int ) )
= ( fract @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% eq_rat(2)
thf(fact_7379_eq__rat_I3_J,axiom,
! [A2: int,C2: int] :
( ( fract @ ( zero_zero @ int ) @ A2 )
= ( fract @ ( zero_zero @ int ) @ C2 ) ) ).
% eq_rat(3)
thf(fact_7380_rat__number__collapse_I6_J,axiom,
! [K: int] :
( ( fract @ K @ ( zero_zero @ int ) )
= ( zero_zero @ rat ) ) ).
% rat_number_collapse(6)
thf(fact_7381_rat__number__collapse_I1_J,axiom,
! [K: int] :
( ( fract @ ( zero_zero @ int ) @ K )
= ( zero_zero @ rat ) ) ).
% rat_number_collapse(1)
thf(fact_7382_Rat__induct__pos,axiom,
! [P: rat > $o,Q3: rat] :
( ! [A6: int,B5: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ( P @ ( fract @ A6 @ B5 ) ) )
=> ( P @ Q3 ) ) ).
% Rat_induct_pos
thf(fact_7383_Zero__rat__def,axiom,
( ( zero_zero @ rat )
= ( fract @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% Zero_rat_def
thf(fact_7384_rat__number__expand_I3_J,axiom,
( ( numeral_numeral @ rat )
= ( ^ [K3: num] : ( fract @ ( numeral_numeral @ int @ K3 ) @ ( one_one @ int ) ) ) ) ).
% rat_number_expand(3)
thf(fact_7385_rat__number__collapse_I3_J,axiom,
! [W: num] :
( ( fract @ ( numeral_numeral @ int @ W ) @ ( one_one @ int ) )
= ( numeral_numeral @ rat @ W ) ) ).
% rat_number_collapse(3)
thf(fact_7386_Fract__of__nat__eq,axiom,
! [K: nat] :
( ( fract @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) )
= ( semiring_1_of_nat @ rat @ K ) ) ).
% Fract_of_nat_eq
thf(fact_7387_of__rat__rat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [B2: int,A2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( field_char_0_of_rat @ A @ ( fract @ A2 @ B2 ) )
= ( divide_divide @ A @ ( ring_1_of_int @ A @ A2 ) @ ( ring_1_of_int @ A @ B2 ) ) ) ) ) ).
% of_rat_rat
thf(fact_7388_Fract__less__zero__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( zero_zero @ rat ) )
= ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% Fract_less_zero_iff
thf(fact_7389_zero__less__Fract__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ ( fract @ A2 @ B2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% zero_less_Fract_iff
thf(fact_7390_Fract_Oabs__eq,axiom,
( fract
= ( ^ [Xa4: int,X: int] :
( abs_Rat
@ ( if @ ( product_prod @ int @ int )
@ ( X
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ Xa4 @ X ) ) ) ) ) ).
% Fract.abs_eq
thf(fact_7391_positive__rat,axiom,
! [A2: int,B2: int] :
( ( positive @ ( fract @ A2 @ B2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ A2 @ B2 ) ) ) ).
% positive_rat
thf(fact_7392_Fract__less__one__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( one_one @ rat ) )
= ( ord_less @ int @ A2 @ B2 ) ) ) ).
% Fract_less_one_iff
thf(fact_7393_one__less__Fract__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( one_one @ rat ) @ ( fract @ A2 @ B2 ) )
= ( ord_less @ int @ B2 @ A2 ) ) ) ).
% one_less_Fract_iff
thf(fact_7394_Fract__add__one,axiom,
! [N: int,M: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( fract @ ( plus_plus @ int @ M @ N ) @ N )
= ( plus_plus @ rat @ ( fract @ M @ N ) @ ( one_one @ rat ) ) ) ) ).
% Fract_add_one
thf(fact_7395_Fract_Otransfer,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat )
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ rat
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ pcr_rat )
@ ^ [A5: int,B4: int] :
( if @ ( product_prod @ int @ int )
@ ( B4
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A5 @ B4 ) )
@ fract ) ).
% Fract.transfer
thf(fact_7396_Fract__le__zero__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( fract @ A2 @ B2 ) @ ( zero_zero @ rat ) )
= ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% Fract_le_zero_iff
thf(fact_7397_zero__le__Fract__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ ( fract @ A2 @ B2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% zero_le_Fract_iff
thf(fact_7398_Fract__le__one__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( fract @ A2 @ B2 ) @ ( one_one @ rat ) )
= ( ord_less_eq @ int @ A2 @ B2 ) ) ) ).
% Fract_le_one_iff
thf(fact_7399_one__le__Fract__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( one_one @ rat ) @ ( fract @ A2 @ B2 ) )
= ( ord_less_eq @ int @ B2 @ A2 ) ) ) ).
% one_le_Fract_iff
thf(fact_7400_rat__number__collapse_I4_J,axiom,
! [W: num] :
( ( fract @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ W ) ) ) ).
% rat_number_collapse(4)
thf(fact_7401_quotient__of__def,axiom,
( quotient_of
= ( ^ [X: rat] :
( the @ ( product_prod @ int @ int )
@ ^ [Pair: product_prod @ int @ int] :
( ( X
= ( fract @ ( product_fst @ int @ int @ Pair ) @ ( product_snd @ int @ int @ Pair ) ) )
& ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ Pair ) )
& ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ Pair ) @ ( product_snd @ int @ int @ Pair ) ) ) ) ) ) ).
% quotient_of_def
thf(fact_7402_less__eq__enat__def,axiom,
( ( ord_less_eq @ extended_enat )
= ( ^ [M3: extended_enat] :
( extended_case_enat @ $o
@ ^ [N1: nat] :
( extended_case_enat @ $o
@ ^ [M1: nat] : ( ord_less_eq @ nat @ M1 @ N1 )
@ $false
@ M3 )
@ $true ) ) ) ).
% less_eq_enat_def
thf(fact_7403_coprime__power__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( power_power @ A @ A2 @ N ) @ B2 )
= ( ( algebr8660921524188924756oprime @ A @ A2 @ B2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% coprime_power_left_iff
thf(fact_7404_coprime__power__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( algebr8660921524188924756oprime @ A @ A2 @ ( power_power @ A @ B2 @ N ) )
= ( ( algebr8660921524188924756oprime @ A @ A2 @ B2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% coprime_power_right_iff
thf(fact_7405_coprime__mod__right__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( algebr8660921524188924756oprime @ A @ A2 @ ( modulo_modulo @ A @ B2 @ A2 ) )
= ( algebr8660921524188924756oprime @ A @ A2 @ B2 ) ) ) ) ).
% coprime_mod_right_iff
thf(fact_7406_coprime__mod__left__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( algebr8660921524188924756oprime @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 )
= ( algebr8660921524188924756oprime @ A @ A2 @ B2 ) ) ) ) ).
% coprime_mod_left_iff
thf(fact_7407_coprime__0__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A] :
( ( algebr8660921524188924756oprime @ A @ A2 @ ( zero_zero @ A ) )
= ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% coprime_0_right_iff
thf(fact_7408_coprime__0__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A] :
( ( algebr8660921524188924756oprime @ A @ ( zero_zero @ A ) @ A2 )
= ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% coprime_0_left_iff
thf(fact_7409_coprime__left__2__iff__odd,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ( algebr8660921524188924756oprime @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% coprime_left_2_iff_odd
thf(fact_7410_coprime__right__2__iff__odd,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ( algebr8660921524188924756oprime @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% coprime_right_2_iff_odd
thf(fact_7411_normalize__stable,axiom,
! [Q3: int,P6: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Q3 )
=> ( ( algebr8660921524188924756oprime @ int @ P6 @ Q3 )
=> ( ( normalize @ ( product_Pair @ int @ int @ P6 @ Q3 ) )
= ( product_Pair @ int @ int @ P6 @ Q3 ) ) ) ) ).
% normalize_stable
thf(fact_7412_Rat__cases,axiom,
! [Q3: rat] :
~ ! [A6: int,B5: int] :
( ( Q3
= ( fract @ A6 @ B5 ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ~ ( algebr8660921524188924756oprime @ int @ A6 @ B5 ) ) ) ).
% Rat_cases
thf(fact_7413_Rat__induct,axiom,
! [P: rat > $o,Q3: rat] :
( ! [A6: int,B5: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ( ( algebr8660921524188924756oprime @ int @ A6 @ B5 )
=> ( P @ ( fract @ A6 @ B5 ) ) ) )
=> ( P @ Q3 ) ) ).
% Rat_induct
thf(fact_7414_div__gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( ( A2
!= ( zero_zero @ A ) )
| ( B2
!= ( zero_zero @ A ) ) )
=> ( algebr8660921524188924756oprime @ A @ ( divide_divide @ A @ A2 @ ( gcd_gcd @ A @ A2 @ B2 ) ) @ ( divide_divide @ A @ B2 @ ( gcd_gcd @ A @ A2 @ B2 ) ) ) ) ) ).
% div_gcd_coprime
thf(fact_7415_gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,A3: A,B3: A] :
( ( ( gcd_gcd @ A @ A2 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A2
= ( times_times @ A @ A3 @ ( gcd_gcd @ A @ A2 @ B2 ) ) )
=> ( ( B2
= ( times_times @ A @ B3 @ ( gcd_gcd @ A @ A2 @ B2 ) ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B3 ) ) ) ) ) ).
% gcd_coprime
thf(fact_7416_gcd__coprime__exists,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( ( gcd_gcd @ A @ A2 @ B2 )
!= ( zero_zero @ A ) )
=> ? [A11: A,B10: A] :
( ( A2
= ( times_times @ A @ A11 @ ( gcd_gcd @ A @ A2 @ B2 ) ) )
& ( B2
= ( times_times @ A @ B10 @ ( gcd_gcd @ A @ A2 @ B2 ) ) )
& ( algebr8660921524188924756oprime @ A @ A11 @ B10 ) ) ) ) ).
% gcd_coprime_exists
thf(fact_7417_Rat__cases__nonzero,axiom,
! [Q3: rat] :
( ! [A6: int,B5: int] :
( ( Q3
= ( fract @ A6 @ B5 ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ( ( A6
!= ( zero_zero @ int ) )
=> ~ ( algebr8660921524188924756oprime @ int @ A6 @ B5 ) ) ) )
=> ( Q3
= ( zero_zero @ rat ) ) ) ).
% Rat_cases_nonzero
thf(fact_7418_Rats__cases_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [X3: A] :
( ( member @ A @ X3 @ ( field_char_0_Rats @ A ) )
=> ~ ! [A6: int,B5: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ( ( algebr8660921524188924756oprime @ int @ A6 @ B5 )
=> ( X3
!= ( divide_divide @ A @ ( ring_1_of_int @ A @ A6 ) @ ( ring_1_of_int @ A @ B5 ) ) ) ) ) ) ) ).
% Rats_cases'
thf(fact_7419_quotient__of__unique,axiom,
! [R2: rat] :
? [X4: product_prod @ int @ int] :
( ( R2
= ( fract @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ X4 ) ) )
& ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ X4 ) )
& ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ X4 ) )
& ! [Y5: product_prod @ int @ int] :
( ( ( R2
= ( fract @ ( product_fst @ int @ int @ Y5 ) @ ( product_snd @ int @ int @ Y5 ) ) )
& ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ Y5 ) )
& ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ Y5 ) @ ( product_snd @ int @ int @ Y5 ) ) )
=> ( Y5 = X4 ) ) ) ).
% quotient_of_unique
thf(fact_7420_less__enat__def,axiom,
( ( ord_less @ extended_enat )
= ( ^ [M3: extended_enat,N3: extended_enat] :
( extended_case_enat @ $o
@ ^ [M1: nat] : ( extended_case_enat @ $o @ ( ord_less @ nat @ M1 ) @ $true @ N3 )
@ $false
@ M3 ) ) ) ).
% less_enat_def
thf(fact_7421_set__encode__vimage__Suc,axiom,
! [A4: set @ nat] :
( ( nat_set_encode @ ( vimage @ nat @ nat @ suc @ A4 ) )
= ( divide_divide @ nat @ ( nat_set_encode @ A4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% set_encode_vimage_Suc
thf(fact_7422_set__decode__div__2,axiom,
! [X3: nat] :
( ( nat_set_decode @ ( divide_divide @ nat @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( vimage @ nat @ nat @ suc @ ( nat_set_decode @ X3 ) ) ) ).
% set_decode_div_2
thf(fact_7423_coprime__int__iff,axiom,
! [M: nat,N: nat] :
( ( algebr8660921524188924756oprime @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
= ( algebr8660921524188924756oprime @ nat @ M @ N ) ) ).
% coprime_int_iff
thf(fact_7424_coprime__nat__abs__right__iff,axiom,
! [N: nat,K: int] :
( ( algebr8660921524188924756oprime @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
= ( algebr8660921524188924756oprime @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ).
% coprime_nat_abs_right_iff
thf(fact_7425_coprime__nat__abs__left__iff,axiom,
! [K: int,N: nat] :
( ( algebr8660921524188924756oprime @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
= ( algebr8660921524188924756oprime @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% coprime_nat_abs_left_iff
thf(fact_7426_coprime__Suc__0__left,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ).
% coprime_Suc_0_left
thf(fact_7427_coprime__Suc__0__right,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ).
% coprime_Suc_0_right
thf(fact_7428_vimage__Suc__insert__0,axiom,
! [A4: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert @ nat @ ( zero_zero @ nat ) @ A4 ) )
= ( vimage @ nat @ nat @ suc @ A4 ) ) ).
% vimage_Suc_insert_0
thf(fact_7429_coprime__diff__one__right__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( algebr8660921524188924756oprime @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% coprime_diff_one_right_nat
thf(fact_7430_coprime__diff__one__left__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( algebr8660921524188924756oprime @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ N ) ) ).
% coprime_diff_one_left_nat
thf(fact_7431_card__vimage__inj__on__le,axiom,
! [A: $tType,B: $tType,F2: A > B,D5: set @ A,A4: set @ B] :
( ( inj_on @ A @ B @ F2 @ D5 )
=> ( ( finite_finite @ B @ A4 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A4 ) @ D5 ) ) @ ( finite_card @ B @ A4 ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_7432_Rats__abs__nat__div__natE,axiom,
! [X3: real] :
( ( member @ real @ X3 @ ( field_char_0_Rats @ real ) )
=> ~ ! [M4: nat,N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ( ( abs_abs @ real @ X3 )
= ( divide_divide @ real @ ( semiring_1_of_nat @ real @ M4 ) @ ( semiring_1_of_nat @ real @ N2 ) ) )
=> ~ ( algebr8660921524188924756oprime @ nat @ M4 @ N2 ) ) ) ) ).
% Rats_abs_nat_div_natE
thf(fact_7433_sorted__wrt__iff__nth__Suc__transp,axiom,
! [A: $tType,P: A > A > $o,Xs2: list @ A] :
( ( transp @ A @ P )
=> ( ( sorted_wrt @ A @ P @ Xs2 )
= ( ! [I4: nat] :
( ( ord_less @ nat @ ( suc @ I4 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ ( nth @ A @ Xs2 @ I4 ) @ ( nth @ A @ Xs2 @ ( suc @ I4 ) ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_Suc_transp
thf(fact_7434_span__explicit_H,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_span @ A )
= ( ^ [B4: set @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [F3: A > real] :
( ( Uu3
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( F3 @ V5 ) @ V5 )
@ ( collect @ A
@ ^ [V5: A] :
( ( F3 @ V5 )
!= ( zero_zero @ real ) ) ) ) )
& ( finite_finite @ A
@ ( collect @ A
@ ^ [V5: A] :
( ( F3 @ V5 )
!= ( zero_zero @ real ) ) ) )
& ! [V5: A] :
( ( ( F3 @ V5 )
!= ( zero_zero @ real ) )
=> ( member @ A @ V5 @ B4 ) ) ) ) ) ) ) ).
% span_explicit'
thf(fact_7435_span__insert__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A] :
( ( real_Vector_span @ A @ ( insert @ A @ ( zero_zero @ A ) @ S2 ) )
= ( real_Vector_span @ A @ S2 ) ) ) ).
% span_insert_0
thf(fact_7436_span__empty,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_span @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% span_empty
thf(fact_7437_span__delete__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A] :
( ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( real_Vector_span @ A @ S2 ) ) ) ).
% span_delete_0
thf(fact_7438_transp__realrel,axiom,
transp @ ( nat > rat ) @ realrel ).
% transp_realrel
thf(fact_7439_span__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A] : ( member @ A @ ( zero_zero @ A ) @ ( real_Vector_span @ A @ S2 ) ) ) ).
% span_0
thf(fact_7440_span__induct__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X3: A,S2: set @ A,H: A > $o] :
( ( member @ A @ X3 @ ( real_Vector_span @ A @ S2 ) )
=> ( ( H @ ( zero_zero @ A ) )
=> ( ! [C3: real,X4: A,Y4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( H @ Y4 )
=> ( H @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ X4 ) @ Y4 ) ) ) )
=> ( H @ X3 ) ) ) ) ) ).
% span_induct_alt
thf(fact_7441_span__image__scale,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A,C2: A > real] :
( ( finite_finite @ A @ S2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( C2 @ X4 )
!= ( zero_zero @ real ) ) )
=> ( ( real_Vector_span @ A
@ ( image @ A @ A
@ ^ [X: A] : ( real_V8093663219630862766scaleR @ A @ ( C2 @ X ) @ X )
@ S2 ) )
= ( real_Vector_span @ A @ S2 ) ) ) ) ) ).
% span_image_scale
thf(fact_7442_independent__span__bound,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [T6: set @ A,S2: set @ A] :
( ( finite_finite @ A @ T6 )
=> ( ~ ( real_V358717886546972837endent @ A @ S2 )
=> ( ( ord_less_eq @ ( set @ A ) @ S2 @ ( real_Vector_span @ A @ T6 ) )
=> ( ( finite_finite @ A @ S2 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ S2 ) @ ( finite_card @ A @ T6 ) ) ) ) ) ) ) ).
% independent_span_bound
thf(fact_7443_span__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_span @ A )
= ( ^ [B9: set @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [F3: A > real] :
( ( Uu3
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X: A] : ( real_V8093663219630862766scaleR @ A @ ( F3 @ X ) @ X )
@ ( collect @ A
@ ^ [X: A] :
( ( F3 @ X )
!= ( zero_zero @ real ) ) ) ) )
& ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( F3 @ X )
!= ( zero_zero @ real ) ) )
@ B9 )
& ( finite_finite @ A
@ ( collect @ A
@ ^ [X: A] :
( ( F3 @ X )
!= ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% span_alt
thf(fact_7444_representation__def,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V7696804695334737415tation @ A )
= ( ^ [Basis2: set @ A,V5: A] :
( if @ ( A > real )
@ ( ~ ( real_V358717886546972837endent @ A @ Basis2 )
& ( member @ A @ V5 @ ( real_Vector_span @ A @ Basis2 ) ) )
@ ( fChoice @ ( A > real )
@ ^ [F3: A > real] :
( ! [W3: A] :
( ( ( F3 @ W3 )
!= ( zero_zero @ real ) )
=> ( member @ A @ W3 @ Basis2 ) )
& ( finite_finite @ A
@ ( collect @ A
@ ^ [W3: A] :
( ( F3 @ W3 )
!= ( zero_zero @ real ) ) ) )
& ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [W3: A] : ( real_V8093663219630862766scaleR @ A @ ( F3 @ W3 ) @ W3 )
@ ( collect @ A
@ ^ [W3: A] :
( ( F3 @ W3 )
!= ( zero_zero @ real ) ) ) )
= V5 ) ) )
@ ^ [B4: A] : ( zero_zero @ real ) ) ) ) ) ).
% representation_def
thf(fact_7445_transp__gr,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X: A,Y: A] : ( ord_less @ A @ Y @ X ) ) ) ).
% transp_gr
thf(fact_7446_representation__ne__zero,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V2: A,B2: A] :
( ( ( real_V7696804695334737415tation @ A @ Basis @ V2 @ B2 )
!= ( zero_zero @ real ) )
=> ( member @ A @ B2 @ Basis ) ) ) ).
% representation_ne_zero
thf(fact_7447_representation__zero,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A] :
( ( real_V7696804695334737415tation @ A @ Basis @ ( zero_zero @ A ) )
= ( ^ [B4: A] : ( zero_zero @ real ) ) ) ) ).
% representation_zero
thf(fact_7448_finite__representation,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V2: A] :
( finite_finite @ A
@ ( collect @ A
@ ^ [B4: A] :
( ( real_V7696804695334737415tation @ A @ Basis @ V2 @ B4 )
!= ( zero_zero @ real ) ) ) ) ) ).
% finite_representation
thf(fact_7449_representation__basis,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,B2: A] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ( member @ A @ B2 @ Basis )
=> ( ( real_V7696804695334737415tation @ A @ Basis @ B2 )
= ( ^ [V5: A] : ( if @ real @ ( V5 = B2 ) @ ( one_one @ real ) @ ( zero_zero @ real ) ) ) ) ) ) ) ).
% representation_basis
thf(fact_7450_sum__nonzero__representation__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V2: A] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ( member @ A @ V2 @ ( real_Vector_span @ A @ Basis ) )
=> ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [B4: A] : ( real_V8093663219630862766scaleR @ A @ ( real_V7696804695334737415tation @ A @ Basis @ V2 @ B4 ) @ B4 )
@ ( collect @ A
@ ^ [B4: A] :
( ( real_V7696804695334737415tation @ A @ Basis @ V2 @ B4 )
!= ( zero_zero @ real ) ) ) )
= V2 ) ) ) ) ).
% sum_nonzero_representation_eq
thf(fact_7451_transp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A @ ( ord_less_eq @ A ) ) ) ).
% transp_le
thf(fact_7452_transp__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A @ ( ord_less @ A ) ) ) ).
% transp_less
thf(fact_7453_representation__eqI,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V2: A,F2: A > real] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ( member @ A @ V2 @ ( real_Vector_span @ A @ Basis ) )
=> ( ! [B5: A] :
( ( ( F2 @ B5 )
!= ( zero_zero @ real ) )
=> ( member @ A @ B5 @ Basis ) )
=> ( ( finite_finite @ A
@ ( collect @ A
@ ^ [B4: A] :
( ( F2 @ B4 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [B4: A] : ( real_V8093663219630862766scaleR @ A @ ( F2 @ B4 ) @ B4 )
@ ( collect @ A
@ ^ [B4: A] :
( ( F2 @ B4 )
!= ( zero_zero @ real ) ) ) )
= V2 )
=> ( ( real_V7696804695334737415tation @ A @ Basis @ V2 )
= F2 ) ) ) ) ) ) ) ).
% representation_eqI
thf(fact_7454_transp__ge,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X: A,Y: A] : ( ord_less_eq @ A @ Y @ X ) ) ) ).
% transp_ge
thf(fact_7455_dim__le__card,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [V: set @ A,W4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ W4 ) )
=> ( ( finite_finite @ A @ W4 )
=> ( ord_less_eq @ nat @ ( real_Vector_dim @ A @ V ) @ ( finite_card @ A @ W4 ) ) ) ) ) ).
% dim_le_card
thf(fact_7456_span__card__ge__dim,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B6: set @ A,V: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ V )
=> ( ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ B6 ) )
=> ( ( finite_finite @ A @ B6 )
=> ( ord_less_eq @ nat @ ( real_Vector_dim @ A @ V ) @ ( finite_card @ A @ B6 ) ) ) ) ) ) ).
% span_card_ge_dim
thf(fact_7457_dim__le__card_H,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A] :
( ( finite_finite @ A @ S )
=> ( ord_less_eq @ nat @ ( real_Vector_dim @ A @ S ) @ ( finite_card @ A @ S ) ) ) ) ).
% dim_le_card'
thf(fact_7458_dim__def,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_dim @ A )
= ( ^ [V6: set @ A] :
( if @ nat
@ ? [B4: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ B4 )
& ( ( real_Vector_span @ A @ B4 )
= ( real_Vector_span @ A @ V6 ) ) )
@ ( finite_card @ A
@ ( fChoice @ ( set @ A )
@ ^ [B4: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ B4 )
& ( ( real_Vector_span @ A @ B4 )
= ( real_Vector_span @ A @ V6 ) ) ) ) )
@ ( zero_zero @ nat ) ) ) ) ) ).
% dim_def
thf(fact_7459_linear__indep__image__lemma,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B,B6: set @ A,X3: A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( finite_finite @ A @ B6 )
=> ( ~ ( real_V358717886546972837endent @ B @ ( image @ A @ B @ F2 @ B6 ) )
=> ( ( inj_on @ A @ B @ F2 @ B6 )
=> ( ( member @ A @ X3 @ ( real_Vector_span @ A @ B6 ) )
=> ( ( ( F2 @ X3 )
= ( zero_zero @ B ) )
=> ( X3
= ( zero_zero @ A ) ) ) ) ) ) ) ) ) ).
% linear_indep_image_lemma
thf(fact_7460_lexordp__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ) ) ) ).
% lexordp_conv_lexord
thf(fact_7461_lexordp__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Xs2: list @ A,Y3: A,Ys: list @ A] :
( ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys ) )
= ( ( ord_less @ A @ X3 @ Y3 )
| ( ~ ( ord_less @ A @ Y3 @ X3 )
& ( ord_lexordp @ A @ Xs2 @ Ys ) ) ) ) ) ).
% lexordp_simps(3)
thf(fact_7462_linear__eq__0__on__span,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [F2: A > B,B2: set @ A,X3: A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ B2 )
=> ( ( F2 @ X4 )
= ( zero_zero @ B ) ) )
=> ( ( member @ A @ X3 @ ( real_Vector_span @ A @ B2 ) )
=> ( ( F2 @ X3 )
= ( zero_zero @ B ) ) ) ) ) ) ).
% linear_eq_0_on_span
thf(fact_7463_lexordp_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y3: A,Xs2: list @ A,Ys: list @ A] :
( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X3 )
=> ( ( ord_lexordp @ A @ Xs2 @ Ys )
=> ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys ) ) ) ) ) ) ).
% lexordp.Cons_eq
thf(fact_7464_lexordp_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y3: A,Xs2: list @ A,Ys: list @ A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys ) ) ) ) ).
% lexordp.Cons
thf(fact_7465_linear__injective__0,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( ! [X: A] :
( ( ( F2 @ X )
= ( zero_zero @ B ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% linear_injective_0
thf(fact_7466_lexordp__append__leftD,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs2: list @ A,Us: list @ A,Vs: list @ A] :
( ( ord_lexordp @ A @ ( append @ A @ Xs2 @ Us ) @ ( append @ A @ Xs2 @ Vs ) )
=> ( ! [A6: A] :
~ ( ord_less @ A @ A6 @ A6 )
=> ( ord_lexordp @ A @ Us @ Vs ) ) ) ) ).
% lexordp_append_leftD
thf(fact_7467_module__hom__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ( real_Vector_linear @ A @ B
@ ^ [X: A] : ( zero_zero @ B ) ) ) ).
% module_hom_zero
thf(fact_7468_lexordp__irreflexive,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs2: list @ A] :
( ! [X4: A] :
~ ( ord_less @ A @ X4 @ X4 )
=> ~ ( ord_lexordp @ A @ Xs2 @ Xs2 ) ) ) ).
% lexordp_irreflexive
thf(fact_7469_linear__0,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [F2: A > B] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( F2 @ ( zero_zero @ A ) )
= ( zero_zero @ B ) ) ) ) ).
% linear_0
thf(fact_7470_lexordp__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,Ys: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp @ A @ Xs2 @ Ys )
=> ( ! [Y4: A,Ys4: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y4 @ Ys4 ) )
=> ( ! [X4: A,Xs3: list @ A,Y4: A,Ys4: list @ A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( P @ ( cons @ A @ X4 @ Xs3 ) @ ( cons @ A @ Y4 @ Ys4 ) ) )
=> ( ! [X4: A,Xs3: list @ A,Ys4: list @ A] :
( ( ord_lexordp @ A @ Xs3 @ Ys4 )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons @ A @ X4 @ Xs3 ) @ ( cons @ A @ X4 @ Ys4 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ) ).
% lexordp_induct
thf(fact_7471_lexordp__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,Ys: list @ A] :
( ( ord_lexordp @ A @ Xs2 @ Ys )
=> ( ( ( Xs2
= ( nil @ A ) )
=> ! [Y4: A,Ys5: list @ A] :
( Ys
!= ( cons @ A @ Y4 @ Ys5 ) ) )
=> ( ! [X4: A] :
( ? [Xs4: list @ A] :
( Xs2
= ( cons @ A @ X4 @ Xs4 ) )
=> ! [Y4: A] :
( ? [Ys5: list @ A] :
( Ys
= ( cons @ A @ Y4 @ Ys5 ) )
=> ~ ( ord_less @ A @ X4 @ Y4 ) ) )
=> ~ ! [X4: A,Xs4: list @ A] :
( ( Xs2
= ( cons @ A @ X4 @ Xs4 ) )
=> ! [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X4 @ Ys5 ) )
=> ~ ( ord_lexordp @ A @ Xs4 @ Ys5 ) ) ) ) ) ) ) ).
% lexordp_cases
thf(fact_7472_lexordp_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [A12: list @ A,A23: list @ A] :
( ? [Y: A,Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y @ Ys3 ) ) )
| ? [X: A,Y: A,Xs: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X @ Xs ) )
& ( A23
= ( cons @ A @ Y @ Ys3 ) )
& ( ord_less @ A @ X @ Y ) )
| ? [X: A,Y: A,Xs: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X @ Xs ) )
& ( A23
= ( cons @ A @ Y @ Ys3 ) )
& ~ ( ord_less @ A @ X @ Y )
& ~ ( ord_less @ A @ Y @ X )
& ( ord_lexordp @ A @ Xs @ Ys3 ) ) ) ) ) ) ).
% lexordp.simps
thf(fact_7473_lexordp_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A22: list @ A] :
( ( ord_lexordp @ A @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Y4: A,Ys4: list @ A] :
( A22
!= ( cons @ A @ Y4 @ Ys4 ) ) )
=> ( ! [X4: A] :
( ? [Xs3: list @ A] :
( A1
= ( cons @ A @ X4 @ Xs3 ) )
=> ! [Y4: A] :
( ? [Ys4: list @ A] :
( A22
= ( cons @ A @ Y4 @ Ys4 ) )
=> ~ ( ord_less @ A @ X4 @ Y4 ) ) )
=> ~ ! [X4: A,Y4: A,Xs3: list @ A] :
( ( A1
= ( cons @ A @ X4 @ Xs3 ) )
=> ! [Ys4: list @ A] :
( ( A22
= ( cons @ A @ Y4 @ Ys4 ) )
=> ( ~ ( ord_less @ A @ X4 @ Y4 )
=> ( ~ ( ord_less @ A @ Y4 @ X4 )
=> ~ ( ord_lexordp @ A @ Xs3 @ Ys4 ) ) ) ) ) ) ) ) ) ).
% lexordp.cases
thf(fact_7474_lexordp__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs: list @ A,Ys3: list @ A] :
( ? [X: A,Vs2: list @ A] :
( Ys3
= ( append @ A @ Xs @ ( cons @ A @ X @ Vs2 ) ) )
| ? [Us2: list @ A,A5: A,B4: A,Vs2: list @ A,Ws: list @ A] :
( ( ord_less @ A @ A5 @ B4 )
& ( Xs
= ( append @ A @ Us2 @ ( cons @ A @ A5 @ Vs2 ) ) )
& ( Ys3
= ( append @ A @ Us2 @ ( cons @ A @ B4 @ Ws ) ) ) ) ) ) ) ) ).
% lexordp_iff
thf(fact_7475_lexordp__append__left__rightI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y3: A,Us: list @ A,Xs2: list @ A,Ys: list @ A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_lexordp @ A @ ( append @ A @ Us @ ( cons @ A @ X3 @ Xs2 ) ) @ ( append @ A @ Us @ ( cons @ A @ Y3 @ Ys ) ) ) ) ) ).
% lexordp_append_left_rightI
thf(fact_7476_ord__class_Olexordp__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P5: ( list @ A ) > ( list @ A ) > $o,X17: list @ A,X23: list @ A] :
( ? [Y: A,Ys3: list @ A] :
( ( X17
= ( nil @ A ) )
& ( X23
= ( cons @ A @ Y @ Ys3 ) ) )
| ? [X: A,Y: A,Xs: list @ A,Ys3: list @ A] :
( ( X17
= ( cons @ A @ X @ Xs ) )
& ( X23
= ( cons @ A @ Y @ Ys3 ) )
& ( ord_less @ A @ X @ Y ) )
| ? [X: A,Y: A,Xs: list @ A,Ys3: list @ A] :
( ( X17
= ( cons @ A @ X @ Xs ) )
& ( X23
= ( cons @ A @ Y @ Ys3 ) )
& ~ ( ord_less @ A @ X @ Y )
& ~ ( ord_less @ A @ Y @ X )
& ( P5 @ Xs @ Ys3 ) ) ) ) ) ) ).
% ord_class.lexordp_def
thf(fact_7477_bounded__bilinear_Ointro,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod: A > B > C] :
( ! [A6: A,A11: A,B5: B] :
( ( Prod @ ( plus_plus @ A @ A6 @ A11 ) @ B5 )
= ( plus_plus @ C @ ( Prod @ A6 @ B5 ) @ ( Prod @ A11 @ B5 ) ) )
=> ( ! [A6: A,B5: B,B10: B] :
( ( Prod @ A6 @ ( plus_plus @ B @ B5 @ B10 ) )
= ( plus_plus @ C @ ( Prod @ A6 @ B5 ) @ ( Prod @ A6 @ B10 ) ) )
=> ( ! [R: real,A6: A,B5: B] :
( ( Prod @ ( real_V8093663219630862766scaleR @ A @ R @ A6 ) @ B5 )
= ( real_V8093663219630862766scaleR @ C @ R @ ( Prod @ A6 @ B5 ) ) )
=> ( ! [A6: A,R: real,B5: B] :
( ( Prod @ A6 @ ( real_V8093663219630862766scaleR @ B @ R @ B5 ) )
= ( real_V8093663219630862766scaleR @ C @ R @ ( Prod @ A6 @ B5 ) ) )
=> ( ? [K8: real] :
! [A6: A,B5: B] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( Prod @ A6 @ B5 ) ) @ ( times_times @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A6 ) @ ( real_V7770717601297561774m_norm @ B @ B5 ) ) @ K8 ) )
=> ( real_V2442710119149674383linear @ A @ B @ C @ Prod ) ) ) ) ) ) ) ).
% bounded_bilinear.intro
thf(fact_7478_bounded__bilinear_Ozero__left,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [Prod: A > B > C,B2: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( Prod @ ( zero_zero @ A ) @ B2 )
= ( zero_zero @ C ) ) ) ) ).
% bounded_bilinear.zero_left
thf(fact_7479_bounded__bilinear_Ozero__right,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [Prod: A > B > C,A2: A] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( Prod @ A2 @ ( zero_zero @ B ) )
= ( zero_zero @ C ) ) ) ) ).
% bounded_bilinear.zero_right
thf(fact_7480_bounded__bilinear_Obounded,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod: A > B > C] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ? [K9: real] :
! [A8: A,B8: B] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( Prod @ A8 @ B8 ) ) @ ( times_times @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A8 ) @ ( real_V7770717601297561774m_norm @ B @ B8 ) ) @ K9 ) ) ) ) ).
% bounded_bilinear.bounded
thf(fact_7481_bounded__bilinear_Otendsto__zero,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod: A > B > C,F2: D > A,F4: filter @ D,G: D > B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( ( filterlim @ D @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( filterlim @ D @ C
@ ^ [X: D] : ( Prod @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F4 ) ) ) ) ) ).
% bounded_bilinear.tendsto_zero
thf(fact_7482_bounded__bilinear_Otendsto__left__zero,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod: A > B > C,F2: D > A,F4: filter @ D,C2: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D @ C
@ ^ [X: D] : ( Prod @ ( F2 @ X ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F4 ) ) ) ) ).
% bounded_bilinear.tendsto_left_zero
thf(fact_7483_bounded__bilinear_Otendsto__right__zero,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod: A > B > C,F2: D > B,F4: filter @ D,C2: A] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( filterlim @ D @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( filterlim @ D @ C
@ ^ [X: D] : ( Prod @ C2 @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F4 ) ) ) ) ).
% bounded_bilinear.tendsto_right_zero
thf(fact_7484_lfp__funpow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,N: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( complete_lattice_lfp @ A @ ( compow @ ( A > A ) @ ( suc @ N ) @ F2 ) )
= ( complete_lattice_lfp @ A @ F2 ) ) ) ) ).
% lfp_funpow
thf(fact_7485_bounded__bilinear_Ononneg__bounded,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod: A > B > C] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ? [K9: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ K9 )
& ! [A8: A,B8: B] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( Prod @ A8 @ B8 ) ) @ ( times_times @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A8 ) @ ( real_V7770717601297561774m_norm @ B @ B8 ) ) @ K9 ) ) ) ) ) ).
% bounded_bilinear.nonneg_bounded
thf(fact_7486_lfp__Kleene__iter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ( compow @ ( A > A ) @ ( suc @ K ) @ F2 @ ( bot_bot @ A ) )
= ( compow @ ( A > A ) @ K @ F2 @ ( bot_bot @ A ) ) )
=> ( ( complete_lattice_lfp @ A @ F2 )
= ( compow @ ( A > A ) @ K @ F2 @ ( bot_bot @ A ) ) ) ) ) ) ).
% lfp_Kleene_iter
thf(fact_7487_bounded__bilinear_Opos__bounded,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod: A > B > C] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ? [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
& ! [A8: A,B8: B] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( Prod @ A8 @ B8 ) ) @ ( times_times @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A8 ) @ ( real_V7770717601297561774m_norm @ B @ B8 ) ) @ K9 ) ) ) ) ) ).
% bounded_bilinear.pos_bounded
thf(fact_7488_bounded__bilinear__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ( ( real_V2442710119149674383linear @ A @ B @ C )
= ( ^ [Prod2: A > B > C] :
( ! [A5: A,A15: A,B4: B] :
( ( Prod2 @ ( plus_plus @ A @ A5 @ A15 ) @ B4 )
= ( plus_plus @ C @ ( Prod2 @ A5 @ B4 ) @ ( Prod2 @ A15 @ B4 ) ) )
& ! [A5: A,B4: B,B11: B] :
( ( Prod2 @ A5 @ ( plus_plus @ B @ B4 @ B11 ) )
= ( plus_plus @ C @ ( Prod2 @ A5 @ B4 ) @ ( Prod2 @ A5 @ B11 ) ) )
& ! [R5: real,A5: A,B4: B] :
( ( Prod2 @ ( real_V8093663219630862766scaleR @ A @ R5 @ A5 ) @ B4 )
= ( real_V8093663219630862766scaleR @ C @ R5 @ ( Prod2 @ A5 @ B4 ) ) )
& ! [A5: A,R5: real,B4: B] :
( ( Prod2 @ A5 @ ( real_V8093663219630862766scaleR @ B @ R5 @ B4 ) )
= ( real_V8093663219630862766scaleR @ C @ R5 @ ( Prod2 @ A5 @ B4 ) ) )
& ? [K6: real] :
! [A5: A,B4: B] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( Prod2 @ A5 @ B4 ) ) @ ( times_times @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A5 ) @ ( real_V7770717601297561774m_norm @ B @ B4 ) ) @ K6 ) ) ) ) ) ) ).
% bounded_bilinear_def
thf(fact_7489_lfp__ordinal__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,P: A > $o] :
( ( order_mono @ A @ A @ F2 )
=> ( ! [S5: A] :
( ( P @ S5 )
=> ( ( ord_less_eq @ A @ S5 @ ( complete_lattice_lfp @ A @ F2 ) )
=> ( P @ ( F2 @ S5 ) ) ) )
=> ( ! [M8: set @ A] :
( ! [X6: A] :
( ( member @ A @ X6 @ M8 )
=> ( P @ X6 ) )
=> ( P @ ( complete_Sup_Sup @ A @ M8 ) ) )
=> ( P @ ( complete_lattice_lfp @ A @ F2 ) ) ) ) ) ) ).
% lfp_ordinal_induct
thf(fact_7490_lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,P: A] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ ( F2 @ ( inf_inf @ A @ ( complete_lattice_lfp @ A @ F2 ) @ P ) ) @ P )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F2 ) @ P ) ) ) ) ).
% lfp_induct
thf(fact_7491_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X9: $o > A,Y8: $o > A] :
( ( ord_less_eq @ A @ ( X9 @ $false ) @ ( Y8 @ $false ) )
& ( ord_less_eq @ A @ ( X9 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_7492_lfp__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,G: A > A] :
( ! [Z9: A] : ( ord_less_eq @ A @ ( F2 @ Z9 ) @ ( G @ Z9 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F2 ) @ ( complete_lattice_lfp @ A @ G ) ) ) ) ).
% lfp_mono
thf(fact_7493_lfp__lowerbound,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,A4: A] :
( ( ord_less_eq @ A @ ( F2 @ A4 ) @ A4 )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F2 ) @ A4 ) ) ) ).
% lfp_lowerbound
thf(fact_7494_lfp__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,A4: A] :
( ! [U4: A] :
( ( ord_less_eq @ A @ ( F2 @ U4 ) @ U4 )
=> ( ord_less_eq @ A @ A4 @ U4 ) )
=> ( ord_less_eq @ A @ A4 @ ( complete_lattice_lfp @ A @ F2 ) ) ) ) ).
% lfp_greatest
thf(fact_7495_lfp__lfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A > A] :
( ! [X4: A,Y4: A,W2: A,Z3: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ( ord_less_eq @ A @ W2 @ Z3 )
=> ( ord_less_eq @ A @ ( F2 @ X4 @ W2 ) @ ( F2 @ Y4 @ Z3 ) ) ) )
=> ( ( complete_lattice_lfp @ A
@ ^ [X: A] : ( complete_lattice_lfp @ A @ ( F2 @ X ) ) )
= ( complete_lattice_lfp @ A
@ ^ [X: A] : ( F2 @ X @ X ) ) ) ) ) ).
% lfp_lfp
thf(fact_7496_lfp__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F4: A > A,X3: A] :
( ( order_mono @ A @ A @ F4 )
=> ( ( ( F4 @ X3 )
= X3 )
=> ( ! [Z3: A] :
( ( ( F4 @ Z3 )
= Z3 )
=> ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ( complete_lattice_lfp @ A @ F4 )
= X3 ) ) ) ) ) ).
% lfp_eqI
thf(fact_7497_lfp__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_lattice_lfp @ A )
= ( ^ [F3: A > A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [U2: A] : ( ord_less_eq @ A @ ( F3 @ U2 ) @ U2 ) ) ) ) ) ) ).
% lfp_def
thf(fact_7498_def__lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: A,F2: A > A,P: A] :
( ( A4
= ( complete_lattice_lfp @ A @ F2 ) )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ ( F2 @ ( inf_inf @ A @ A4 @ P ) ) @ P )
=> ( ord_less_eq @ A @ A4 @ P ) ) ) ) ) ).
% def_lfp_induct
thf(fact_7499_lfp__transfer__bounded,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [P: A > $o,F2: A > A,Alpha: A > B,G: B > B] :
( ( P @ ( bot_bot @ A ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( P @ ( F2 @ X4 ) ) )
=> ( ! [M8: nat > A] :
( ! [I2: nat] : ( P @ ( M8 @ I2 ) )
=> ( P @ ( complete_Sup_Sup @ A @ ( image @ nat @ A @ M8 @ ( top_top @ ( set @ nat ) ) ) ) ) )
=> ( ! [M8: nat > A] :
( ( order_mono @ nat @ A @ M8 )
=> ( ! [I2: nat] : ( P @ ( M8 @ I2 ) )
=> ( ( Alpha @ ( complete_Sup_Sup @ A @ ( image @ nat @ A @ M8 @ ( top_top @ ( set @ nat ) ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image @ nat @ B
@ ^ [I4: nat] : ( Alpha @ ( M8 @ I4 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) )
=> ( ( order_sup_continuous @ A @ A @ F2 )
=> ( ( order_sup_continuous @ B @ B @ G )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( ( ord_less_eq @ A @ X4 @ ( complete_lattice_lfp @ A @ F2 ) )
=> ( ( Alpha @ ( F2 @ X4 ) )
= ( G @ ( Alpha @ X4 ) ) ) ) )
=> ( ! [X4: B] : ( ord_less_eq @ B @ ( Alpha @ ( bot_bot @ A ) ) @ ( G @ X4 ) )
=> ( ( Alpha @ ( complete_lattice_lfp @ A @ F2 ) )
= ( complete_lattice_lfp @ B @ G ) ) ) ) ) ) ) ) ) ) ) ).
% lfp_transfer_bounded
thf(fact_7500_cSUP__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A4: set @ C,B6: C > ( set @ D ),F2: D > B] :
( ( A4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ A4 )
=> ( ( B6 @ X4 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit941137186595557371_above @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image @ C @ ( set @ B )
@ ^ [X: C] : ( image @ D @ B @ F2 @ ( B6 @ X ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup @ B @ ( image @ D @ B @ F2 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ B6 @ A4 ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image @ C @ B
@ ^ [X: C] : ( complete_Sup_Sup @ B @ ( image @ D @ B @ F2 @ ( B6 @ X ) ) )
@ A4 ) ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_7501_bdd__above__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: set @ A] : ( condit941137186595557371_above @ A @ A4 ) ) ).
% bdd_above_top
thf(fact_7502_bdd__above_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A,M7: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ X4 @ M7 ) )
=> ( condit941137186595557371_above @ A @ A4 ) ) ) ).
% bdd_above.I
thf(fact_7503_bdd__above__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit941137186595557371_above @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_above_empty
thf(fact_7504_bdd__above__insert,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A2: A,A4: set @ A] :
( ( condit941137186595557371_above @ A @ ( insert @ A @ A2 @ A4 ) )
= ( condit941137186595557371_above @ A @ A4 ) ) ) ).
% bdd_above_insert
thf(fact_7505_bdd__above__Icc,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] : ( condit941137186595557371_above @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) ) ).
% bdd_above_Icc
thf(fact_7506_bdd__above__Un,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( condit941137186595557371_above @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
= ( ( condit941137186595557371_above @ A @ A4 )
& ( condit941137186595557371_above @ A @ B6 ) ) ) ) ).
% bdd_above_Un
thf(fact_7507_bdd__above__Ico,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] : ( condit941137186595557371_above @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) ) ).
% bdd_above_Ico
thf(fact_7508_bdd__above__Iio,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A] : ( condit941137186595557371_above @ A @ ( set_ord_lessThan @ A @ B2 ) ) ) ).
% bdd_above_Iio
thf(fact_7509_bdd__above__Iic,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A] : ( condit941137186595557371_above @ A @ ( set_ord_atMost @ A @ B2 ) ) ) ).
% bdd_above_Iic
thf(fact_7510_bdd__above__Ioo,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] : ( condit941137186595557371_above @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) ).
% bdd_above_Ioo
thf(fact_7511_bdd__above__Ioc,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] : ( condit941137186595557371_above @ A @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) ) ).
% bdd_above_Ioc
thf(fact_7512_bdd__above__image__sup,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [F2: B > A,G: B > A,A4: set @ B] :
( ( condit941137186595557371_above @ A
@ ( image @ B @ A
@ ^ [X: B] : ( sup_sup @ A @ ( F2 @ X ) @ ( G @ X ) )
@ A4 ) )
= ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A4 ) )
& ( condit941137186595557371_above @ A @ ( image @ B @ A @ G @ A4 ) ) ) ) ) ).
% bdd_above_image_sup
thf(fact_7513_bdd__above__UN,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [I6: set @ B,A4: B > ( set @ A )] :
( ( finite_finite @ B @ I6 )
=> ( ( condit941137186595557371_above @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A4 @ I6 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ I6 )
=> ( condit941137186595557371_above @ A @ ( A4 @ X ) ) ) ) ) ) ) ).
% bdd_above_UN
thf(fact_7514_bdd__above__nat,axiom,
( ( condit941137186595557371_above @ nat )
= ( finite_finite @ nat ) ) ).
% bdd_above_nat
thf(fact_7515_bdd__above__finite,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( condit941137186595557371_above @ A @ A4 ) ) ) ).
% bdd_above_finite
thf(fact_7516_bdd__above__image__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit941137186595557371_above @ A @ A4 )
=> ( condit941137186595557371_above @ B @ ( image @ A @ B @ F2 @ A4 ) ) ) ) ) ).
% bdd_above_image_mono
thf(fact_7517_cSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A4: set @ B,X3: B,U: A] :
( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( member @ B @ X3 @ A4 )
=> ( ( ord_less_eq @ A @ U @ ( F2 @ X3 ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_7518_cSUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: B,A4: set @ B,F2: B > A] :
( ( member @ B @ X3 @ A4 )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ) ) ).
% cSUP_upper
thf(fact_7519_bdd__above_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A4: set @ B,F2: B > A,M7: A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ M7 ) )
=> ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ).
% bdd_above.I2
thf(fact_7520_cSup__upper,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A,X8: set @ A] :
( ( member @ A @ X3 @ X8 )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ).
% cSup_upper
thf(fact_7521_cSup__upper2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A,X8: set @ A,Y3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ord_less_eq @ A @ Y3 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ) ).
% cSup_upper2
thf(fact_7522_less__cSup__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Y3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ( ord_less @ A @ Y3 @ ( complete_Sup_Sup @ A @ X8 ) )
= ( ? [X: A] :
( ( member @ A @ X @ X8 )
& ( ord_less @ A @ Y3 @ X ) ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_7523_cSup__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B6: set @ A,A4: set @ A] :
( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A4 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ B6 )
=> ? [X6: A] :
( ( member @ A @ X6 @ A4 )
& ( ord_less_eq @ A @ B5 @ X6 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B6 ) @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ) ).
% cSup_mono
thf(fact_7524_cSup__le__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S2: set @ A,A2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S2 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S2 ) @ A2 )
= ( ! [X: A] :
( ( member @ A @ X @ S2 )
=> ( ord_less_eq @ A @ X @ A2 ) ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_7525_bdd__above_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A] :
( ( condit941137186595557371_above @ A @ A4 )
=> ~ ! [M8: A] :
~ ! [X6: A] :
( ( member @ A @ X6 @ A4 )
=> ( ord_less_eq @ A @ X6 @ M8 ) ) ) ) ).
% bdd_above.E
thf(fact_7526_bdd__above_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit941137186595557371_above @ A )
= ( ^ [A9: set @ A] :
? [M9: A] :
! [X: A] :
( ( member @ A @ X @ A9 )
=> ( ord_less_eq @ A @ X @ M9 ) ) ) ) ) ).
% bdd_above.unfold
thf(fact_7527_bdd__above__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B6: set @ A,A4: set @ A] :
( ( condit941137186595557371_above @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( condit941137186595557371_above @ A @ A4 ) ) ) ) ).
% bdd_above_mono
thf(fact_7528_bdd__above__Int2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B6: set @ A,A4: set @ A] :
( ( condit941137186595557371_above @ A @ B6 )
=> ( condit941137186595557371_above @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ) ).
% bdd_above_Int2
thf(fact_7529_bdd__above__Int1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( condit941137186595557371_above @ A @ A4 )
=> ( condit941137186595557371_above @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ) ).
% bdd_above_Int1
thf(fact_7530_cSUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A4: set @ B,Y3: A,I: B] :
( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ Y3 )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ ( F2 @ I ) @ Y3 ) ) ) ) ) ).
% cSUP_lessD
thf(fact_7531_lfp__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [Alpha: A > B,F2: A > A,G: B > B] :
( ( order_sup_continuous @ A @ B @ Alpha )
=> ( ( order_sup_continuous @ A @ A @ F2 )
=> ( ( order_sup_continuous @ B @ B @ G )
=> ( ! [X4: B] : ( ord_less_eq @ B @ ( Alpha @ ( bot_bot @ A ) ) @ ( G @ X4 ) )
=> ( ! [X4: A] :
( ( ord_less_eq @ A @ X4 @ ( complete_lattice_lfp @ A @ F2 ) )
=> ( ( Alpha @ ( F2 @ X4 ) )
= ( G @ ( Alpha @ X4 ) ) ) )
=> ( ( Alpha @ ( complete_lattice_lfp @ A @ F2 ) )
= ( complete_lattice_lfp @ B @ G ) ) ) ) ) ) ) ) ).
% lfp_transfer
thf(fact_7532_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,G: C > A,B6: set @ C,F2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ C @ A @ G @ B6 ) )
=> ( ! [N2: B] :
( ( member @ B @ N2 @ A4 )
=> ? [X6: C] :
( ( member @ C @ X6 @ B6 )
& ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B6 ) ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_7533_cSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,U: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ U )
= ( ! [X: B] :
( ( member @ B @ X @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X ) @ U ) ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_7534_cSup__subset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_7535_cSup__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A2: A] :
( ( condit941137186595557371_above @ A @ X8 )
=> ( ( ( X8
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ A2 @ X8 ) )
= A2 ) )
& ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ A2 @ X8 ) )
= ( sup_sup @ A @ A2 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_7536_cSup__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ A2 @ X8 ) )
= ( sup_sup @ A @ A2 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ) ).
% cSup_insert
thf(fact_7537_cSup__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A4 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B6 )
=> ( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_7538_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A4: set @ B,F2: B > A,A2: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) )
= ( ? [X: B] :
( ( member @ B @ X @ A4 )
& ( ord_less @ A @ A2 @ ( F2 @ X ) ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_7539_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,G: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ G @ A4 ) )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A4 ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [A5: B] : ( sup_sup @ A @ ( F2 @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_7540_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,G: B > A,B6: set @ B,F2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ G @ B6 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B6 ) ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_7541_cSup__inter__less__eq,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( condit941137186595557371_above @ A @ A4 )
=> ( ( condit941137186595557371_above @ A @ B6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) @ ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_7542_cSUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,A2: B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( insert @ B @ A2 @ A4 ) ) )
= ( sup_sup @ A @ ( F2 @ A2 ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_7543_cSUP__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,B6: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ B6 ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ B6 ) ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_7544_cSup__cInf,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S2: set @ A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S2 )
=> ( ( complete_Sup_Sup @ A @ S2 )
= ( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [X: A] :
! [Y: A] :
( ( member @ A @ Y @ S2 )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_7545_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A4: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit941137186595557371_above @ A @ ( image @ C @ A @ A4 @ I6 ) )
=> ( ( I6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image @ C @ B
@ ^ [X: C] : ( F2 @ ( A4 @ X ) )
@ I6 ) )
@ ( F2 @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ A4 @ I6 ) ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_7546_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit941137186595557371_above @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image @ A @ B @ F2 @ A4 ) ) @ ( F2 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ) ) ).
% mono_cSup
thf(fact_7547_cINF__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A4: set @ C,B6: C > ( set @ D ),F2: D > B] :
( ( A4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ A4 )
=> ( ( B6 @ X4 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit1013018076250108175_below @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image @ C @ ( set @ B )
@ ^ [X: C] : ( image @ D @ B @ F2 @ ( B6 @ X ) )
@ A4 ) ) )
=> ( ( complete_Inf_Inf @ B @ ( image @ D @ B @ F2 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ B6 @ A4 ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image @ C @ B
@ ^ [X: C] : ( complete_Inf_Inf @ B @ ( image @ D @ B @ F2 @ ( B6 @ X ) ) )
@ A4 ) ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_7548_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit1013018076250108175_below @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ A4 ) ) @ ( complete_Inf_Inf @ B @ ( image @ A @ B @ F2 @ A4 ) ) ) ) ) ) ) ).
% mono_cInf
thf(fact_7549_bdd__below__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A4: set @ A] : ( condit1013018076250108175_below @ A @ A4 ) ) ).
% bdd_below_bot
thf(fact_7550_bdd__belowI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A,M: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ M @ X4 ) )
=> ( condit1013018076250108175_below @ A @ A4 ) ) ) ).
% bdd_belowI
thf(fact_7551_bdd__below_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A,M7: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ M7 @ X4 ) )
=> ( condit1013018076250108175_below @ A @ A4 ) ) ) ).
% bdd_below.I
thf(fact_7552_bdd__below__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit1013018076250108175_below @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_below_empty
thf(fact_7553_bdd__below__insert,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A2: A,A4: set @ A] :
( ( condit1013018076250108175_below @ A @ ( insert @ A @ A2 @ A4 ) )
= ( condit1013018076250108175_below @ A @ A4 ) ) ) ).
% bdd_below_insert
thf(fact_7554_bdd__below__Icc,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] : ( condit1013018076250108175_below @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) ) ).
% bdd_below_Icc
thf(fact_7555_bdd__below__Un,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( condit1013018076250108175_below @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
= ( ( condit1013018076250108175_below @ A @ A4 )
& ( condit1013018076250108175_below @ A @ B6 ) ) ) ) ).
% bdd_below_Un
thf(fact_7556_bdd__below__Ico,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] : ( condit1013018076250108175_below @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) ) ).
% bdd_below_Ico
thf(fact_7557_bdd__below__Ioi,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] : ( condit1013018076250108175_below @ A @ ( set_ord_greaterThan @ A @ A2 ) ) ) ).
% bdd_below_Ioi
thf(fact_7558_bdd__below__Ioo,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] : ( condit1013018076250108175_below @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) ).
% bdd_below_Ioo
thf(fact_7559_bdd__below__Ioc,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] : ( condit1013018076250108175_below @ A @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) ) ).
% bdd_below_Ioc
thf(fact_7560_bdd__below__Ici,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] : ( condit1013018076250108175_below @ A @ ( set_ord_atLeast @ A @ A2 ) ) ) ).
% bdd_below_Ici
thf(fact_7561_bdd__below__image__inf,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [F2: B > A,G: B > A,A4: set @ B] :
( ( condit1013018076250108175_below @ A
@ ( image @ B @ A
@ ^ [X: B] : ( inf_inf @ A @ ( F2 @ X ) @ ( G @ X ) )
@ A4 ) )
= ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A4 ) )
& ( condit1013018076250108175_below @ A @ ( image @ B @ A @ G @ A4 ) ) ) ) ) ).
% bdd_below_image_inf
thf(fact_7562_bdd__above__uminus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X8: set @ A] :
( ( condit941137186595557371_above @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ X8 ) )
= ( condit1013018076250108175_below @ A @ X8 ) ) ) ).
% bdd_above_uminus
thf(fact_7563_bdd__below__uminus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X8: set @ A] :
( ( condit1013018076250108175_below @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ X8 ) )
= ( condit941137186595557371_above @ A @ X8 ) ) ) ).
% bdd_below_uminus
thf(fact_7564_bdd__below__UN,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [I6: set @ B,A4: B > ( set @ A )] :
( ( finite_finite @ B @ I6 )
=> ( ( condit1013018076250108175_below @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A4 @ I6 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ I6 )
=> ( condit1013018076250108175_below @ A @ ( A4 @ X ) ) ) ) ) ) ) ).
% bdd_below_UN
thf(fact_7565_cINF__lower,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A4: set @ B,X3: B] :
( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( F2 @ X3 ) ) ) ) ) ).
% cINF_lower
thf(fact_7566_cINF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A4: set @ B,X3: B,U: A] :
( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( member @ B @ X3 @ A4 )
=> ( ( ord_less_eq @ A @ ( F2 @ X3 ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ U ) ) ) ) ) ).
% cINF_lower2
thf(fact_7567_bdd__belowI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A4: set @ B,M: A,F2: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ M @ ( F2 @ X4 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ).
% bdd_belowI2
thf(fact_7568_bdd__below_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A4: set @ B,M7: A,F2: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ M7 @ ( F2 @ X4 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ).
% bdd_below.I2
thf(fact_7569_le__cInf__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S2: set @ A,A2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S2 )
=> ( ( ord_less_eq @ A @ A2 @ ( complete_Inf_Inf @ A @ S2 ) )
= ( ! [X: A] :
( ( member @ A @ X @ S2 )
=> ( ord_less_eq @ A @ A2 @ X ) ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_7570_cInf__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B6: set @ A,A4: set @ A] :
( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A4 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ B6 )
=> ? [X6: A] :
( ( member @ A @ X6 @ A4 )
& ( ord_less_eq @ A @ X6 @ B5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ) ) ).
% cInf_mono
thf(fact_7571_bdd__below__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B6: set @ A,A4: set @ A] :
( ( condit1013018076250108175_below @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( condit1013018076250108175_below @ A @ A4 ) ) ) ) ).
% bdd_below_mono
thf(fact_7572_bdd__below_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A] :
( ( condit1013018076250108175_below @ A @ A4 )
=> ~ ! [M8: A] :
~ ! [X6: A] :
( ( member @ A @ X6 @ A4 )
=> ( ord_less_eq @ A @ M8 @ X6 ) ) ) ) ).
% bdd_below.E
thf(fact_7573_bdd__below_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit1013018076250108175_below @ A )
= ( ^ [A9: set @ A] :
? [M9: A] :
! [X: A] :
( ( member @ A @ X @ A9 )
=> ( ord_less_eq @ A @ M9 @ X ) ) ) ) ) ).
% bdd_below.unfold
thf(fact_7574_cInf__lower,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A,X8: set @ A] :
( ( member @ A @ X3 @ X8 )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X3 ) ) ) ) ).
% cInf_lower
thf(fact_7575_cInf__lower2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A,X8: set @ A,Y3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Y3 ) ) ) ) ) ).
% cInf_lower2
thf(fact_7576_bdd__below__Int1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( condit1013018076250108175_below @ A @ A4 )
=> ( condit1013018076250108175_below @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ) ).
% bdd_below_Int1
thf(fact_7577_bdd__below__Int2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B6: set @ A,A4: set @ A] :
( ( condit1013018076250108175_below @ A @ B6 )
=> ( condit1013018076250108175_below @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ) ).
% bdd_below_Int2
thf(fact_7578_cInf__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Y3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Y3 )
= ( ? [X: A] :
( ( member @ A @ X @ X8 )
& ( ord_less @ A @ X @ Y3 ) ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_7579_bdd__below__image__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit1013018076250108175_below @ A @ A4 )
=> ( condit1013018076250108175_below @ B @ ( image @ A @ B @ F2 @ A4 ) ) ) ) ) ).
% bdd_below_image_mono
thf(fact_7580_bdd__below__finite,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( condit1013018076250108175_below @ A @ A4 ) ) ) ).
% bdd_below_finite
thf(fact_7581_bdd__below__image__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( condit941137186595557371_above @ A @ A4 )
=> ( condit1013018076250108175_below @ B @ ( image @ A @ B @ F2 @ A4 ) ) ) ) ) ).
% bdd_below_image_antimono
thf(fact_7582_bdd__above__image__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( condit1013018076250108175_below @ A @ A4 )
=> ( condit941137186595557371_above @ B @ ( image @ A @ B @ F2 @ A4 ) ) ) ) ) ).
% bdd_above_image_antimono
thf(fact_7583_less__cINF__D,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A4: set @ B,Y3: A,I: B] :
( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less @ A @ Y3 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ Y3 @ ( F2 @ I ) ) ) ) ) ) ).
% less_cINF_D
thf(fact_7584_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B6: set @ B,F2: C > A,A4: set @ C,G: B > A] :
( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ C @ A @ F2 @ A4 ) )
=> ( ! [M4: B] :
( ( member @ B @ M4 @ B6 )
=> ? [X6: C] :
( ( member @ C @ X6 @ A4 )
& ( ord_less_eq @ A @ ( F2 @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B6 ) ) ) ) ) ) ) ).
% cINF_mono
thf(fact_7585_le__cINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,U: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A4 )
=> ( ord_less_eq @ A @ U @ ( F2 @ X ) ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_7586_cInf__superset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ B6 ) @ ( complete_Inf_Inf @ A @ A4 ) ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_7587_cInf__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A2: A] :
( ( condit1013018076250108175_below @ A @ X8 )
=> ( ( ( X8
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ A2 @ X8 ) )
= A2 ) )
& ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ A2 @ X8 ) )
= ( inf_inf @ A @ A2 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_7588_cInf__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ A2 @ X8 ) )
= ( inf_inf @ A @ A2 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ) ).
% cInf_insert
thf(fact_7589_cInf__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A4 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B6 )
=> ( ( complete_Inf_Inf @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_7590_cINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A4: set @ B,F2: B > A,A2: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ A2 )
= ( ? [X: B] :
( ( member @ B @ X @ A4 )
& ( ord_less @ A @ ( F2 @ X ) @ A2 ) ) ) ) ) ) ) ).
% cINF_less_iff
thf(fact_7591_cINF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,G: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ G @ A4 ) )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ A4 ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [A5: B] : ( inf_inf @ A @ ( F2 @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_7592_cSUP__eq__cINF__D,axiom,
! [B: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F2: C > B,A4: set @ C,A2: C] :
( ( ( complete_Sup_Sup @ B @ ( image @ C @ B @ F2 @ A4 ) )
= ( complete_Inf_Inf @ B @ ( image @ C @ B @ F2 @ A4 ) ) )
=> ( ( condit941137186595557371_above @ B @ ( image @ C @ B @ F2 @ A4 ) )
=> ( ( condit1013018076250108175_below @ B @ ( image @ C @ B @ F2 @ A4 ) )
=> ( ( member @ C @ A2 @ A4 )
=> ( ( F2 @ A2 )
= ( complete_Inf_Inf @ B @ ( image @ C @ B @ F2 @ A4 ) ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_7593_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,G: B > A,B6: set @ B,F2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ G @ B6 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ B6 )
=> ( ord_less_eq @ A @ ( G @ X4 ) @ ( F2 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B6 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_7594_less__eq__cInf__inter,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( condit1013018076250108175_below @ A @ A4 )
=> ( ( condit1013018076250108175_below @ A @ B6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B6 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_7595_cINF__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,A2: B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ ( insert @ B @ A2 @ A4 ) ) )
= ( inf_inf @ A @ ( F2 @ A2 ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ) ) ) ).
% cINF_insert
thf(fact_7596_cINF__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,B6: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ B6 ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ B6 ) ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_7597_cInf__le__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A4 )
=> ( ( condit1013018076250108175_below @ A @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_7598_cInf__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S2: set @ A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S2 )
=> ( ( complete_Inf_Inf @ A @ S2 )
= ( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [X: A] :
! [Y: A] :
( ( member @ A @ Y @ S2 )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_7599_mono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A4: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit1013018076250108175_below @ A @ ( image @ C @ A @ A4 @ I6 ) )
=> ( ( I6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ A4 @ I6 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image @ C @ B
@ ^ [X: C] : ( F2 @ ( A4 @ X ) )
@ I6 ) ) ) ) ) ) ) ).
% mono_cINF
thf(fact_7600_Max_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic4895041142388067077er_set @ A @ ( ord_max @ A )
@ ^ [X: A,Y: A] : ( ord_less_eq @ A @ Y @ X )
@ ^ [X: A,Y: A] : ( ord_less @ A @ Y @ X ) ) ) ).
% Max.semilattice_order_set_axioms
thf(fact_7601_Gcd__fin__def,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( bounde2362111253966948842tice_F @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ) ).
% Gcd_fin_def
thf(fact_7602_Inf__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( lattic4895041142388067077er_set @ A @ ( inf_inf @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_7603_Min_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic4895041142388067077er_set @ A @ ( ord_min @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% Min.semilattice_order_set_axioms
thf(fact_7604_Sup__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( lattic4895041142388067077er_set @ A @ ( sup_sup @ A )
@ ^ [X: A,Y: A] : ( ord_less_eq @ A @ Y @ X )
@ ^ [X: A,Y: A] : ( ord_less @ A @ Y @ X ) ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_7605_num__of__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ num @ num @ code_pcr_integer
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
@ code_num_of_integer ) ).
% num_of_integer.transfer
thf(fact_7606_compute__powr__real,axiom,
( powr_real
= ( ^ [B4: real,I4: real] :
( if @ real @ ( ord_less_eq @ real @ B4 @ ( zero_zero @ real ) )
@ ( abort @ real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( zero_zero @ literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ^ [Uu3: product_unit] : ( powr_real @ B4 @ I4 ) )
@ ( if @ real
@ ( ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ I4 ) )
= I4 )
@ ( if @ real @ ( ord_less_eq @ real @ ( zero_zero @ real ) @ I4 ) @ ( power_power @ real @ B4 @ ( nat2 @ ( archim6421214686448440834_floor @ real @ I4 ) ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ B4 @ ( nat2 @ ( archim6421214686448440834_floor @ real @ ( uminus_uminus @ real @ I4 ) ) ) ) ) )
@ ( abort @ real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( zero_zero @ literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ^ [Uu3: product_unit] : ( powr_real @ B4 @ I4 ) ) ) ) ) ) ).
% compute_powr_real
thf(fact_7607_less__eq__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ ( int > $o ) @ ( code_integer > $o ) @ code_pcr_integer
@ ( bNF_rel_fun @ int @ code_integer @ $o @ $o @ code_pcr_integer
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( ord_less_eq @ int )
@ ( ord_less_eq @ code_integer ) ) ).
% less_eq_integer.transfer
thf(fact_7608_Code__Target__Nat_ONat_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ nat @ nat @ code_pcr_integer
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ nat2
@ code_Target_Nat ) ).
% Code_Target_Nat.Nat.transfer
thf(fact_7609_nat__of__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ nat @ nat @ code_pcr_integer
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ nat2
@ code_nat_of_integer ) ).
% nat_of_integer.transfer
thf(fact_7610_less__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ ( int > $o ) @ ( code_integer > $o ) @ code_pcr_integer
@ ( bNF_rel_fun @ int @ code_integer @ $o @ $o @ code_pcr_integer
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( ord_less @ int )
@ ( ord_less @ code_integer ) ) ).
% less_integer.transfer
thf(fact_7611_zero__integer_Otransfer,axiom,
code_pcr_integer @ ( zero_zero @ int ) @ ( zero_zero @ code_integer ) ).
% zero_integer.transfer
thf(fact_7612_String_Oempty__neq__Literal,axiom,
! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,S: literal] :
( ( zero_zero @ literal )
!= ( literal2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ S ) ) ).
% String.empty_neq_Literal
thf(fact_7613_uniformity__Abort,axiom,
! [A: $tType] :
( ( topolo4638772830378233104ormity @ A )
=> ( ( topolo7806501430040627800ormity @ A )
= ( abstract_filter @ ( product_prod @ A @ A )
@ ^ [U2: product_unit] :
( abort @ ( filter @ ( product_prod @ A @ A ) ) @ ( literal2 @ $true @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( zero_zero @ literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ^ [V5: product_unit] : ( topolo7806501430040627800ormity @ A ) ) ) ) ) ).
% uniformity_Abort
thf(fact_7614_int__of__integer__integer__of__nat,axiom,
! [N: nat] :
( ( code_int_of_integer @ ( code_integer_of_nat @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ).
% int_of_integer_integer_of_nat
thf(fact_7615_integer__of__nat_Orep__eq,axiom,
! [X3: nat] :
( ( code_int_of_integer @ ( code_integer_of_nat @ X3 ) )
= ( semiring_1_of_nat @ int @ X3 ) ) ).
% integer_of_nat.rep_eq
thf(fact_7616_integer__of__nat__0,axiom,
( ( code_integer_of_nat @ ( zero_zero @ nat ) )
= ( zero_zero @ code_integer ) ) ).
% integer_of_nat_0
thf(fact_7617_integer__of__nat_Oabs__eq,axiom,
( code_integer_of_nat
= ( ^ [X: nat] : ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X ) ) ) ) ).
% integer_of_nat.abs_eq
thf(fact_7618_integer__of__nat_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ int @ code_integer
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ code_pcr_integer
@ ( semiring_1_of_nat @ int )
@ code_integer_of_nat ) ).
% integer_of_nat.transfer
thf(fact_7619_integer__of__nat__numeral,axiom,
! [N: num] :
( ( code_integer_of_nat @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ code_integer @ N ) ) ).
% integer_of_nat_numeral
thf(fact_7620_divmod__nat__code,axiom,
( divmod_nat
= ( ^ [M3: nat,N3: nat] :
( product_map_prod @ code_integer @ nat @ code_integer @ nat @ code_nat_of_integer @ code_nat_of_integer
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( ( code_integer_of_nat @ M3 )
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( ( code_integer_of_nat @ N3 )
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( code_integer_of_nat @ M3 ) )
@ ( code_divmod_abs @ ( code_integer_of_nat @ M3 ) @ ( code_integer_of_nat @ N3 ) ) ) ) ) ) ) ).
% divmod_nat_code
thf(fact_7621_integer__of__nat__def,axiom,
( code_integer_of_nat
= ( map_fun @ nat @ nat @ int @ code_integer @ ( id @ nat ) @ code_integer_of_int @ ( semiring_1_of_nat @ int ) ) ) ).
% integer_of_nat_def
thf(fact_7622_length__code,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% length_code
thf(fact_7623_pairs__le__eq__Sigma,axiom,
! [M: nat] :
( ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ M ) ) )
= ( product_Sigma @ nat @ nat @ ( set_ord_atMost @ nat @ M )
@ ^ [R5: nat] : ( set_ord_atMost @ nat @ ( minus_minus @ nat @ M @ R5 ) ) ) ) ).
% pairs_le_eq_Sigma
thf(fact_7624_card__def,axiom,
! [B: $tType] :
( ( finite_card @ B )
= ( finite_folding_F @ B @ nat
@ ^ [Uu3: B] : suc
@ ( zero_zero @ nat ) ) ) ).
% card_def
thf(fact_7625_numeral__le__enat__iff,axiom,
! [M: num,N: nat] :
( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M ) @ N ) ) ).
% numeral_le_enat_iff
thf(fact_7626_enat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( extended_enat2 @ Nat )
= ( extended_enat2 @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% enat.inject
thf(fact_7627_enat_Osimps_I4_J,axiom,
! [T: $tType,F1: nat > T,F22: T,Nat: nat] :
( ( extended_case_enat @ T @ F1 @ F22 @ ( extended_enat2 @ Nat ) )
= ( F1 @ Nat ) ) ).
% enat.simps(4)
thf(fact_7628_enat__ord__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% enat_ord_simps(2)
thf(fact_7629_plus__enat__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( plus_plus @ nat @ M @ N ) ) ) ).
% plus_enat_simps(1)
thf(fact_7630_enat__ord__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% enat_ord_simps(1)
thf(fact_7631_idiff__enat__0__right,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ N @ ( extended_enat2 @ ( zero_zero @ nat ) ) )
= N ) ).
% idiff_enat_0_right
thf(fact_7632_idiff__enat__0,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( extended_enat2 @ ( zero_zero @ nat ) ) @ N )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% idiff_enat_0
thf(fact_7633_idiff__enat__enat,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus @ extended_enat @ ( extended_enat2 @ A2 ) @ ( extended_enat2 @ B2 ) )
= ( extended_enat2 @ ( minus_minus @ nat @ A2 @ B2 ) ) ) ).
% idiff_enat_enat
thf(fact_7634_times__enat__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( times_times @ nat @ M @ N ) ) ) ).
% times_enat_simps(1)
thf(fact_7635_max__enat__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_max @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( ord_max @ nat @ M @ N ) ) ) ).
% max_enat_simps(1)
thf(fact_7636_min__enat__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_min @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( ord_min @ nat @ M @ N ) ) ) ).
% min_enat_simps(1)
thf(fact_7637_numeral__less__enat__iff,axiom,
! [M: num,N: nat] :
( ( ord_less @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ N ) ) ).
% numeral_less_enat_iff
thf(fact_7638_finite__enat__bounded,axiom,
! [A4: set @ extended_enat,N: nat] :
( ! [Y4: extended_enat] :
( ( member @ extended_enat @ Y4 @ A4 )
=> ( ord_less_eq @ extended_enat @ Y4 @ ( extended_enat2 @ N ) ) )
=> ( finite_finite @ extended_enat @ A4 ) ) ).
% finite_enat_bounded
thf(fact_7639_enat__ile,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less_eq @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K2: nat] :
( N
= ( extended_enat2 @ K2 ) ) ) ).
% enat_ile
thf(fact_7640_chain__incr,axiom,
! [A: $tType,Y7: A > extended_enat,K: nat] :
( ! [I3: A] :
? [J4: A] : ( ord_less @ extended_enat @ ( Y7 @ I3 ) @ ( Y7 @ J4 ) )
=> ? [J2: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K ) @ ( Y7 @ J2 ) ) ) ).
% chain_incr
thf(fact_7641_enat__iless,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K2: nat] :
( N
= ( extended_enat2 @ K2 ) ) ) ).
% enat_iless
thf(fact_7642_less__enatE,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ~ ! [K2: nat] :
( ( N
= ( extended_enat2 @ K2 ) )
=> ~ ( ord_less @ nat @ K2 @ M ) ) ) ).
% less_enatE
thf(fact_7643_numeral__eq__enat,axiom,
( ( numeral_numeral @ extended_enat )
= ( ^ [K3: num] : ( extended_enat2 @ ( numeral_numeral @ nat @ K3 ) ) ) ) ).
% numeral_eq_enat
thf(fact_7644_one__enat__def,axiom,
( ( one_one @ extended_enat )
= ( extended_enat2 @ ( one_one @ nat ) ) ) ).
% one_enat_def
thf(fact_7645_enat__1__iff_I1_J,axiom,
! [X3: nat] :
( ( ( extended_enat2 @ X3 )
= ( one_one @ extended_enat ) )
= ( X3
= ( one_one @ nat ) ) ) ).
% enat_1_iff(1)
thf(fact_7646_enat__1__iff_I2_J,axiom,
! [X3: nat] :
( ( ( one_one @ extended_enat )
= ( extended_enat2 @ X3 ) )
= ( X3
= ( one_one @ nat ) ) ) ).
% enat_1_iff(2)
thf(fact_7647_of__nat__eq__enat,axiom,
( ( semiring_1_of_nat @ extended_enat )
= extended_enat2 ) ).
% of_nat_eq_enat
thf(fact_7648_zero__enat__def,axiom,
( ( zero_zero @ extended_enat )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% zero_enat_def
thf(fact_7649_enat__0__iff_I1_J,axiom,
! [X3: nat] :
( ( ( extended_enat2 @ X3 )
= ( zero_zero @ extended_enat ) )
= ( X3
= ( zero_zero @ nat ) ) ) ).
% enat_0_iff(1)
thf(fact_7650_enat__0__iff_I2_J,axiom,
! [X3: nat] :
( ( ( zero_zero @ extended_enat )
= ( extended_enat2 @ X3 ) )
= ( X3
= ( zero_zero @ nat ) ) ) ).
% enat_0_iff(2)
thf(fact_7651_Suc__ile__eq,axiom,
! [M: nat,N: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ ( suc @ M ) ) @ N )
= ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% Suc_ile_eq
thf(fact_7652_iadd__le__enat__iff,axiom,
! [X3: extended_enat,Y3: extended_enat,N: nat] :
( ( ord_less_eq @ extended_enat @ ( plus_plus @ extended_enat @ X3 @ Y3 ) @ ( extended_enat2 @ N ) )
= ( ? [Y9: nat,X10: nat] :
( ( X3
= ( extended_enat2 @ X10 ) )
& ( Y3
= ( extended_enat2 @ Y9 ) )
& ( ord_less_eq @ nat @ ( plus_plus @ nat @ X10 @ Y9 ) @ N ) ) ) ) ).
% iadd_le_enat_iff
thf(fact_7653_elimnum,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) ) ) ).
% elimnum
thf(fact_7654_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,L: nat] :
( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ L ) )
= ( vEBT_Node @ Info @ Deg
@ ( take @ vEBT_VEBT @ ( divide_divide @ nat @ L @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ TreeList2 ) )
@ ( vEBT_VEBT_elim_dead @ Summary @ ( extended_enat2 @ ( divide_divide @ nat @ L @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.elim_dead.simps(3)
thf(fact_7655_VEBT__internal_Oelim__dead_Oelims,axiom,
! [X3: vEBT_VEBT,Xa: extended_enat,Y3: vEBT_VEBT] :
( ( ( vEBT_VEBT_elim_dead @ X3 @ Xa )
= Y3 )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( Y3
!= ( vEBT_Leaf @ A6 @ B5 ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( Xa
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( Y3
!= ( vEBT_Node @ Info2 @ Deg2
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ TreeList3 )
@ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extend4730790105801354508finity @ extended_enat ) ) ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
=> ! [L4: nat] :
( ( Xa
= ( extended_enat2 @ L4 ) )
=> ( Y3
!= ( vEBT_Node @ Info2 @ Deg2
@ ( take @ vEBT_VEBT @ ( divide_divide @ nat @ L4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ TreeList3 ) )
@ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extended_enat2 @ ( divide_divide @ nat @ L4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.elim_dead.elims
thf(fact_7656_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( vEBT_Node @ Info @ Deg
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ TreeList2 )
@ ( vEBT_VEBT_elim_dead @ Summary @ ( extend4730790105801354508finity @ extended_enat ) ) ) ) ).
% VEBT_internal.elim_dead.simps(2)
thf(fact_7657_elimcomplete,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) ) ) ).
% elimcomplete
thf(fact_7658_not__enat__eq,axiom,
! [X3: extended_enat] :
( ( ! [Y: nat] :
( X3
!= ( extended_enat2 @ Y ) ) )
= ( X3
= ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% not_enat_eq
thf(fact_7659_not__infinity__eq,axiom,
! [X3: extended_enat] :
( ( X3
!= ( extend4730790105801354508finity @ extended_enat ) )
= ( ? [I4: nat] :
( X3
= ( extended_enat2 @ I4 ) ) ) ) ).
% not_infinity_eq
thf(fact_7660_enat__ord__simps_I6_J,axiom,
! [Q3: extended_enat] :
~ ( ord_less @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ Q3 ) ).
% enat_ord_simps(6)
thf(fact_7661_enat__ord__simps_I4_J,axiom,
! [Q3: extended_enat] :
( ( ord_less @ extended_enat @ Q3 @ ( extend4730790105801354508finity @ extended_enat ) )
= ( Q3
!= ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% enat_ord_simps(4)
thf(fact_7662_plus__enat__simps_I3_J,axiom,
! [Q3: extended_enat] :
( ( plus_plus @ extended_enat @ Q3 @ ( extend4730790105801354508finity @ extended_enat ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ).
% plus_enat_simps(3)
thf(fact_7663_plus__enat__simps_I2_J,axiom,
! [Q3: extended_enat] :
( ( plus_plus @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ Q3 )
= ( extend4730790105801354508finity @ extended_enat ) ) ).
% plus_enat_simps(2)
thf(fact_7664_enat__ord__code_I3_J,axiom,
! [Q3: extended_enat] : ( ord_less_eq @ extended_enat @ Q3 @ ( extend4730790105801354508finity @ extended_enat ) ) ).
% enat_ord_code(3)
thf(fact_7665_enat__ord__simps_I5_J,axiom,
! [Q3: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ Q3 )
= ( Q3
= ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% enat_ord_simps(5)
thf(fact_7666_idiff__infinity,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ N )
= ( extend4730790105801354508finity @ extended_enat ) ) ).
% idiff_infinity
thf(fact_7667_enat_Osimps_I5_J,axiom,
! [T: $tType,F1: nat > T,F22: T] :
( ( extended_case_enat @ T @ F1 @ F22 @ ( extend4730790105801354508finity @ extended_enat ) )
= F22 ) ).
% enat.simps(5)
thf(fact_7668_times__enat__simps_I2_J,axiom,
( ( times_times @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ).
% times_enat_simps(2)
thf(fact_7669_max__enat__simps_I5_J,axiom,
! [Q3: extended_enat] :
( ( ord_max @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ Q3 )
= ( extend4730790105801354508finity @ extended_enat ) ) ).
% max_enat_simps(5)
thf(fact_7670_max__enat__simps_I4_J,axiom,
! [Q3: extended_enat] :
( ( ord_max @ extended_enat @ Q3 @ ( extend4730790105801354508finity @ extended_enat ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ).
% max_enat_simps(4)
thf(fact_7671_min__enat__simps_I4_J,axiom,
! [Q3: extended_enat] :
( ( ord_min @ extended_enat @ Q3 @ ( extend4730790105801354508finity @ extended_enat ) )
= Q3 ) ).
% min_enat_simps(4)
thf(fact_7672_min__enat__simps_I5_J,axiom,
! [Q3: extended_enat] :
( ( ord_min @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ Q3 )
= Q3 ) ).
% min_enat_simps(5)
thf(fact_7673_idiff__self,axiom,
! [N: extended_enat] :
( ( N
!= ( extend4730790105801354508finity @ extended_enat ) )
=> ( ( minus_minus @ extended_enat @ N @ N )
= ( zero_zero @ extended_enat ) ) ) ).
% idiff_self
thf(fact_7674_add__diff__cancel__enat,axiom,
! [X3: extended_enat,Y3: extended_enat] :
( ( X3
!= ( extend4730790105801354508finity @ extended_enat ) )
=> ( ( minus_minus @ extended_enat @ ( plus_plus @ extended_enat @ X3 @ Y3 ) @ X3 )
= Y3 ) ) ).
% add_diff_cancel_enat
thf(fact_7675_idiff__infinity__right,axiom,
! [A2: nat] :
( ( minus_minus @ extended_enat @ ( extended_enat2 @ A2 ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) ).
% idiff_infinity_right
thf(fact_7676_times__enat__simps_I3_J,axiom,
! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ N ) )
= ( zero_zero @ extended_enat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ N ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ) ).
% times_enat_simps(3)
thf(fact_7677_times__enat__simps_I4_J,axiom,
! [M: nat] :
( ( ( M
= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) )
& ( ( M
!= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ) ).
% times_enat_simps(4)
thf(fact_7678_Sup__enat__def,axiom,
( ( complete_Sup_Sup @ extended_enat )
= ( ^ [A9: set @ extended_enat] :
( if @ extended_enat
@ ( A9
= ( bot_bot @ ( set @ extended_enat ) ) )
@ ( zero_zero @ extended_enat )
@ ( if @ extended_enat @ ( finite_finite @ extended_enat @ A9 ) @ ( lattic643756798349783984er_Max @ extended_enat @ A9 ) @ ( extend4730790105801354508finity @ extended_enat ) ) ) ) ) ).
% Sup_enat_def
thf(fact_7679_enat__add__left__cancel__le,axiom,
! [A2: extended_enat,B2: extended_enat,C2: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( plus_plus @ extended_enat @ A2 @ B2 ) @ ( plus_plus @ extended_enat @ A2 @ C2 ) )
= ( ( A2
= ( extend4730790105801354508finity @ extended_enat ) )
| ( ord_less_eq @ extended_enat @ B2 @ C2 ) ) ) ).
% enat_add_left_cancel_le
thf(fact_7680_enat__ord__simps_I3_J,axiom,
! [Q3: extended_enat] : ( ord_less_eq @ extended_enat @ Q3 @ ( extend4730790105801354508finity @ extended_enat ) ) ).
% enat_ord_simps(3)
thf(fact_7681_enat__add__left__cancel__less,axiom,
! [A2: extended_enat,B2: extended_enat,C2: extended_enat] :
( ( ord_less @ extended_enat @ ( plus_plus @ extended_enat @ A2 @ B2 ) @ ( plus_plus @ extended_enat @ A2 @ C2 ) )
= ( ( A2
!= ( extend4730790105801354508finity @ extended_enat ) )
& ( ord_less @ extended_enat @ B2 @ C2 ) ) ) ).
% enat_add_left_cancel_less
thf(fact_7682_enat__add__left__cancel,axiom,
! [A2: extended_enat,B2: extended_enat,C2: extended_enat] :
( ( ( plus_plus @ extended_enat @ A2 @ B2 )
= ( plus_plus @ extended_enat @ A2 @ C2 ) )
= ( ( A2
= ( extend4730790105801354508finity @ extended_enat ) )
| ( B2 = C2 ) ) ) ).
% enat_add_left_cancel
thf(fact_7683_plus__eq__infty__iff__enat,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( plus_plus @ extended_enat @ M @ N )
= ( extend4730790105801354508finity @ extended_enat ) )
= ( ( M
= ( extend4730790105801354508finity @ extended_enat ) )
| ( N
= ( extend4730790105801354508finity @ extended_enat ) ) ) ) ).
% plus_eq_infty_iff_enat
thf(fact_7684_infinity__ne__i1,axiom,
( ( extend4730790105801354508finity @ extended_enat )
!= ( one_one @ extended_enat ) ) ).
% infinity_ne_i1
thf(fact_7685_top__enat__def,axiom,
( ( top_top @ extended_enat )
= ( extend4730790105801354508finity @ extended_enat ) ) ).
% top_enat_def
thf(fact_7686_numeral__ne__infinity,axiom,
! [K: num] :
( ( numeral_numeral @ extended_enat @ K )
!= ( extend4730790105801354508finity @ extended_enat ) ) ).
% numeral_ne_infinity
thf(fact_7687_Inf__enat__def,axiom,
( ( complete_Inf_Inf @ extended_enat )
= ( ^ [A9: set @ extended_enat] :
( if @ extended_enat
@ ( A9
= ( bot_bot @ ( set @ extended_enat ) ) )
@ ( extend4730790105801354508finity @ extended_enat )
@ ( ord_Least @ extended_enat
@ ^ [X: extended_enat] : ( member @ extended_enat @ X @ A9 ) ) ) ) ) ).
% Inf_enat_def
thf(fact_7688_infinity__ne__i0,axiom,
( ( extend4730790105801354508finity @ extended_enat )
!= ( zero_zero @ extended_enat ) ) ).
% infinity_ne_i0
thf(fact_7689_imult__is__infinity,axiom,
! [A2: extended_enat,B2: extended_enat] :
( ( ( times_times @ extended_enat @ A2 @ B2 )
= ( extend4730790105801354508finity @ extended_enat ) )
= ( ( ( A2
= ( extend4730790105801354508finity @ extended_enat ) )
& ( B2
!= ( zero_zero @ extended_enat ) ) )
| ( ( B2
= ( extend4730790105801354508finity @ extended_enat ) )
& ( A2
!= ( zero_zero @ extended_enat ) ) ) ) ) ).
% imult_is_infinity
thf(fact_7690_enat__ord__code_I4_J,axiom,
! [M: nat] : ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extend4730790105801354508finity @ extended_enat ) ) ).
% enat_ord_code(4)
thf(fact_7691_less__infinityE,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ N @ ( extend4730790105801354508finity @ extended_enat ) )
=> ~ ! [K2: nat] :
( N
!= ( extended_enat2 @ K2 ) ) ) ).
% less_infinityE
thf(fact_7692_infinity__ilessE,axiom,
! [M: nat] :
~ ( ord_less @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ M ) ) ).
% infinity_ilessE
thf(fact_7693_enat_Odistinct_I2_J,axiom,
! [Nat5: nat] :
( ( extend4730790105801354508finity @ extended_enat )
!= ( extended_enat2 @ Nat5 ) ) ).
% enat.distinct(2)
thf(fact_7694_enat_Odistinct_I1_J,axiom,
! [Nat: nat] :
( ( extended_enat2 @ Nat )
!= ( extend4730790105801354508finity @ extended_enat ) ) ).
% enat.distinct(1)
thf(fact_7695_enat_Oexhaust,axiom,
! [Y3: extended_enat] :
( ! [Nat4: nat] :
( Y3
!= ( extended_enat2 @ Nat4 ) )
=> ( Y3
= ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% enat.exhaust
thf(fact_7696_enat2__cases,axiom,
! [Y3: extended_enat,Ya: extended_enat] :
( ( ? [Nat4: nat] :
( Y3
= ( extended_enat2 @ Nat4 ) )
=> ! [Nata: nat] :
( Ya
!= ( extended_enat2 @ Nata ) ) )
=> ( ( ? [Nat4: nat] :
( Y3
= ( extended_enat2 @ Nat4 ) )
=> ( Ya
!= ( extend4730790105801354508finity @ extended_enat ) ) )
=> ( ( ( Y3
= ( extend4730790105801354508finity @ extended_enat ) )
=> ! [Nat4: nat] :
( Ya
!= ( extended_enat2 @ Nat4 ) ) )
=> ~ ( ( Y3
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( Ya
!= ( extend4730790105801354508finity @ extended_enat ) ) ) ) ) ) ).
% enat2_cases
thf(fact_7697_enat3__cases,axiom,
! [Y3: extended_enat,Ya: extended_enat,Yb: extended_enat] :
( ( ? [Nat4: nat] :
( Y3
= ( extended_enat2 @ Nat4 ) )
=> ( ? [Nata: nat] :
( Ya
= ( extended_enat2 @ Nata ) )
=> ! [Natb: nat] :
( Yb
!= ( extended_enat2 @ Natb ) ) ) )
=> ( ( ? [Nat4: nat] :
( Y3
= ( extended_enat2 @ Nat4 ) )
=> ( ? [Nata: nat] :
( Ya
= ( extended_enat2 @ Nata ) )
=> ( Yb
!= ( extend4730790105801354508finity @ extended_enat ) ) ) )
=> ( ( ? [Nat4: nat] :
( Y3
= ( extended_enat2 @ Nat4 ) )
=> ( ( Ya
= ( extend4730790105801354508finity @ extended_enat ) )
=> ! [Nata: nat] :
( Yb
!= ( extended_enat2 @ Nata ) ) ) )
=> ( ( ? [Nat4: nat] :
( Y3
= ( extended_enat2 @ Nat4 ) )
=> ( ( Ya
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( Yb
!= ( extend4730790105801354508finity @ extended_enat ) ) ) )
=> ( ( ( Y3
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( ? [Nat4: nat] :
( Ya
= ( extended_enat2 @ Nat4 ) )
=> ! [Nata: nat] :
( Yb
!= ( extended_enat2 @ Nata ) ) ) )
=> ( ( ( Y3
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( ? [Nat4: nat] :
( Ya
= ( extended_enat2 @ Nat4 ) )
=> ( Yb
!= ( extend4730790105801354508finity @ extended_enat ) ) ) )
=> ( ( ( Y3
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( ( Ya
= ( extend4730790105801354508finity @ extended_enat ) )
=> ! [Nat4: nat] :
( Yb
!= ( extended_enat2 @ Nat4 ) ) ) )
=> ~ ( ( Y3
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( ( Ya
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( Yb
!= ( extend4730790105801354508finity @ extended_enat ) ) ) ) ) ) ) ) ) ) ) ).
% enat3_cases
thf(fact_7698_enat__ex__split,axiom,
( ( ^ [P2: extended_enat > $o] :
? [X5: extended_enat] : ( P2 @ X5 ) )
= ( ^ [P3: extended_enat > $o] :
( ( P3 @ ( extend4730790105801354508finity @ extended_enat ) )
| ? [X: nat] : ( P3 @ ( extended_enat2 @ X ) ) ) ) ) ).
% enat_ex_split
thf(fact_7699_infinity__ileE,axiom,
! [M: nat] :
~ ( ord_less_eq @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ M ) ) ).
% infinity_ileE
thf(fact_7700_enat__ord__code_I5_J,axiom,
! [N: nat] :
~ ( ord_less_eq @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ N ) ) ).
% enat_ord_code(5)
thf(fact_7701_plus__enat__def,axiom,
( ( plus_plus @ extended_enat )
= ( ^ [M3: extended_enat,N3: extended_enat] :
( extended_case_enat @ extended_enat
@ ^ [O: nat] :
( extended_case_enat @ extended_enat
@ ^ [P5: nat] : ( extended_enat2 @ ( plus_plus @ nat @ O @ P5 ) )
@ ( extend4730790105801354508finity @ extended_enat )
@ N3 )
@ ( extend4730790105801354508finity @ extended_enat )
@ M3 ) ) ) ).
% plus_enat_def
thf(fact_7702_imult__infinity__right,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
=> ( ( times_times @ extended_enat @ N @ ( extend4730790105801354508finity @ extended_enat ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% imult_infinity_right
thf(fact_7703_imult__infinity,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
=> ( ( times_times @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ N )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% imult_infinity
thf(fact_7704_diff__enat__def,axiom,
( ( minus_minus @ extended_enat )
= ( ^ [A5: extended_enat,B4: extended_enat] :
( extended_case_enat @ extended_enat
@ ^ [X: nat] :
( extended_case_enat @ extended_enat
@ ^ [Y: nat] : ( extended_enat2 @ ( minus_minus @ nat @ X @ Y ) )
@ ( zero_zero @ extended_enat )
@ B4 )
@ ( extend4730790105801354508finity @ extended_enat )
@ A5 ) ) ) ).
% diff_enat_def
thf(fact_7705_times__enat__def,axiom,
( ( times_times @ extended_enat )
= ( ^ [M3: extended_enat,N3: extended_enat] :
( extended_case_enat @ extended_enat
@ ^ [O: nat] :
( extended_case_enat @ extended_enat
@ ^ [P5: nat] : ( extended_enat2 @ ( times_times @ nat @ O @ P5 ) )
@ ( if @ extended_enat
@ ( O
= ( zero_zero @ nat ) )
@ ( zero_zero @ extended_enat )
@ ( extend4730790105801354508finity @ extended_enat ) )
@ N3 )
@ ( if @ extended_enat
@ ( N3
= ( zero_zero @ extended_enat ) )
@ ( zero_zero @ extended_enat )
@ ( extend4730790105801354508finity @ extended_enat ) )
@ M3 ) ) ) ).
% times_enat_def
thf(fact_7706_VEBT__internal_Oelim__dead_Opelims,axiom,
! [X3: vEBT_VEBT,Xa: extended_enat,Y3: vEBT_VEBT] :
( ( ( vEBT_VEBT_elim_dead @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ extended_enat ) @ vEBT_V312737461966249ad_rel @ ( product_Pair @ vEBT_VEBT @ extended_enat @ X3 @ Xa ) )
=> ( ! [A6: $o,B5: $o] :
( ( X3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Y3
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ extended_enat ) @ vEBT_V312737461966249ad_rel @ ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( Xa
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( ( Y3
= ( vEBT_Node @ Info2 @ Deg2
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ TreeList3 )
@ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extend4730790105801354508finity @ extended_enat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ extended_enat ) @ vEBT_V312737461966249ad_rel @ ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) @ ( extend4730790105801354508finity @ extended_enat ) ) ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
=> ! [L4: nat] :
( ( Xa
= ( extended_enat2 @ L4 ) )
=> ( ( Y3
= ( vEBT_Node @ Info2 @ Deg2
@ ( take @ vEBT_VEBT @ ( divide_divide @ nat @ L4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ TreeList3 ) )
@ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extended_enat2 @ ( divide_divide @ nat @ L4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ extended_enat ) @ vEBT_V312737461966249ad_rel @ ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) @ ( extended_enat2 @ L4 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.elim_dead.pelims
thf(fact_7707_the__enat_Osimps,axiom,
! [N: nat] :
( ( extended_the_enat @ ( extended_enat2 @ N ) )
= N ) ).
% the_enat.simps
thf(fact_7708_eSuc__Max,axiom,
! [A4: set @ extended_enat] :
( ( finite_finite @ extended_enat @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ extended_enat ) ) )
=> ( ( extended_eSuc @ ( lattic643756798349783984er_Max @ extended_enat @ A4 ) )
= ( lattic643756798349783984er_Max @ extended_enat @ ( image @ extended_enat @ extended_enat @ extended_eSuc @ A4 ) ) ) ) ) ).
% eSuc_Max
thf(fact_7709_enat_Osimps_I7_J,axiom,
! [T: $tType,F1: nat > T,F22: T] :
( ( extended_rec_enat @ T @ F1 @ F22 @ ( extend4730790105801354508finity @ extended_enat ) )
= F22 ) ).
% enat.simps(7)
thf(fact_7710_eSuc__inject,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( extended_eSuc @ M )
= ( extended_eSuc @ N ) )
= ( M = N ) ) ).
% eSuc_inject
thf(fact_7711_eSuc__infinity,axiom,
( ( extended_eSuc @ ( extend4730790105801354508finity @ extended_enat ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ).
% eSuc_infinity
thf(fact_7712_eSuc__mono,axiom,
! [N: extended_enat,M: extended_enat] :
( ( ord_less @ extended_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
= ( ord_less @ extended_enat @ N @ M ) ) ).
% eSuc_mono
thf(fact_7713_eSuc__ile__mono,axiom,
! [N: extended_enat,M: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
= ( ord_less_eq @ extended_enat @ N @ M ) ) ).
% eSuc_ile_mono
thf(fact_7714_eSuc__minus__eSuc,axiom,
! [N: extended_enat,M: extended_enat] :
( ( minus_minus @ extended_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
= ( minus_minus @ extended_enat @ N @ M ) ) ).
% eSuc_minus_eSuc
thf(fact_7715_enat_Osimps_I6_J,axiom,
! [T: $tType,F1: nat > T,F22: T,Nat: nat] :
( ( extended_rec_enat @ T @ F1 @ F22 @ ( extended_enat2 @ Nat ) )
= ( F1 @ Nat ) ) ).
% enat.simps(6)
thf(fact_7716_iless__eSuc0,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ N @ ( extended_eSuc @ ( zero_zero @ extended_enat ) ) )
= ( N
= ( zero_zero @ extended_enat ) ) ) ).
% iless_eSuc0
thf(fact_7717_eSuc__minus__1,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( extended_eSuc @ N ) @ ( one_one @ extended_enat ) )
= N ) ).
% eSuc_minus_1
thf(fact_7718_eSuc__numeral,axiom,
! [K: num] :
( ( extended_eSuc @ ( numeral_numeral @ extended_enat @ K ) )
= ( numeral_numeral @ extended_enat @ ( plus_plus @ num @ K @ one2 ) ) ) ).
% eSuc_numeral
thf(fact_7719_iless__Suc__eq,axiom,
! [M: nat,N: extended_enat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_eSuc @ N ) )
= ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% iless_Suc_eq
thf(fact_7720_ile__eSuc,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ N @ ( extended_eSuc @ N ) ) ).
% ile_eSuc
thf(fact_7721_ileI1,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_less @ extended_enat @ M @ N )
=> ( ord_less_eq @ extended_enat @ ( extended_eSuc @ M ) @ N ) ) ).
% ileI1
thf(fact_7722_sup__continuous__eSuc,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F2: A > extended_enat] :
( ( order_sup_continuous @ A @ extended_enat @ F2 )
=> ( order_sup_continuous @ A @ extended_enat
@ ^ [X: A] : ( extended_eSuc @ ( F2 @ X ) ) ) ) ) ).
% sup_continuous_eSuc
thf(fact_7723_eSuc__plus__1,axiom,
( extended_eSuc
= ( ^ [N3: extended_enat] : ( plus_plus @ extended_enat @ N3 @ ( one_one @ extended_enat ) ) ) ) ).
% eSuc_plus_1
thf(fact_7724_plus__1__eSuc_I1_J,axiom,
! [Q3: extended_enat] :
( ( plus_plus @ extended_enat @ ( one_one @ extended_enat ) @ Q3 )
= ( extended_eSuc @ Q3 ) ) ).
% plus_1_eSuc(1)
thf(fact_7725_plus__1__eSuc_I2_J,axiom,
! [Q3: extended_enat] :
( ( plus_plus @ extended_enat @ Q3 @ ( one_one @ extended_enat ) )
= ( extended_eSuc @ Q3 ) ) ).
% plus_1_eSuc(2)
thf(fact_7726_mono__eSuc,axiom,
order_mono @ extended_enat @ extended_enat @ extended_eSuc ).
% mono_eSuc
thf(fact_7727_iadd__Suc,axiom,
! [M: extended_enat,N: extended_enat] :
( ( plus_plus @ extended_enat @ ( extended_eSuc @ M ) @ N )
= ( extended_eSuc @ ( plus_plus @ extended_enat @ M @ N ) ) ) ).
% iadd_Suc
thf(fact_7728_iadd__Suc__right,axiom,
! [M: extended_enat,N: extended_enat] :
( ( plus_plus @ extended_enat @ M @ ( extended_eSuc @ N ) )
= ( extended_eSuc @ ( plus_plus @ extended_enat @ M @ N ) ) ) ).
% iadd_Suc_right
thf(fact_7729_eSuc__max,axiom,
! [X3: extended_enat,Y3: extended_enat] :
( ( extended_eSuc @ ( ord_max @ extended_enat @ X3 @ Y3 ) )
= ( ord_max @ extended_enat @ ( extended_eSuc @ X3 ) @ ( extended_eSuc @ Y3 ) ) ) ).
% eSuc_max
thf(fact_7730_mult__eSuc__right,axiom,
! [M: extended_enat,N: extended_enat] :
( ( times_times @ extended_enat @ M @ ( extended_eSuc @ N ) )
= ( plus_plus @ extended_enat @ M @ ( times_times @ extended_enat @ M @ N ) ) ) ).
% mult_eSuc_right
thf(fact_7731_mult__eSuc,axiom,
! [M: extended_enat,N: extended_enat] :
( ( times_times @ extended_enat @ ( extended_eSuc @ M ) @ N )
= ( plus_plus @ extended_enat @ N @ ( times_times @ extended_enat @ M @ N ) ) ) ).
% mult_eSuc
thf(fact_7732_zero__ne__eSuc,axiom,
! [N: extended_enat] :
( ( zero_zero @ extended_enat )
!= ( extended_eSuc @ N ) ) ).
% zero_ne_eSuc
thf(fact_7733_i0__iless__eSuc,axiom,
! [N: extended_enat] : ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ ( extended_eSuc @ N ) ) ).
% i0_iless_eSuc
thf(fact_7734_one__eSuc,axiom,
( ( one_one @ extended_enat )
= ( extended_eSuc @ ( zero_zero @ extended_enat ) ) ) ).
% one_eSuc
thf(fact_7735_not__eSuc__ilei0,axiom,
! [N: extended_enat] :
~ ( ord_less_eq @ extended_enat @ ( extended_eSuc @ N ) @ ( zero_zero @ extended_enat ) ) ).
% not_eSuc_ilei0
thf(fact_7736_enat__eSuc__iff,axiom,
! [Y3: nat,X3: extended_enat] :
( ( ( extended_enat2 @ Y3 )
= ( extended_eSuc @ X3 ) )
= ( ? [N3: nat] :
( ( Y3
= ( suc @ N3 ) )
& ( ( extended_enat2 @ N3 )
= X3 ) ) ) ) ).
% enat_eSuc_iff
thf(fact_7737_eSuc__enat__iff,axiom,
! [X3: extended_enat,Y3: nat] :
( ( ( extended_eSuc @ X3 )
= ( extended_enat2 @ Y3 ) )
= ( ? [N3: nat] :
( ( Y3
= ( suc @ N3 ) )
& ( X3
= ( extended_enat2 @ N3 ) ) ) ) ) ).
% eSuc_enat_iff
thf(fact_7738_eSuc__enat,axiom,
! [N: nat] :
( ( extended_eSuc @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( suc @ N ) ) ) ).
% eSuc_enat
thf(fact_7739_eSuc__Sup,axiom,
! [A4: set @ extended_enat] :
( ( A4
!= ( bot_bot @ ( set @ extended_enat ) ) )
=> ( ( extended_eSuc @ ( complete_Sup_Sup @ extended_enat @ A4 ) )
= ( complete_Sup_Sup @ extended_enat @ ( image @ extended_enat @ extended_enat @ extended_eSuc @ A4 ) ) ) ) ).
% eSuc_Sup
thf(fact_7740_case__enat__def,axiom,
! [T: $tType] :
( ( extended_case_enat @ T )
= ( extended_rec_enat @ T ) ) ).
% case_enat_def
thf(fact_7741_eSuc__def,axiom,
( extended_eSuc
= ( extended_case_enat @ extended_enat
@ ^ [N3: nat] : ( extended_enat2 @ ( suc @ N3 ) )
@ ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% eSuc_def
thf(fact_7742_rec__enat__def,axiom,
! [T: $tType] :
( ( extended_rec_enat @ T )
= ( ^ [F12: nat > T,F23: T,X: extended_enat] : ( the @ T @ ( extend4933016492236175606t_enat @ T @ F12 @ F23 @ X ) ) ) ) ).
% rec_enat_def
thf(fact_7743_sub_Otransfer,axiom,
( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > code_integer )
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ ( bNF_rel_fun @ num @ num @ int @ code_integer
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ code_pcr_integer )
@ ^ [M3: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M3 ) @ ( numeral_numeral @ int @ N3 ) )
@ code_sub ) ).
% sub.transfer
thf(fact_7744_Code__Numeral_Osub__code_I1_J,axiom,
( ( code_sub @ one2 @ one2 )
= ( zero_zero @ code_integer ) ) ).
% Code_Numeral.sub_code(1)
thf(fact_7745_sub_Orep__eq,axiom,
! [X3: num,Xa: num] :
( ( code_int_of_integer @ ( code_sub @ X3 @ Xa ) )
= ( minus_minus @ int @ ( numeral_numeral @ int @ X3 ) @ ( numeral_numeral @ int @ Xa ) ) ) ).
% sub.rep_eq
thf(fact_7746_sub_Oabs__eq,axiom,
( code_sub
= ( ^ [Xa4: num,X: num] : ( code_integer_of_int @ ( minus_minus @ int @ ( numeral_numeral @ int @ Xa4 ) @ ( numeral_numeral @ int @ X ) ) ) ) ) ).
% sub.abs_eq
thf(fact_7747_Code__Numeral_Osub__code_I9_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( minus_minus @ code_integer @ ( code_dup @ ( code_sub @ M @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% Code_Numeral.sub_code(9)
thf(fact_7748_Code__Numeral_Osub__code_I8_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( plus_plus @ code_integer @ ( code_dup @ ( code_sub @ M @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% Code_Numeral.sub_code(8)
thf(fact_7749_Code__Numeral_Odup__code_I1_J,axiom,
( ( code_dup @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% Code_Numeral.dup_code(1)
thf(fact_7750_Code__Numeral_Osub__code_I6_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( code_dup @ ( code_sub @ M @ N ) ) ) ).
% Code_Numeral.sub_code(6)
thf(fact_7751_Code__Numeral_Osub__code_I4_J,axiom,
! [N: num] :
( ( code_sub @ one2 @ ( bit0 @ N ) )
= ( code_Neg @ ( bitM @ N ) ) ) ).
% Code_Numeral.sub_code(4)
thf(fact_7752_less__than__iff,axiom,
! [X3: nat,Y3: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X3 @ Y3 ) @ less_than )
= ( ord_less @ nat @ X3 @ Y3 ) ) ).
% less_than_iff
thf(fact_7753_Code__Numeral_Odup__code_I3_J,axiom,
! [N: num] :
( ( code_dup @ ( code_Neg @ N ) )
= ( code_Neg @ ( bit0 @ N ) ) ) ).
% Code_Numeral.dup_code(3)
thf(fact_7754_less__eq__integer__code_I7_J,axiom,
! [K: num] : ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( zero_zero @ code_integer ) ) ).
% less_eq_integer_code(7)
thf(fact_7755_less__eq__integer__code_I3_J,axiom,
! [L: num] :
~ ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Neg @ L ) ) ).
% less_eq_integer_code(3)
thf(fact_7756_less__integer__code_I3_J,axiom,
! [L: num] :
~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Neg @ L ) ) ).
% less_integer_code(3)
thf(fact_7757_less__integer__code_I7_J,axiom,
! [K: num] : ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(7)
thf(fact_7758_less__integer__code_I9_J,axiom,
! [K: num,L: num] :
( ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
= ( ord_less @ num @ L @ K ) ) ).
% less_integer_code(9)
thf(fact_7759_less__eq__integer__code_I9_J,axiom,
! [K: num,L: num] :
( ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
= ( ord_less_eq @ num @ L @ K ) ) ).
% less_eq_integer_code(9)
thf(fact_7760_Code__Numeral_Osub__code_I5_J,axiom,
! [N: num] :
( ( code_sub @ one2 @ ( bit1 @ N ) )
= ( code_Neg @ ( bit0 @ N ) ) ) ).
% Code_Numeral.sub_code(5)
thf(fact_7761_pair__less__def,axiom,
( fun_pair_less
= ( lex_prod @ nat @ nat @ less_than @ less_than ) ) ).
% pair_less_def
thf(fact_7762_Gcd__nat__set__eq__fold,axiom,
! [Xs2: list @ nat] :
( ( gcd_Gcd @ nat @ ( set2 @ nat @ Xs2 ) )
= ( fold @ nat @ nat @ ( gcd_gcd @ nat ) @ Xs2 @ ( zero_zero @ nat ) ) ) ).
% Gcd_nat_set_eq_fold
thf(fact_7763_Gcd__set__eq__fold,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [Xs2: list @ A] :
( ( gcd_Gcd @ A @ ( set2 @ A @ Xs2 ) )
= ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs2 @ ( zero_zero @ A ) ) ) ) ).
% Gcd_set_eq_fold
thf(fact_7764_Gcd__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs2: list @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Xs2 ) )
= ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs2 @ ( zero_zero @ A ) ) ) ) ).
% Gcd_fin.set_eq_fold
thf(fact_7765_Quotient__real,axiom,
quotient @ ( nat > rat ) @ real @ realrel @ real2 @ rep_real @ cr_real ).
% Quotient_real
thf(fact_7766_Code__Numeral_Osub__code_I2_J,axiom,
! [M: num] :
( ( code_sub @ ( bit0 @ M ) @ one2 )
= ( code_Pos @ ( bitM @ M ) ) ) ).
% Code_Numeral.sub_code(2)
thf(fact_7767_Code__Numeral_Odup__code_I2_J,axiom,
! [N: num] :
( ( code_dup @ ( code_Pos @ N ) )
= ( code_Pos @ ( bit0 @ N ) ) ) ).
% Code_Numeral.dup_code(2)
thf(fact_7768_less__eq__integer__code_I4_J,axiom,
! [K: num] :
~ ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( zero_zero @ code_integer ) ) ).
% less_eq_integer_code(4)
thf(fact_7769_less__eq__integer__code_I2_J,axiom,
! [L: num] : ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Pos @ L ) ) ).
% less_eq_integer_code(2)
thf(fact_7770_less__integer__code_I2_J,axiom,
! [L: num] : ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Pos @ L ) ) ).
% less_integer_code(2)
thf(fact_7771_less__integer__code_I4_J,axiom,
! [K: num] :
~ ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(4)
thf(fact_7772_less__integer__code_I5_J,axiom,
! [K: num,L: num] :
( ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
= ( ord_less @ num @ K @ L ) ) ).
% less_integer_code(5)
thf(fact_7773_one__integer__code,axiom,
( ( one_one @ code_integer )
= ( code_Pos @ one2 ) ) ).
% one_integer_code
thf(fact_7774_Pos__fold_I2_J,axiom,
! [K: num] :
( ( numeral_numeral @ code_integer @ ( bit0 @ K ) )
= ( code_Pos @ ( bit0 @ K ) ) ) ).
% Pos_fold(2)
thf(fact_7775_Pos__fold_I1_J,axiom,
( ( numeral_numeral @ code_integer @ one2 )
= ( code_Pos @ one2 ) ) ).
% Pos_fold(1)
thf(fact_7776_less__eq__integer__code_I5_J,axiom,
! [K: num,L: num] :
( ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
= ( ord_less_eq @ num @ K @ L ) ) ).
% less_eq_integer_code(5)
thf(fact_7777_Gcd__int__set__eq__fold,axiom,
! [Xs2: list @ int] :
( ( gcd_Gcd @ int @ ( set2 @ int @ Xs2 ) )
= ( fold @ int @ int @ ( gcd_gcd @ int ) @ Xs2 @ ( zero_zero @ int ) ) ) ).
% Gcd_int_set_eq_fold
thf(fact_7778_Code__Numeral_Osub__code_I3_J,axiom,
! [M: num] :
( ( code_sub @ ( bit1 @ M ) @ one2 )
= ( code_Pos @ ( bit0 @ M ) ) ) ).
% Code_Numeral.sub_code(3)
thf(fact_7779_Zfun__imp__Zfun,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F4: filter @ A,G: A > C,K5: real] :
( ( zfun @ A @ B @ F2 @ F4 )
=> ( ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G @ X ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) @ K5 ) )
@ F4 )
=> ( zfun @ A @ C @ G @ F4 ) ) ) ) ).
% Zfun_imp_Zfun
thf(fact_7780_semilattice__order__set_Osubset__imp,axiom,
! [A: $tType,F2: A > A > A,Less_eq2: A > A > $o,Less: A > A > $o,A4: set @ A,B6: set @ A] :
( ( lattic4895041142388067077er_set @ A @ F2 @ Less_eq2 @ Less )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B6 )
=> ( Less_eq2 @ ( lattic1715443433743089157tice_F @ A @ F2 @ B6 ) @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_7781_Max__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( lattic1715443433743089157tice_F @ A @ ( ord_max @ A ) ) ) ) ).
% Max_def
thf(fact_7782_semilattice__order__set_OcoboundedI,axiom,
! [A: $tType,F2: A > A > A,Less_eq2: A > A > $o,Less: A > A > $o,A4: set @ A,A2: A] :
( ( lattic4895041142388067077er_set @ A @ F2 @ Less_eq2 @ Less )
=> ( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ A2 @ A4 )
=> ( Less_eq2 @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) @ A2 ) ) ) ) ).
% semilattice_order_set.coboundedI
thf(fact_7783_Zfun__le,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G: A > B,F4: filter @ A,F2: A > C] :
( ( zfun @ A @ B @ G @ F4 )
=> ( ! [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( F2 @ X4 ) ) @ ( real_V7770717601297561774m_norm @ B @ ( G @ X4 ) ) )
=> ( zfun @ A @ C @ F2 @ F4 ) ) ) ) ).
% Zfun_le
thf(fact_7784_Inf__fin__def,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( lattic1715443433743089157tice_F @ A @ ( inf_inf @ A ) ) ) ) ).
% Inf_fin_def
thf(fact_7785_semilattice__set_OF_Ocong,axiom,
! [A: $tType] :
( ( lattic1715443433743089157tice_F @ A )
= ( lattic1715443433743089157tice_F @ A ) ) ).
% semilattice_set.F.cong
thf(fact_7786_Min__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798350308766er_Min @ A )
= ( lattic1715443433743089157tice_F @ A @ ( ord_min @ A ) ) ) ) ).
% Min_def
thf(fact_7787_Sup__fin__def,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( lattic1715443433743089157tice_F @ A @ ( sup_sup @ A ) ) ) ) ).
% Sup_fin_def
thf(fact_7788_Zfun__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F4: filter @ A] :
( zfun @ A @ B
@ ^ [X: A] : ( zero_zero @ B )
@ F4 ) ) ).
% Zfun_zero
thf(fact_7789_semilattice__order__set_OboundedE,axiom,
! [A: $tType,F2: A > A > A,Less_eq2: A > A > $o,Less: A > A > $o,A4: set @ A,X3: A] :
( ( lattic4895041142388067077er_set @ A @ F2 @ Less_eq2 @ Less )
=> ( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( Less_eq2 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) )
=> ! [A8: A] :
( ( member @ A @ A8 @ A4 )
=> ( Less_eq2 @ X3 @ A8 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_7790_semilattice__order__set_OboundedI,axiom,
! [A: $tType,F2: A > A > A,Less_eq2: A > A > $o,Less: A > A > $o,A4: set @ A,X3: A] :
( ( lattic4895041142388067077er_set @ A @ F2 @ Less_eq2 @ Less )
=> ( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ( Less_eq2 @ X3 @ A6 ) )
=> ( Less_eq2 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_7791_semilattice__order__set_Obounded__iff,axiom,
! [A: $tType,F2: A > A > A,Less_eq2: A > A > $o,Less: A > A > $o,A4: set @ A,X3: A] :
( ( lattic4895041142388067077er_set @ A @ F2 @ Less_eq2 @ Less )
=> ( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( Less_eq2 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( Less_eq2 @ X3 @ X ) ) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
thf(fact_7792_ZfunD,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A,R2: real] :
( ( zfun @ A @ B @ F2 @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) @ R2 )
@ F4 ) ) ) ) ).
% ZfunD
thf(fact_7793_ZfunI,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ! [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) @ R )
@ F4 ) )
=> ( zfun @ A @ B @ F2 @ F4 ) ) ) ).
% ZfunI
thf(fact_7794_Zfun__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ( ( zfun @ A @ B )
= ( ^ [F3: A > B,F8: filter @ A] :
! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X ) ) @ R5 )
@ F8 ) ) ) ) ) ).
% Zfun_def
thf(fact_7795_semilattice__set_Oeq__fold_H,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ A4 )
= ( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X: A,Y: option @ A] : ( some @ A @ ( case_option @ A @ A @ X @ ( F2 @ X ) @ Y ) )
@ ( none @ A )
@ A4 ) ) ) ) ).
% semilattice_set.eq_fold'
thf(fact_7796_semilattice__set_Oremove,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ A4 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ A4 )
= ( F2 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_7797_semilattice__set_Oin__idem,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( F2 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) )
= ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) ) ) ) ) ).
% semilattice_set.in_idem
thf(fact_7798_Sup__fin_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( lattic149705377957585745ce_set @ A @ ( sup_sup @ A ) ) ) ).
% Sup_fin.semilattice_set_axioms
thf(fact_7799_Min_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic149705377957585745ce_set @ A @ ( ord_min @ A ) ) ) ).
% Min.semilattice_set_axioms
thf(fact_7800_Max_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic149705377957585745ce_set @ A @ ( ord_max @ A ) ) ) ).
% Max.semilattice_set_axioms
thf(fact_7801_Inf__fin_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( lattic149705377957585745ce_set @ A @ ( inf_inf @ A ) ) ) ).
% Inf_fin.semilattice_set_axioms
thf(fact_7802_semilattice__order__set_Oaxioms_I2_J,axiom,
! [A: $tType,F2: A > A > A,Less_eq2: A > A > $o,Less: A > A > $o] :
( ( lattic4895041142388067077er_set @ A @ F2 @ Less_eq2 @ Less )
=> ( lattic149705377957585745ce_set @ A @ F2 ) ) ).
% semilattice_order_set.axioms(2)
thf(fact_7803_semilattice__set_Osingleton,axiom,
! [A: $tType,F2: A > A > A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% semilattice_set.singleton
thf(fact_7804_semilattice__set_Ohom__commute,axiom,
! [A: $tType,F2: A > A > A,H: A > A,N4: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ! [X4: A,Y4: A] :
( ( H @ ( F2 @ X4 @ Y4 ) )
= ( F2 @ ( H @ X4 ) @ ( H @ Y4 ) ) )
=> ( ( finite_finite @ A @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic1715443433743089157tice_F @ A @ F2 @ N4 ) )
= ( lattic1715443433743089157tice_F @ A @ F2 @ ( image @ A @ A @ H @ N4 ) ) ) ) ) ) ) ).
% semilattice_set.hom_commute
thf(fact_7805_semilattice__set_Osubset,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A,B6: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
=> ( ( F2 @ ( lattic1715443433743089157tice_F @ A @ F2 @ B6 ) @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) )
= ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) ) ) ) ) ) ).
% semilattice_set.subset
thf(fact_7806_semilattice__set_Oinsert__not__elem,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite @ A @ A4 )
=> ( ~ ( member @ A @ X3 @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( insert @ A @ X3 @ A4 ) )
= ( F2 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_7807_semilattice__set_Oinsert,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( insert @ A @ X3 @ A4 ) )
= ( F2 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) ) ) ) ) ) ).
% semilattice_set.insert
thf(fact_7808_semilattice__set_Oclosed,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y4: A] : ( member @ A @ ( F2 @ X4 @ Y4 ) @ ( insert @ A @ X4 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) @ A4 ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_7809_semilattice__set_Ounion,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A,B6: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite @ A @ B6 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
= ( F2 @ ( lattic1715443433743089157tice_F @ A @ F2 @ A4 ) @ ( lattic1715443433743089157tice_F @ A @ F2 @ B6 ) ) ) ) ) ) ) ) ).
% semilattice_set.union
thf(fact_7810_semilattice__set_Oinfinite,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ~ ( finite_finite @ A @ A4 )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% semilattice_set.infinite
thf(fact_7811_semilattice__set_Oeq__fold,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( insert @ A @ X3 @ A4 ) )
= ( finite_fold @ A @ A @ F2 @ X3 @ A4 ) ) ) ) ).
% semilattice_set.eq_fold
thf(fact_7812_semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( insert @ A @ X3 @ A4 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( insert @ A @ X3 @ A4 ) )
= ( F2 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_7813_not__in__connected__cases,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S2: set @ A,X3: A] :
( ( topolo1966860045006549960nected @ A @ S2 )
=> ( ~ ( member @ A @ X3 @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( condit941137186595557371_above @ A @ S2 )
=> ~ ! [Y5: A] :
( ( member @ A @ Y5 @ S2 )
=> ( ord_less_eq @ A @ Y5 @ X3 ) ) )
=> ~ ( ( condit1013018076250108175_below @ A @ S2 )
=> ~ ! [Y5: A] :
( ( member @ A @ Y5 @ S2 )
=> ( ord_less_eq @ A @ X3 @ Y5 ) ) ) ) ) ) ) ) ).
% not_in_connected_cases
thf(fact_7814_cr__int__def,axiom,
( cr_int
= ( ^ [X: product_prod @ nat @ nat] :
( ^ [Y6: int,Z2: int] : Y6 = Z2
@ ( abs_Integ @ X ) ) ) ) ).
% cr_int_def
thf(fact_7815_connected__iff__interval,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ( ( topolo1966860045006549960nected @ A )
= ( ^ [U5: set @ A] :
! [X: A] :
( ( member @ A @ X @ U5 )
=> ! [Y: A] :
( ( member @ A @ Y @ U5 )
=> ! [Z6: A] :
( ( ord_less_eq @ A @ X @ Z6 )
=> ( ( ord_less_eq @ A @ Z6 @ Y )
=> ( member @ A @ Z6 @ U5 ) ) ) ) ) ) ) ) ).
% connected_iff_interval
thf(fact_7816_connectedI__interval,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [U3: set @ A] :
( ! [X4: A,Y4: A,Z3: A] :
( ( member @ A @ X4 @ U3 )
=> ( ( member @ A @ Y4 @ U3 )
=> ( ( ord_less_eq @ A @ X4 @ Z3 )
=> ( ( ord_less_eq @ A @ Z3 @ Y4 )
=> ( member @ A @ Z3 @ U3 ) ) ) ) )
=> ( topolo1966860045006549960nected @ A @ U3 ) ) ) ).
% connectedI_interval
thf(fact_7817_connectedD__interval,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [U3: set @ A,X3: A,Y3: A,Z: A] :
( ( topolo1966860045006549960nected @ A @ U3 )
=> ( ( member @ A @ X3 @ U3 )
=> ( ( member @ A @ Y3 @ U3 )
=> ( ( ord_less_eq @ A @ X3 @ Z )
=> ( ( ord_less_eq @ A @ Z @ Y3 )
=> ( member @ A @ Z @ U3 ) ) ) ) ) ) ) ).
% connectedD_interval
thf(fact_7818_int_Opcr__cr__eq,axiom,
pcr_int = cr_int ).
% int.pcr_cr_eq
thf(fact_7819_Quotient__int,axiom,
quotient @ ( product_prod @ nat @ nat ) @ int @ intrel @ abs_Integ @ rep_Integ @ cr_int ).
% Quotient_int
thf(fact_7820_strict__sorted__equal__Uniq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( uniq @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
& ( ( set2 @ A @ Xs )
= A4 ) ) ) ) ).
% strict_sorted_equal_Uniq
thf(fact_7821_gfp__Kleene__iter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ( compow @ ( A > A ) @ ( suc @ K ) @ F2 @ ( top_top @ A ) )
= ( compow @ ( A > A ) @ K @ F2 @ ( top_top @ A ) ) )
=> ( ( complete_lattice_gfp @ A @ F2 )
= ( compow @ ( A > A ) @ K @ F2 @ ( top_top @ A ) ) ) ) ) ) ).
% gfp_Kleene_iter
thf(fact_7822_list__ex__length,axiom,
! [A: $tType] :
( ( list_ex @ A )
= ( ^ [P3: A > $o,Xs: list @ A] :
? [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P3 @ ( nth @ A @ Xs @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_7823_gfp__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F4: A > A,X3: A] :
( ( order_mono @ A @ A @ F4 )
=> ( ( ( F4 @ X3 )
= X3 )
=> ( ! [Z3: A] :
( ( ( F4 @ Z3 )
= Z3 )
=> ( ord_less_eq @ A @ Z3 @ X3 ) )
=> ( ( complete_lattice_gfp @ A @ F4 )
= X3 ) ) ) ) ) ).
% gfp_eqI
thf(fact_7824_gfp__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_lattice_gfp @ A )
= ( ^ [F3: A > A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [U2: A] : ( ord_less_eq @ A @ U2 @ ( F3 @ U2 ) ) ) ) ) ) ) ).
% gfp_def
thf(fact_7825_gfp__gfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A > A] :
( ! [X4: A,Y4: A,W2: A,Z3: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ( ord_less_eq @ A @ W2 @ Z3 )
=> ( ord_less_eq @ A @ ( F2 @ X4 @ W2 ) @ ( F2 @ Y4 @ Z3 ) ) ) )
=> ( ( complete_lattice_gfp @ A
@ ^ [X: A] : ( complete_lattice_gfp @ A @ ( F2 @ X ) ) )
= ( complete_lattice_gfp @ A
@ ^ [X: A] : ( F2 @ X @ X ) ) ) ) ) ).
% gfp_gfp
thf(fact_7826_gfp__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,G: A > A] :
( ! [Z9: A] : ( ord_less_eq @ A @ ( F2 @ Z9 ) @ ( G @ Z9 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F2 ) @ ( complete_lattice_gfp @ A @ G ) ) ) ) ).
% gfp_mono
thf(fact_7827_gfp__upperbound,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X8: A,F2: A > A] :
( ( ord_less_eq @ A @ X8 @ ( F2 @ X8 ) )
=> ( ord_less_eq @ A @ X8 @ ( complete_lattice_gfp @ A @ F2 ) ) ) ) ).
% gfp_upperbound
thf(fact_7828_gfp__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,X8: A] :
( ! [U4: A] :
( ( ord_less_eq @ A @ U4 @ ( F2 @ U4 ) )
=> ( ord_less_eq @ A @ U4 @ X8 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F2 ) @ X8 ) ) ) ).
% gfp_least
thf(fact_7829_coinduct__lemma,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X8: A,F2: A > A] :
( ( ord_less_eq @ A @ X8 @ ( F2 @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F2 ) ) ) )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F2 ) ) @ ( F2 @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F2 ) ) ) ) ) ) ) ).
% coinduct_lemma
thf(fact_7830_def__coinduct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: A,F2: A > A,X8: A] :
( ( A4
= ( complete_lattice_gfp @ A @ F2 ) )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ X8 @ ( F2 @ ( sup_sup @ A @ X8 @ A4 ) ) )
=> ( ord_less_eq @ A @ X8 @ A4 ) ) ) ) ) ).
% def_coinduct
thf(fact_7831_coinduct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,X8: A] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ X8 @ ( F2 @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F2 ) ) ) )
=> ( ord_less_eq @ A @ X8 @ ( complete_lattice_gfp @ A @ F2 ) ) ) ) ) ).
% coinduct
thf(fact_7832_gfp__ordinal__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,P: A > $o] :
( ( order_mono @ A @ A @ F2 )
=> ( ! [S5: A] :
( ( P @ S5 )
=> ( ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F2 ) @ S5 )
=> ( P @ ( F2 @ S5 ) ) ) )
=> ( ! [M8: set @ A] :
( ! [X6: A] :
( ( member @ A @ X6 @ M8 )
=> ( P @ X6 ) )
=> ( P @ ( complete_Inf_Inf @ A @ M8 ) ) )
=> ( P @ ( complete_lattice_gfp @ A @ F2 ) ) ) ) ) ) ).
% gfp_ordinal_induct
thf(fact_7833_gfp__funpow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,N: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( complete_lattice_gfp @ A @ ( compow @ ( A > A ) @ ( suc @ N ) @ F2 ) )
= ( complete_lattice_gfp @ A @ F2 ) ) ) ) ).
% gfp_funpow
thf(fact_7834_lfp__le__gfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A] :
( ( order_mono @ A @ A @ F2 )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F2 ) @ ( complete_lattice_gfp @ A @ F2 ) ) ) ) ).
% lfp_le_gfp
thf(fact_7835_gfp__transfer__bounded,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [P: A > $o,F2: A > A,Alpha: A > B,G: B > B] :
( ( P @ ( F2 @ ( top_top @ A ) ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( P @ ( F2 @ X4 ) ) )
=> ( ! [M8: nat > A] :
( ( order_antimono @ nat @ A @ M8 )
=> ( ! [I2: nat] : ( P @ ( M8 @ I2 ) )
=> ( P @ ( complete_Inf_Inf @ A @ ( image @ nat @ A @ M8 @ ( top_top @ ( set @ nat ) ) ) ) ) ) )
=> ( ! [M8: nat > A] :
( ( order_antimono @ nat @ A @ M8 )
=> ( ! [I2: nat] : ( P @ ( M8 @ I2 ) )
=> ( ( Alpha @ ( complete_Inf_Inf @ A @ ( image @ nat @ A @ M8 @ ( top_top @ ( set @ nat ) ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image @ nat @ B
@ ^ [I4: nat] : ( Alpha @ ( M8 @ I4 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) )
=> ( ( order_inf_continuous @ A @ A @ F2 )
=> ( ( order_inf_continuous @ B @ B @ G )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( ( Alpha @ ( F2 @ X4 ) )
= ( G @ ( Alpha @ X4 ) ) ) )
=> ( ! [X4: B] : ( ord_less_eq @ B @ ( G @ X4 ) @ ( Alpha @ ( F2 @ ( top_top @ A ) ) ) )
=> ( ( Alpha @ ( complete_lattice_gfp @ A @ F2 ) )
= ( complete_lattice_gfp @ B @ G ) ) ) ) ) ) ) ) ) ) ) ).
% gfp_transfer_bounded
thf(fact_7836_and_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( semilattice_neutr @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% and.semilattice_neutr_axioms
thf(fact_7837_gcd__nat_Osemilattice__neutr__axioms,axiom,
semilattice_neutr @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% gcd_nat.semilattice_neutr_axioms
thf(fact_7838_or_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( semilattice_neutr @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% or.semilattice_neutr_axioms
thf(fact_7839_max__nat_Osemilattice__neutr__axioms,axiom,
semilattice_neutr @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% max_nat.semilattice_neutr_axioms
thf(fact_7840_cclfp__lowerbound,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F2: A > A,A4: A] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ ( F2 @ A4 ) @ A4 )
=> ( ord_less_eq @ A @ ( order_532582986084564980_cclfp @ A @ F2 ) @ A4 ) ) ) ) ).
% cclfp_lowerbound
thf(fact_7841_sum__list__def,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( groups_monoid_F @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_def
thf(fact_7842_construct__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ( ( real_V4425403222259421789struct @ A @ B )
= ( ^ [B9: set @ A,G2: A > B,V5: A] :
( groups7311177749621191930dd_sum @ A @ B
@ ^ [B4: A] : ( real_V8093663219630862766scaleR @ B @ ( real_V7696804695334737415tation @ A @ ( real_V4986007116245087402_basis @ A @ B9 ) @ V5 @ B4 ) @ ( if @ B @ ( member @ A @ B4 @ B9 ) @ ( G2 @ B4 ) @ ( zero_zero @ B ) ) )
@ ( collect @ A
@ ^ [B4: A] :
( ( real_V7696804695334737415tation @ A @ ( real_V4986007116245087402_basis @ A @ B9 ) @ V5 @ B4 )
!= ( zero_zero @ real ) ) ) ) ) ) ) ).
% construct_def
thf(fact_7843_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ) ).
% natLess_def
thf(fact_7844_construct__outside,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [B6: set @ A,V2: A,F2: A > B] :
( ~ ( real_V358717886546972837endent @ A @ B6 )
=> ( ( member @ A @ V2 @ ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ ( real_V4986007116245087402_basis @ A @ B6 ) @ B6 ) ) )
=> ( ( real_V4425403222259421789struct @ A @ B @ B6 @ F2 @ V2 )
= ( zero_zero @ B ) ) ) ) ) ).
% construct_outside
thf(fact_7845_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ A ),As2: A > B] :
! [I4: A,J3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I4 @ J3 ) @ R5 )
=> ( ord_less_eq @ B @ ( As2 @ I4 ) @ ( As2 @ J3 ) ) ) ) ) ) ).
% relChain_def
thf(fact_7846_Restr__natLeq,axiom,
! [N: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat
@ ( collect @ nat
@ ^ [X: nat] : ( ord_less @ nat @ X @ N ) )
@ ^ [Uu3: nat] :
( collect @ nat
@ ^ [X: nat] : ( ord_less @ nat @ X @ N ) ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X: nat,Y: nat] :
( ( ord_less @ nat @ X @ N )
& ( ord_less @ nat @ Y @ N )
& ( ord_less_eq @ nat @ X @ Y ) ) ) ) ) ).
% Restr_natLeq
thf(fact_7847_natLeq__def,axiom,
( bNF_Ca8665028551170535155natLeq
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less_eq @ nat ) ) ) ) ).
% natLeq_def
thf(fact_7848_Restr__natLeq2,axiom,
! [N: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat @ ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N )
@ ^ [Uu3: nat] : ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X: nat,Y: nat] :
( ( ord_less @ nat @ X @ N )
& ( ord_less @ nat @ Y @ N )
& ( ord_less_eq @ nat @ X @ Y ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_7849_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( folding_insort_key @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ ( set @ A ) )
@ ^ [X: A] : X ) ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_7850_natLeq__underS__less,axiom,
! [N: nat] :
( ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N )
= ( collect @ nat
@ ^ [X: nat] : ( ord_less @ nat @ X @ N ) ) ) ).
% natLeq_underS_less
thf(fact_7851_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F2: ( A > B ) > C,G: C] :
( ( F2
= ( ^ [X: A > B] : G ) )
=> ( ( F2
@ ^ [X: A] : ( zero_zero @ B ) )
= G ) ) ) ).
% fun_cong_unused_0
thf(fact_7852_add_Ogroup__axioms,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( group @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A ) ) ) ).
% add.group_axioms
thf(fact_7853_group_Oinverse__distrib__swap,axiom,
! [A: $tType,F2: A > A > A,Z: A,Inverse: A > A,A2: A,B2: A] :
( ( group @ A @ F2 @ Z @ Inverse )
=> ( ( Inverse @ ( F2 @ A2 @ B2 ) )
= ( F2 @ ( Inverse @ B2 ) @ ( Inverse @ A2 ) ) ) ) ).
% group.inverse_distrib_swap
thf(fact_7854_group_Ogroup__left__neutral,axiom,
! [A: $tType,F2: A > A > A,Z: A,Inverse: A > A,A2: A] :
( ( group @ A @ F2 @ Z @ Inverse )
=> ( ( F2 @ Z @ A2 )
= A2 ) ) ).
% group.group_left_neutral
thf(fact_7855_group_Oinverse__neutral,axiom,
! [A: $tType,F2: A > A > A,Z: A,Inverse: A > A] :
( ( group @ A @ F2 @ Z @ Inverse )
=> ( ( Inverse @ Z )
= Z ) ) ).
% group.inverse_neutral
thf(fact_7856_group_Oinverse__inverse,axiom,
! [A: $tType,F2: A > A > A,Z: A,Inverse: A > A,A2: A] :
( ( group @ A @ F2 @ Z @ Inverse )
=> ( ( Inverse @ ( Inverse @ A2 ) )
= A2 ) ) ).
% group.inverse_inverse
thf(fact_7857_group_Oinverse__unique,axiom,
! [A: $tType,F2: A > A > A,Z: A,Inverse: A > A,A2: A,B2: A] :
( ( group @ A @ F2 @ Z @ Inverse )
=> ( ( ( F2 @ A2 @ B2 )
= Z )
=> ( ( Inverse @ A2 )
= B2 ) ) ) ).
% group.inverse_unique
thf(fact_7858_group_Oright__inverse,axiom,
! [A: $tType,F2: A > A > A,Z: A,Inverse: A > A,A2: A] :
( ( group @ A @ F2 @ Z @ Inverse )
=> ( ( F2 @ A2 @ ( Inverse @ A2 ) )
= Z ) ) ).
% group.right_inverse
thf(fact_7859_group_Oright__cancel,axiom,
! [A: $tType,F2: A > A > A,Z: A,Inverse: A > A,B2: A,A2: A,C2: A] :
( ( group @ A @ F2 @ Z @ Inverse )
=> ( ( ( F2 @ B2 @ A2 )
= ( F2 @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% group.right_cancel
thf(fact_7860_group_Oleft__inverse,axiom,
! [A: $tType,F2: A > A > A,Z: A,Inverse: A > A,A2: A] :
( ( group @ A @ F2 @ Z @ Inverse )
=> ( ( F2 @ ( Inverse @ A2 ) @ A2 )
= Z ) ) ).
% group.left_inverse
thf(fact_7861_group_Oleft__cancel,axiom,
! [A: $tType,F2: A > A > A,Z: A,Inverse: A > A,A2: A,B2: A,C2: A] :
( ( group @ A @ F2 @ Z @ Inverse )
=> ( ( ( F2 @ A2 @ B2 )
= ( F2 @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% group.left_cancel
thf(fact_7862_map__add__map__of__foldr,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),Ps: list @ ( product_prod @ A @ B )] :
( ( map_add @ A @ B @ M @ ( map_of @ A @ B @ Ps ) )
= ( foldr @ ( product_prod @ A @ B ) @ ( A > ( option @ B ) )
@ ( product_case_prod @ A @ B @ ( ( A > ( option @ B ) ) > A > ( option @ B ) )
@ ^ [K3: A,V5: B,M3: A > ( option @ B )] : ( fun_upd @ A @ ( option @ B ) @ M3 @ K3 @ ( some @ B @ V5 ) ) )
@ Ps
@ M ) ) ).
% map_add_map_of_foldr
thf(fact_7863_iteratesp_Omono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F2: A > A] :
( order_mono @ ( A > $o ) @ ( A > $o )
@ ^ [P5: A > $o,X: A] :
( ? [Y: A] :
( ( X
= ( F2 @ Y ) )
& ( P5 @ Y ) )
| ? [M9: set @ A] :
( ( X
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [Y: A] :
( ( member @ A @ Y @ M9 )
=> ( P5 @ Y ) ) ) ) ) ) ).
% iteratesp.mono
thf(fact_7864_map__add__find__right,axiom,
! [B: $tType,A: $tType,N: B > ( option @ A ),K: B,Xx: A,M: B > ( option @ A )] :
( ( ( N @ K )
= ( some @ A @ Xx ) )
=> ( ( map_add @ B @ A @ M @ N @ K )
= ( some @ A @ Xx ) ) ) ).
% map_add_find_right
thf(fact_7865_map__add__upd,axiom,
! [A: $tType,B: $tType,F2: A > ( option @ B ),G: A > ( option @ B ),X3: A,Y3: B] :
( ( map_add @ A @ B @ F2 @ ( fun_upd @ A @ ( option @ B ) @ G @ X3 @ ( some @ B @ Y3 ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_add @ A @ B @ F2 @ G ) @ X3 @ ( some @ B @ Y3 ) ) ) ).
% map_add_upd
thf(fact_7866_map__add__Some__iff,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),N: B > ( option @ A ),K: B,X3: A] :
( ( ( map_add @ B @ A @ M @ N @ K )
= ( some @ A @ X3 ) )
= ( ( ( N @ K )
= ( some @ A @ X3 ) )
| ( ( ( N @ K )
= ( none @ A ) )
& ( ( M @ K )
= ( some @ A @ X3 ) ) ) ) ) ).
% map_add_Some_iff
thf(fact_7867_map__add__SomeD,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),N: B > ( option @ A ),K: B,X3: A] :
( ( ( map_add @ B @ A @ M @ N @ K )
= ( some @ A @ X3 ) )
=> ( ( ( N @ K )
= ( some @ A @ X3 ) )
| ( ( ( N @ K )
= ( none @ A ) )
& ( ( M @ K )
= ( some @ A @ X3 ) ) ) ) ) ).
% map_add_SomeD
thf(fact_7868_map__add__def,axiom,
! [B: $tType,A: $tType] :
( ( map_add @ A @ B )
= ( ^ [M1: A > ( option @ B ),M22: A > ( option @ B ),X: A] : ( case_option @ ( option @ B ) @ B @ ( M1 @ X ) @ ( some @ B ) @ ( M22 @ X ) ) ) ) ).
% map_add_def
thf(fact_7869_ccpo__Sup__upper,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A4: set @ A,X3: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% ccpo_Sup_upper
thf(fact_7870_ccpo__Sup__least,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A4: set @ A,Z: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A4 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ Z ) ) ) ) ).
% ccpo_Sup_least
thf(fact_7871_chain__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X3: A] : ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% chain_singleton
thf(fact_7872_map__add__upd__left,axiom,
! [A: $tType,B: $tType,M: A,E22: A > ( option @ B ),E1: A > ( option @ B ),U1: B] :
( ~ ( member @ A @ M @ ( dom @ A @ B @ E22 ) )
=> ( ( map_add @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ E1 @ M @ ( some @ B @ U1 ) ) @ E22 )
= ( fun_upd @ A @ ( option @ B ) @ ( map_add @ A @ B @ E1 @ E22 ) @ M @ ( some @ B @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_7873_in__chain__finite,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A4: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A4 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A4 ) @ A4 ) ) ) ) ) ).
% in_chain_finite
thf(fact_7874_iteratesp__def,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F3: A > A] :
( complete_lattice_lfp @ ( A > $o )
@ ^ [P5: A > $o,X: A] :
( ? [Y: A] :
( ( X
= ( F3 @ Y ) )
& ( P5 @ Y ) )
| ? [M9: set @ A] :
( ( X
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [Y: A] :
( ( member @ A @ Y @ M9 )
=> ( P5 @ Y ) ) ) ) ) ) ) ) ).
% iteratesp_def
thf(fact_7875_iteratesp_OSup,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [M7: set @ A,F2: A > A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M7 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ M7 )
=> ( comple7512665784863727008ratesp @ A @ F2 @ X4 ) )
=> ( comple7512665784863727008ratesp @ A @ F2 @ ( complete_Sup_Sup @ A @ M7 ) ) ) ) ) ).
% iteratesp.Sup
thf(fact_7876_iteratesp_Ocases,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F2: A > A,A2: A] :
( ( comple7512665784863727008ratesp @ A @ F2 @ A2 )
=> ( ! [X4: A] :
( ( A2
= ( F2 @ X4 ) )
=> ~ ( comple7512665784863727008ratesp @ A @ F2 @ X4 ) )
=> ~ ! [M8: set @ A] :
( ( A2
= ( complete_Sup_Sup @ A @ M8 ) )
=> ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M8 )
=> ~ ! [X6: A] :
( ( member @ A @ X6 @ M8 )
=> ( comple7512665784863727008ratesp @ A @ F2 @ X6 ) ) ) ) ) ) ) ).
% iteratesp.cases
thf(fact_7877_iteratesp_Osimps,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F3: A > A,A5: A] :
( ? [X: A] :
( ( A5
= ( F3 @ X ) )
& ( comple7512665784863727008ratesp @ A @ F3 @ X ) )
| ? [M9: set @ A] :
( ( A5
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [X: A] :
( ( member @ A @ X @ M9 )
=> ( comple7512665784863727008ratesp @ A @ F3 @ X ) ) ) ) ) ) ) ).
% iteratesp.simps
thf(fact_7878_admissible__chfin,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P: A > $o] :
( ! [S5: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ S5 )
=> ( finite_finite @ A @ S5 ) )
=> ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P ) ) ) ).
% admissible_chfin
thf(fact_7879_and_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( monoid @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% and.monoid_axioms
thf(fact_7880_or_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( monoid @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% or.monoid_axioms
thf(fact_7881_gcd__nat_Omonoid__axioms,axiom,
monoid @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% gcd_nat.monoid_axioms
thf(fact_7882_add_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% add.monoid_axioms
thf(fact_7883_max__nat_Omonoid__axioms,axiom,
monoid @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% max_nat.monoid_axioms
thf(fact_7884_xor_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( monoid @ A @ ( bit_se5824344971392196577ns_xor @ A ) @ ( zero_zero @ A ) ) ) ).
% xor.monoid_axioms
thf(fact_7885_mult_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% mult.monoid_axioms
thf(fact_7886_monoid_Oleft__neutral,axiom,
! [A: $tType,F2: A > A > A,Z: A,A2: A] :
( ( monoid @ A @ F2 @ Z )
=> ( ( F2 @ Z @ A2 )
= A2 ) ) ).
% monoid.left_neutral
thf(fact_7887_monoid_Oright__neutral,axiom,
! [A: $tType,F2: A > A > A,Z: A,A2: A] :
( ( monoid @ A @ F2 @ Z )
=> ( ( F2 @ A2 @ Z )
= A2 ) ) ).
% monoid.right_neutral
thf(fact_7888_admissible__disj,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P: A > $o,Q: A > $o] :
( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P )
=> ( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ Q )
=> ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A )
@ ^ [X: A] :
( ( P @ X )
| ( Q @ X ) ) ) ) ) ) ).
% admissible_disj
thf(fact_7889_VEBT__internal_Ooption__shift_Opelims,axiom,
! [A: $tType,X3: A > A > A,Xa: option @ A,Xb: option @ A,Y3: option @ A] :
( ( ( vEBT_V2048590022279873568_shift @ A @ X3 @ Xa @ Xb )
= Y3 )
=> ( ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X3 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( none @ A ) )
=> ( ( Y3
= ( none @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X3 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Xb ) ) ) ) )
=> ( ! [V4: A] :
( ( Xa
= ( some @ A @ V4 ) )
=> ( ( Xb
= ( none @ A ) )
=> ( ( Y3
= ( none @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X3 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V4 ) @ ( none @ A ) ) ) ) ) ) )
=> ~ ! [A6: A] :
( ( Xa
= ( some @ A @ A6 ) )
=> ! [B5: A] :
( ( Xb
= ( some @ A @ B5 ) )
=> ( ( Y3
= ( some @ A @ ( X3 @ A6 @ B5 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X3 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ A6 ) @ ( some @ A @ B5 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.pelims
thf(fact_7890_arg__min__list_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,X3: A,Y3: A,Zs2: list @ A] :
( ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs2 ) ) )
= ( if @ A @ ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_7891_arg__min__list_Oelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X3: A > B,Xa: list @ A,Y3: A] :
( ( ( arg_min_list @ A @ B @ X3 @ Xa )
= Y3 )
=> ( ! [X4: A] :
( ( Xa
= ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( Y3 != X4 ) )
=> ( ! [X4: A,Y4: A,Zs: list @ A] :
( ( Xa
= ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Zs ) ) )
=> ( Y3
!= ( if @ A @ ( ord_less_eq @ B @ ( X3 @ X4 ) @ ( X3 @ ( arg_min_list @ A @ B @ X3 @ ( cons @ A @ Y4 @ Zs ) ) ) ) @ X4 @ ( arg_min_list @ A @ B @ X3 @ ( cons @ A @ Y4 @ Zs ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( Y3
!= ( undefined @ A ) ) ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_7892_infinity__enat__def,axiom,
( ( extend4730790105801354508finity @ extended_enat )
= ( extended_Abs_enat @ ( none @ nat ) ) ) ).
% infinity_enat_def
thf(fact_7893_option_Othe__def,axiom,
! [A: $tType] :
( ( the2 @ A )
= ( case_option @ A @ A @ ( undefined @ A )
@ ^ [X23: A] : X23 ) ) ).
% option.the_def
thf(fact_7894_Abs__enat__cases,axiom,
! [X3: extended_enat] :
~ ! [Y4: option @ nat] :
( ( X3
= ( extended_Abs_enat @ Y4 ) )
=> ~ ( member @ ( option @ nat ) @ Y4 @ ( top_top @ ( set @ ( option @ nat ) ) ) ) ) ).
% Abs_enat_cases
thf(fact_7895_Abs__enat__induct,axiom,
! [P: extended_enat > $o,X3: extended_enat] :
( ! [Y4: option @ nat] :
( ( member @ ( option @ nat ) @ Y4 @ ( top_top @ ( set @ ( option @ nat ) ) ) )
=> ( P @ ( extended_Abs_enat @ Y4 ) ) )
=> ( P @ X3 ) ) ).
% Abs_enat_induct
thf(fact_7896_Abs__enat__inject,axiom,
! [X3: option @ nat,Y3: option @ nat] :
( ( member @ ( option @ nat ) @ X3 @ ( top_top @ ( set @ ( option @ nat ) ) ) )
=> ( ( member @ ( option @ nat ) @ Y3 @ ( top_top @ ( set @ ( option @ nat ) ) ) )
=> ( ( ( extended_Abs_enat @ X3 )
= ( extended_Abs_enat @ Y3 ) )
= ( X3 = Y3 ) ) ) ) ).
% Abs_enat_inject
thf(fact_7897_enat__def,axiom,
( extended_enat2
= ( ^ [N3: nat] : ( extended_Abs_enat @ ( some @ nat @ N3 ) ) ) ) ).
% enat_def
thf(fact_7898_arg__min__list_Opelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X3: A > B,Xa: list @ A,Y3: A] :
( ( ( arg_min_list @ A @ B @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X3 @ Xa ) )
=> ( ! [X4: A] :
( ( Xa
= ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ( Y3 = X4 )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X3 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) )
=> ( ! [X4: A,Y4: A,Zs: list @ A] :
( ( Xa
= ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Zs ) ) )
=> ( ( Y3
= ( if @ A @ ( ord_less_eq @ B @ ( X3 @ X4 ) @ ( X3 @ ( arg_min_list @ A @ B @ X3 @ ( cons @ A @ Y4 @ Zs ) ) ) ) @ X4 @ ( arg_min_list @ A @ B @ X3 @ ( cons @ A @ Y4 @ Zs ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X3 @ ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Zs ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ( Y3
= ( undefined @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X3 @ ( nil @ A ) ) ) ) ) ) ) ) ) ) ).
% arg_min_list.pelims
thf(fact_7899_iterates_OSup,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [M7: set @ A,F2: A > A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M7 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ M7 )
=> ( member @ A @ X4 @ ( comple6359979572994053840erates @ A @ F2 ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ M7 ) @ ( comple6359979572994053840erates @ A @ F2 ) ) ) ) ) ).
% iterates.Sup
thf(fact_7900_iterates_Ocases,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A2: A,F2: A > A] :
( ( member @ A @ A2 @ ( comple6359979572994053840erates @ A @ F2 ) )
=> ( ! [X4: A] :
( ( A2
= ( F2 @ X4 ) )
=> ~ ( member @ A @ X4 @ ( comple6359979572994053840erates @ A @ F2 ) ) )
=> ~ ! [M8: set @ A] :
( ( A2
= ( complete_Sup_Sup @ A @ M8 ) )
=> ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M8 )
=> ~ ! [X6: A] :
( ( member @ A @ X6 @ M8 )
=> ( member @ A @ X6 @ ( comple6359979572994053840erates @ A @ F2 ) ) ) ) ) ) ) ) ).
% iterates.cases
thf(fact_7901_iterates_Osimps,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A2: A,F2: A > A] :
( ( member @ A @ A2 @ ( comple6359979572994053840erates @ A @ F2 ) )
= ( ? [X: A] :
( ( A2
= ( F2 @ X ) )
& ( member @ A @ X @ ( comple6359979572994053840erates @ A @ F2 ) ) )
| ? [M9: set @ A] :
( ( A2
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [X: A] :
( ( member @ A @ X @ M9 )
=> ( member @ A @ X @ ( comple6359979572994053840erates @ A @ F2 ) ) ) ) ) ) ) ).
% iterates.simps
thf(fact_7902_Abs__enat__inverse,axiom,
! [Y3: option @ nat] :
( ( member @ ( option @ nat ) @ Y3 @ ( top_top @ ( set @ ( option @ nat ) ) ) )
=> ( ( extended_Rep_enat @ ( extended_Abs_enat @ Y3 ) )
= Y3 ) ) ).
% Abs_enat_inverse
thf(fact_7903_chain__iterates,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F2: A > A] :
( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F2 )
=> ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( comple6359979572994053840erates @ A @ F2 ) ) ) ) ).
% chain_iterates
thf(fact_7904_iterates__le__f,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X3: A,F2: A > A] :
( ( member @ A @ X3 @ ( comple6359979572994053840erates @ A @ F2 ) )
=> ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F2 )
=> ( ord_less_eq @ A @ X3 @ ( F2 @ X3 ) ) ) ) ) ).
% iterates_le_f
thf(fact_7905_Rep__enat__induct,axiom,
! [Y3: option @ nat,P: ( option @ nat ) > $o] :
( ( member @ ( option @ nat ) @ Y3 @ ( top_top @ ( set @ ( option @ nat ) ) ) )
=> ( ! [X4: extended_enat] : ( P @ ( extended_Rep_enat @ X4 ) )
=> ( P @ Y3 ) ) ) ).
% Rep_enat_induct
thf(fact_7906_Rep__enat__cases,axiom,
! [Y3: option @ nat] :
( ( member @ ( option @ nat ) @ Y3 @ ( top_top @ ( set @ ( option @ nat ) ) ) )
=> ~ ! [X4: extended_enat] :
( Y3
!= ( extended_Rep_enat @ X4 ) ) ) ).
% Rep_enat_cases
thf(fact_7907_Rep__enat,axiom,
! [X3: extended_enat] : ( member @ ( option @ nat ) @ ( extended_Rep_enat @ X3 ) @ ( top_top @ ( set @ ( option @ nat ) ) ) ) ).
% Rep_enat
thf(fact_7908_Rep__enat__inject,axiom,
! [X3: extended_enat,Y3: extended_enat] :
( ( ( extended_Rep_enat @ X3 )
= ( extended_Rep_enat @ Y3 ) )
= ( X3 = Y3 ) ) ).
% Rep_enat_inject
thf(fact_7909_Rep__enat__inverse,axiom,
! [X3: extended_enat] :
( ( extended_Abs_enat @ ( extended_Rep_enat @ X3 ) )
= X3 ) ).
% Rep_enat_inverse
thf(fact_7910_fixp__induct,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P: A > $o,F2: A > A] :
( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P )
=> ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F2 )
=> ( ( P @ ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( P @ ( F2 @ X4 ) ) )
=> ( P @ ( comple115746919287870866o_fixp @ A @ F2 ) ) ) ) ) ) ) ).
% fixp_induct
thf(fact_7911_iterates__fixp,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F2: A > A] :
( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F2 )
=> ( member @ A @ ( comple115746919287870866o_fixp @ A @ F2 ) @ ( comple6359979572994053840erates @ A @ F2 ) ) ) ) ).
% iterates_fixp
thf(fact_7912_fixp__lowerbound,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F2: A > A,Z: A] :
( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F2 )
=> ( ( ord_less_eq @ A @ ( F2 @ Z ) @ Z )
=> ( ord_less_eq @ A @ ( comple115746919287870866o_fixp @ A @ F2 ) @ Z ) ) ) ) ).
% fixp_lowerbound
thf(fact_7913_fixp__unfold,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F2: A > A] :
( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F2 )
=> ( ( comple115746919287870866o_fixp @ A @ F2 )
= ( F2 @ ( comple115746919287870866o_fixp @ A @ F2 ) ) ) ) ) ).
% fixp_unfold
thf(fact_7914_type__definition__enat,axiom,
type_definition @ extended_enat @ ( option @ nat ) @ extended_Rep_enat @ extended_Abs_enat @ ( top_top @ ( set @ ( option @ nat ) ) ) ).
% type_definition_enat
thf(fact_7915_sorted__sort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs2: list @ B] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ ( linorder_sort_key @ B @ A @ F2 @ Xs2 ) ) ) ) ).
% sorted_sort_key
thf(fact_7916_sorted__sort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_sort_key @ A @ A
@ ^ [X: A] : X
@ Xs2 ) ) ) ).
% sorted_sort
thf(fact_7917_sorted__sort__id,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X: A] : X
@ Xs2 )
= Xs2 ) ) ) ).
% sorted_sort_id
thf(fact_7918_list__encode_Oelims,axiom,
! [X3: list @ nat,Y3: nat] :
( ( ( nat_list_encode @ X3 )
= Y3 )
=> ( ( ( X3
= ( nil @ nat ) )
=> ( Y3
!= ( zero_zero @ nat ) ) )
=> ~ ! [X4: nat,Xs3: list @ nat] :
( ( X3
= ( cons @ nat @ X4 @ Xs3 ) )
=> ( Y3
!= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X4 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_7919_continuous__attains__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ? [X4: A] :
( ( member @ A @ X4 @ S )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ S )
=> ( ord_less_eq @ B @ ( F2 @ Xa2 ) @ ( F2 @ X4 ) ) ) ) ) ) ) ) ).
% continuous_attains_sup
thf(fact_7920_compact__attains__inf,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S2: set @ A] :
( ( topolo2193935891317330818ompact @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ S2 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ S2 )
=> ( ord_less_eq @ A @ X4 @ Xa2 ) ) ) ) ) ) ).
% compact_attains_inf
thf(fact_7921_compact__attains__sup,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S2: set @ A] :
( ( topolo2193935891317330818ompact @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ S2 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ S2 )
=> ( ord_less_eq @ A @ Xa2 @ X4 ) ) ) ) ) ) ).
% compact_attains_sup
thf(fact_7922_le__prod__encode__1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq @ nat @ A2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A2 @ B2 ) ) ) ).
% le_prod_encode_1
thf(fact_7923_le__prod__encode__2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq @ nat @ B2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A2 @ B2 ) ) ) ).
% le_prod_encode_2
thf(fact_7924_list__encode_Osimps_I1_J,axiom,
( ( nat_list_encode @ ( nil @ nat ) )
= ( zero_zero @ nat ) ) ).
% list_encode.simps(1)
thf(fact_7925_continuous__attains__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ? [X4: A] :
( ( member @ A @ X4 @ S )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ S )
=> ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( F2 @ Xa2 ) ) ) ) ) ) ) ) ).
% continuous_attains_inf
thf(fact_7926_list__encode_Opelims,axiom,
! [X3: list @ nat,Y3: nat] :
( ( ( nat_list_encode @ X3 )
= Y3 )
=> ( ( accp @ ( list @ nat ) @ nat_list_encode_rel @ X3 )
=> ( ( ( X3
= ( nil @ nat ) )
=> ( ( Y3
= ( zero_zero @ nat ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( nil @ nat ) ) ) )
=> ~ ! [X4: nat,Xs3: list @ nat] :
( ( X3
= ( cons @ nat @ X4 @ Xs3 ) )
=> ( ( Y3
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X4 @ ( nat_list_encode @ Xs3 ) ) ) ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X4 @ Xs3 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_7927_map__conv__bind__option,axiom,
! [A: $tType,B: $tType] :
( ( map_option @ B @ A )
= ( ^ [F3: B > A,X: option @ B] : ( bind @ B @ A @ X @ ( comp @ A @ ( option @ A ) @ B @ ( some @ A ) @ F3 ) ) ) ) ).
% map_conv_bind_option
thf(fact_7928_bind__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,X3: option @ C,F2: C > ( option @ B ),G: B > ( option @ A )] :
( ( bind @ B @ A @ ( bind @ C @ B @ X3 @ F2 ) @ G )
= ( bind @ C @ A @ X3
@ ^ [Y: C] : ( bind @ B @ A @ ( F2 @ Y ) @ G ) ) ) ).
% bind_assoc
thf(fact_7929_bind__runit,axiom,
! [A: $tType,X3: option @ A] :
( ( bind @ A @ A @ X3 @ ( some @ A ) )
= X3 ) ).
% bind_runit
thf(fact_7930_bind__rzero,axiom,
! [B: $tType,A: $tType,X3: option @ B] :
( ( bind @ B @ A @ X3
@ ^ [X: B] : ( none @ A ) )
= ( none @ A ) ) ).
% bind_rzero
thf(fact_7931_bind__map__option,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > B,X3: option @ C,G: B > ( option @ A )] :
( ( bind @ B @ A @ ( map_option @ C @ B @ F2 @ X3 ) @ G )
= ( bind @ C @ A @ X3 @ ( comp @ B @ ( option @ A ) @ C @ G @ F2 ) ) ) ).
% bind_map_option
thf(fact_7932_bind__split__asm,axiom,
! [A: $tType,B: $tType,P: ( option @ A ) > $o,M: option @ B,F2: B > ( option @ A )] :
( ( P @ ( bind @ B @ A @ M @ F2 ) )
= ( ~ ( ( ( M
= ( none @ B ) )
& ~ ( P @ ( none @ A ) ) )
| ? [X: B] :
( ( M
= ( some @ B @ X ) )
& ~ ( P @ ( F2 @ X ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_7933_bind__split,axiom,
! [A: $tType,B: $tType,P: ( option @ A ) > $o,M: option @ B,F2: B > ( option @ A )] :
( ( P @ ( bind @ B @ A @ M @ F2 ) )
= ( ( ( M
= ( none @ B ) )
=> ( P @ ( none @ A ) ) )
& ! [V5: B] :
( ( M
= ( some @ B @ V5 ) )
=> ( P @ ( F2 @ V5 ) ) ) ) ) ).
% bind_split
thf(fact_7934_bind__option__cong__code,axiom,
! [B: $tType,A: $tType,X3: option @ A,Y3: option @ A,F2: A > ( option @ B )] :
( ( X3 = Y3 )
=> ( ( bind @ A @ B @ X3 @ F2 )
= ( bind @ A @ B @ Y3 @ F2 ) ) ) ).
% bind_option_cong_code
thf(fact_7935_bind__eq__Some__conv,axiom,
! [A: $tType,B: $tType,F2: option @ B,G: B > ( option @ A ),X3: A] :
( ( ( bind @ B @ A @ F2 @ G )
= ( some @ A @ X3 ) )
= ( ? [Y: B] :
( ( F2
= ( some @ B @ Y ) )
& ( ( G @ Y )
= ( some @ A @ X3 ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_7936_Option_Obind__cong,axiom,
! [B: $tType,A: $tType,X3: option @ A,Y3: option @ A,F2: A > ( option @ B ),G: A > ( option @ B )] :
( ( X3 = Y3 )
=> ( ! [A6: A] :
( ( Y3
= ( some @ A @ A6 ) )
=> ( ( F2 @ A6 )
= ( G @ A6 ) ) )
=> ( ( bind @ A @ B @ X3 @ F2 )
= ( bind @ A @ B @ Y3 @ G ) ) ) ) ).
% Option.bind_cong
thf(fact_7937_bind_Obind__lunit,axiom,
! [B: $tType,A: $tType,X3: A,F2: A > ( option @ B )] :
( ( bind @ A @ B @ ( some @ A @ X3 ) @ F2 )
= ( F2 @ X3 ) ) ).
% bind.bind_lunit
thf(fact_7938_bind_Obind__lzero,axiom,
! [A: $tType,B: $tType,F2: A > ( option @ B )] :
( ( bind @ A @ B @ ( none @ A ) @ F2 )
= ( none @ B ) ) ).
% bind.bind_lzero
thf(fact_7939_bind__eq__None__conv,axiom,
! [B: $tType,A: $tType,A2: option @ B,F2: B > ( option @ A )] :
( ( ( bind @ B @ A @ A2 @ F2 )
= ( none @ A ) )
= ( ( A2
= ( none @ B ) )
| ( ( F2 @ ( the2 @ B @ A2 ) )
= ( none @ A ) ) ) ) ).
% bind_eq_None_conv
thf(fact_7940_map__option__bind,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,X3: option @ C,G: C > ( option @ B )] :
( ( map_option @ B @ A @ F2 @ ( bind @ C @ B @ X3 @ G ) )
= ( bind @ C @ A @ X3 @ ( comp @ ( option @ B ) @ ( option @ A ) @ C @ ( map_option @ B @ A @ F2 ) @ G ) ) ) ).
% map_option_bind
thf(fact_7941_set__bind__option,axiom,
! [A: $tType,B: $tType,X3: option @ B,F2: B > ( option @ A )] :
( ( set_option @ A @ ( bind @ B @ A @ X3 @ F2 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ ( comp @ ( option @ A ) @ ( set @ A ) @ B @ ( set_option @ A ) @ F2 ) @ ( set_option @ B @ X3 ) ) ) ) ).
% set_bind_option
thf(fact_7942_mono__ccINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F2: A > B,I6: set @ C,A4: C > A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( countable_countable @ C @ I6 )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ A4 @ I6 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image @ C @ B
@ ^ [X: C] : ( F2 @ ( A4 @ X ) )
@ I6 ) ) ) ) ) ) ).
% mono_ccINF
thf(fact_7943_elem__set,axiom,
! [A: $tType,X3: A,Xo: option @ A] :
( ( member @ A @ X3 @ ( set_option @ A @ Xo ) )
= ( Xo
= ( some @ A @ X3 ) ) ) ).
% elem_set
thf(fact_7944_set__empty__eq,axiom,
! [A: $tType,Xo: option @ A] :
( ( ( set_option @ A @ Xo )
= ( bot_bot @ ( set @ A ) ) )
= ( Xo
= ( none @ A ) ) ) ).
% set_empty_eq
thf(fact_7945_bind__option__cong,axiom,
! [B: $tType,A: $tType,X3: option @ A,Y3: option @ A,F2: A > ( option @ B ),G: A > ( option @ B )] :
( ( X3 = Y3 )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set_option @ A @ Y3 ) )
=> ( ( F2 @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( bind @ A @ B @ X3 @ F2 )
= ( bind @ A @ B @ Y3 @ G ) ) ) ) ).
% bind_option_cong
thf(fact_7946_less__ccSup__iff,axiom,
! [A: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [S2: set @ A,A2: A] :
( ( countable_countable @ A @ S2 )
=> ( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ S2 ) )
= ( ? [X: A] :
( ( member @ A @ X @ S2 )
& ( ord_less @ A @ A2 @ X ) ) ) ) ) ) ).
% less_ccSup_iff
thf(fact_7947_option_Osimps_I14_J,axiom,
! [A: $tType] :
( ( set_option @ A @ ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% option.simps(14)
thf(fact_7948_ccInf__less__iff,axiom,
! [A: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [S2: set @ A,A2: A] :
( ( countable_countable @ A @ S2 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S2 ) @ A2 )
= ( ? [X: A] :
( ( member @ A @ X @ S2 )
& ( ord_less @ A @ X @ A2 ) ) ) ) ) ) ).
% ccInf_less_iff
thf(fact_7949_ccInf__superset__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ) ).
% ccInf_superset_mono
thf(fact_7950_ccInf__greatest,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,Z: A] :
( ( countable_countable @ A @ A4 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ Z @ X4 ) )
=> ( ord_less_eq @ A @ Z @ ( complete_Inf_Inf @ A @ A4 ) ) ) ) ) ).
% ccInf_greatest
thf(fact_7951_le__ccInf__iff,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,B2: A] :
( ( countable_countable @ A @ A4 )
=> ( ( ord_less_eq @ A @ B2 @ ( complete_Inf_Inf @ A @ A4 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ B2 @ X ) ) ) ) ) ) ).
% le_ccInf_iff
thf(fact_7952_ccInf__lower2,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,U: A,V2: A] :
( ( countable_countable @ A @ A4 )
=> ( ( member @ A @ U @ A4 )
=> ( ( ord_less_eq @ A @ U @ V2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ V2 ) ) ) ) ) ).
% ccInf_lower2
thf(fact_7953_ccInf__lower,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,X3: A] :
( ( countable_countable @ A @ A4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ X3 ) ) ) ) ).
% ccInf_lower
thf(fact_7954_ccInf__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B6: set @ A,A4: set @ A] :
( ( countable_countable @ A @ B6 )
=> ( ( countable_countable @ A @ A4 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ B6 )
=> ? [X6: A] :
( ( member @ A @ X6 @ A4 )
& ( ord_less_eq @ A @ X6 @ B5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ) ) ).
% ccInf_mono
thf(fact_7955_ccSup__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B6: set @ A,A4: set @ A] :
( ( countable_countable @ A @ B6 )
=> ( ( countable_countable @ A @ A4 )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ? [X6: A] :
( ( member @ A @ X6 @ B6 )
& ( ord_less_eq @ A @ A6 @ X6 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ) ).
% ccSup_mono
thf(fact_7956_ccSup__least,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,Z: A] :
( ( countable_countable @ A @ A4 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ Z ) ) ) ) ).
% ccSup_least
thf(fact_7957_ccSup__upper,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,X3: A] :
( ( countable_countable @ A @ A4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% ccSup_upper
thf(fact_7958_ccSup__le__iff,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,B2: A] :
( ( countable_countable @ A @ A4 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ B2 )
= ( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ X @ B2 ) ) ) ) ) ) ).
% ccSup_le_iff
thf(fact_7959_ccSup__upper2,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,U: A,V2: A] :
( ( countable_countable @ A @ A4 )
=> ( ( member @ A @ U @ A4 )
=> ( ( ord_less_eq @ A @ V2 @ U )
=> ( ord_less_eq @ A @ V2 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ) ).
% ccSup_upper2
thf(fact_7960_ccSup__subset__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B6: set @ A,A4: set @ A] :
( ( countable_countable @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ).
% ccSup_subset_mono
thf(fact_7961_option_Oset__sel,axiom,
! [A: $tType,A2: option @ A] :
( ( A2
!= ( none @ A ) )
=> ( member @ A @ ( the2 @ A @ A2 ) @ ( set_option @ A @ A2 ) ) ) ).
% option.set_sel
thf(fact_7962_option_Oset__cases,axiom,
! [A: $tType,E2: A,A2: option @ A] :
( ( member @ A @ E2 @ ( set_option @ A @ A2 ) )
=> ( A2
= ( some @ A @ E2 ) ) ) ).
% option.set_cases
thf(fact_7963_option_Oset__intros,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( set_option @ A @ ( some @ A @ X2 ) ) ) ).
% option.set_intros
thf(fact_7964_ospec,axiom,
! [A: $tType,A4: option @ A,P: A > $o,X3: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set_option @ A @ A4 ) )
=> ( P @ X4 ) )
=> ( ( A4
= ( some @ A @ X3 ) )
=> ( P @ X3 ) ) ) ).
% ospec
thf(fact_7965_map__option__idI,axiom,
! [A: $tType,X3: option @ A,F2: A > A] :
( ! [Y4: A] :
( ( member @ A @ Y4 @ ( set_option @ A @ X3 ) )
=> ( ( F2 @ Y4 )
= Y4 ) )
=> ( ( map_option @ A @ A @ F2 @ X3 )
= X3 ) ) ).
% map_option_idI
thf(fact_7966_option_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X3: option @ A,Xa: option @ A,F2: A > B,Fa: A > B] :
( ! [Z3: A,Za: A] :
( ( member @ A @ Z3 @ ( set_option @ A @ X3 ) )
=> ( ( member @ A @ Za @ ( set_option @ A @ Xa ) )
=> ( ( ( F2 @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_option @ A @ B @ F2 @ X3 )
= ( map_option @ A @ B @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% option.inj_map_strong
thf(fact_7967_option_Omap__cong0,axiom,
! [B: $tType,A: $tType,X3: option @ A,F2: A > B,G: A > B] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set_option @ A @ X3 ) )
=> ( ( F2 @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_option @ A @ B @ F2 @ X3 )
= ( map_option @ A @ B @ G @ X3 ) ) ) ).
% option.map_cong0
thf(fact_7968_option_Omap__cong,axiom,
! [B: $tType,A: $tType,X3: option @ A,Ya: option @ A,F2: A > B,G: A > B] :
( ( X3 = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set_option @ A @ Ya ) )
=> ( ( F2 @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_option @ A @ B @ F2 @ X3 )
= ( map_option @ A @ B @ G @ Ya ) ) ) ) ).
% option.map_cong
thf(fact_7969_option_Oset__map,axiom,
! [B: $tType,A: $tType,F2: A > B,V2: option @ A] :
( ( set_option @ B @ ( map_option @ A @ B @ F2 @ V2 ) )
= ( image @ A @ B @ F2 @ ( set_option @ A @ V2 ) ) ) ).
% option.set_map
thf(fact_7970_ccINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [A4: set @ B,F2: B > A,A2: A] :
( ( countable_countable @ B @ A4 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ A2 )
= ( ? [X: B] :
( ( member @ B @ X @ A4 )
& ( ord_less @ A @ ( F2 @ X ) @ A2 ) ) ) ) ) ) ).
% ccINF_less_iff
thf(fact_7971_ccSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,B6: set @ C,F2: B > A,G: C > A] :
( ( countable_countable @ B @ A4 )
=> ( ( countable_countable @ C @ B6 )
=> ( ! [N2: B] :
( ( member @ B @ N2 @ A4 )
=> ? [X6: C] :
( ( member @ C @ X6 @ B6 )
& ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B6 ) ) ) ) ) ) ) ).
% ccSUP_mono
thf(fact_7972_ccSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,F2: B > A,U: A] :
( ( countable_countable @ B @ A4 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ U ) ) ) ) ).
% ccSUP_least
thf(fact_7973_ccSUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,I: B,F2: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ) ) ).
% ccSUP_upper
thf(fact_7974_ccSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,F2: B > A,U: A] :
( ( countable_countable @ B @ A4 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ U )
= ( ! [X: B] :
( ( member @ B @ X @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X ) @ U ) ) ) ) ) ) ).
% ccSUP_le_iff
thf(fact_7975_ccSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,I: B,U: A,F2: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( member @ B @ I @ A4 )
=> ( ( ord_less_eq @ A @ U @ ( F2 @ I ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ) ) ) ).
% ccSUP_upper2
thf(fact_7976_less__ccSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [A4: set @ B,A2: A,F2: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) )
= ( ? [X: B] :
( ( member @ B @ X @ A4 )
& ( ord_less @ A @ A2 @ ( F2 @ X ) ) ) ) ) ) ) ).
% less_ccSUP_iff
thf(fact_7977_ccINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,U: A,F2: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ord_less_eq @ A @ U @ ( F2 @ I3 ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ) ) ).
% ccINF_greatest
thf(fact_7978_le__ccINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,U: A,F2: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A4 )
=> ( ord_less_eq @ A @ U @ ( F2 @ X ) ) ) ) ) ) ) ).
% le_ccINF_iff
thf(fact_7979_ccINF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,I: B,F2: B > A,U: A] :
( ( countable_countable @ B @ A4 )
=> ( ( member @ B @ I @ A4 )
=> ( ( ord_less_eq @ A @ ( F2 @ I ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ U ) ) ) ) ) ).
% ccINF_lower2
thf(fact_7980_ccINF__lower,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,I: B,F2: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( F2 @ I ) ) ) ) ) ).
% ccINF_lower
thf(fact_7981_ccINF__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,B6: set @ C,F2: B > A,G: C > A] :
( ( countable_countable @ B @ A4 )
=> ( ( countable_countable @ C @ B6 )
=> ( ! [M4: C] :
( ( member @ C @ M4 @ B6 )
=> ? [X6: B] :
( ( member @ B @ X6 @ A4 )
& ( ord_less_eq @ A @ ( F2 @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ G @ B6 ) ) ) ) ) ) ) ).
% ccINF_mono
thf(fact_7982_option_Osimps_I15_J,axiom,
! [A: $tType,X2: A] :
( ( set_option @ A @ ( some @ A @ X2 ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% option.simps(15)
thf(fact_7983_ccSup__inter__less__eq,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ( countable_countable @ A @ B6 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ) ).
% ccSup_inter_less_eq
thf(fact_7984_less__eq__ccInf__inter,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ( countable_countable @ A @ B6 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B6 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ) ) ) ).
% less_eq_ccInf_inter
thf(fact_7985_ccSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B6: set @ B,A4: set @ B,F2: B > A,G: B > A] :
( ( countable_countable @ B @ B6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B6 ) ) ) ) ) ) ) ).
% ccSUP_subset_mono
thf(fact_7986_ccINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,B6: set @ B,F2: B > A,G: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ A4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ B6 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B6 ) ) ) ) ) ) ) ).
% ccINF_superset_mono
thf(fact_7987_mono__ccSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F2: A > B,I6: set @ C,A4: C > A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( countable_countable @ C @ I6 )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image @ C @ B
@ ^ [X: C] : ( F2 @ ( A4 @ X ) )
@ I6 ) )
@ ( F2 @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ A4 @ I6 ) ) ) ) ) ) ) ).
% mono_ccSUP
thf(fact_7988_mono__ccSup,axiom,
! [B: $tType,A: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( countable_countable @ A @ A4 )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image @ A @ B @ F2 @ A4 ) ) @ ( F2 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ) ).
% mono_ccSup
thf(fact_7989_mono__ccInf,axiom,
! [B: $tType,A: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( countable_countable @ A @ A4 )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ A4 ) ) @ ( complete_Inf_Inf @ B @ ( image @ A @ B @ F2 @ A4 ) ) ) ) ) ) ).
% mono_ccInf
thf(fact_7990_option_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,A5: option @ A,B4: option @ B] :
? [Z6: option @ ( product_prod @ A @ B )] :
( ( member @ ( option @ ( product_prod @ A @ B ) ) @ Z6
@ ( collect @ ( option @ ( product_prod @ A @ B ) )
@ ^ [X: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( map_option @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z6 )
= A5 )
& ( ( map_option @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z6 )
= B4 ) ) ) ) ).
% option.in_rel
thf(fact_7991_combine__options__def,axiom,
! [A: $tType] :
( ( combine_options @ A )
= ( ^ [F3: A > A > A,X: option @ A,Y: option @ A] :
( case_option @ ( option @ A ) @ A @ Y
@ ^ [Z6: A] :
( case_option @ ( option @ A ) @ A @ ( some @ A @ Z6 )
@ ^ [Aa2: A] : ( some @ A @ ( F3 @ Z6 @ Aa2 ) )
@ Y )
@ X ) ) ) ).
% combine_options_def
thf(fact_7992_rel__option__None1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X3: option @ B] :
( ( rel_option @ A @ B @ P @ ( none @ A ) @ X3 )
= ( X3
= ( none @ B ) ) ) ).
% rel_option_None1
thf(fact_7993_rel__option__None2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X3: option @ A] :
( ( rel_option @ A @ B @ P @ X3 @ ( none @ B ) )
= ( X3
= ( none @ A ) ) ) ).
% rel_option_None2
thf(fact_7994_combine__options__simps_I3_J,axiom,
! [A: $tType,F2: A > A > A,A2: A,B2: A] :
( ( combine_options @ A @ F2 @ ( some @ A @ A2 ) @ ( some @ A @ B2 ) )
= ( some @ A @ ( F2 @ A2 @ B2 ) ) ) ).
% combine_options_simps(3)
thf(fact_7995_combine__options__simps_I2_J,axiom,
! [A: $tType,F2: A > A > A,X3: option @ A] :
( ( combine_options @ A @ F2 @ X3 @ ( none @ A ) )
= X3 ) ).
% combine_options_simps(2)
thf(fact_7996_combine__options__simps_I1_J,axiom,
! [A: $tType,F2: A > A > A,Y3: option @ A] :
( ( combine_options @ A @ F2 @ ( none @ A ) @ Y3 )
= Y3 ) ).
% combine_options_simps(1)
thf(fact_7997_rel__option__reflI,axiom,
! [A: $tType,Y3: option @ A,P: A > A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set_option @ A @ Y3 ) )
=> ( P @ X4 @ X4 ) )
=> ( rel_option @ A @ A @ P @ Y3 @ Y3 ) ) ).
% rel_option_reflI
thf(fact_7998_option_Orel__refl__strong,axiom,
! [A: $tType,X3: option @ A,Ra: A > A > $o] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set_option @ A @ X3 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( rel_option @ A @ A @ Ra @ X3 @ X3 ) ) ).
% option.rel_refl_strong
thf(fact_7999_option_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,X3: option @ A,Y3: option @ B,Ra: A > B > $o] :
( ( rel_option @ A @ B @ R3 @ X3 @ Y3 )
=> ( ! [Z3: A,Yb2: B] :
( ( member @ A @ Z3 @ ( set_option @ A @ X3 ) )
=> ( ( member @ B @ Yb2 @ ( set_option @ B @ Y3 ) )
=> ( ( R3 @ Z3 @ Yb2 )
=> ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( rel_option @ A @ B @ Ra @ X3 @ Y3 ) ) ) ).
% option.rel_mono_strong
thf(fact_8000_option_Orel__cong,axiom,
! [A: $tType,B: $tType,X3: option @ A,Ya: option @ A,Y3: option @ B,Xa: option @ B,R3: A > B > $o,Ra: A > B > $o] :
( ( X3 = Ya )
=> ( ( Y3 = Xa )
=> ( ! [Z3: A,Yb2: B] :
( ( member @ A @ Z3 @ ( set_option @ A @ Ya ) )
=> ( ( member @ B @ Yb2 @ ( set_option @ B @ Xa ) )
=> ( ( R3 @ Z3 @ Yb2 )
= ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( ( rel_option @ A @ B @ R3 @ X3 @ Y3 )
= ( rel_option @ A @ B @ Ra @ Ya @ Xa ) ) ) ) ) ).
% option.rel_cong
thf(fact_8001_rel__option__iff,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,X: option @ A,Y: option @ B] :
( product_case_prod @ ( option @ A ) @ ( option @ B ) @ $o
@ ^ [A5: option @ A,B4: option @ B] :
( case_option @ $o @ A
@ ( case_option @ $o @ B @ $true
@ ^ [C4: B] : $false
@ B4 )
@ ^ [Z6: A] : ( case_option @ $o @ B @ $false @ ( R6 @ Z6 ) @ B4 )
@ A5 )
@ ( product_Pair @ ( option @ A ) @ ( option @ B ) @ X @ Y ) ) ) ) ).
% rel_option_iff
thf(fact_8002_rel__option__inf,axiom,
! [B: $tType,A: $tType,A4: A > B > $o,B6: A > B > $o] :
( ( inf_inf @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( rel_option @ A @ B @ A4 ) @ ( rel_option @ A @ B @ B6 ) )
= ( rel_option @ A @ B @ ( inf_inf @ ( A > B > $o ) @ A4 @ B6 ) ) ) ).
% rel_option_inf
thf(fact_8003_option_Orel__transp,axiom,
! [A: $tType,R3: A > A > $o] :
( ( transp @ A @ R3 )
=> ( transp @ ( option @ A ) @ ( rel_option @ A @ A @ R3 ) ) ) ).
% option.rel_transp
thf(fact_8004_option_Obi__total__rel,axiom,
! [B: $tType,A: $tType,R3: A > B > $o] :
( ( bi_total @ A @ B @ R3 )
=> ( bi_total @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ B @ R3 ) ) ) ).
% option.bi_total_rel
thf(fact_8005_option_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X3: option @ A,G: B > C,Y3: option @ B] :
( ( rel_option @ A @ C @ Sa @ X3 @ ( map_option @ B @ C @ G @ Y3 ) )
= ( rel_option @ A @ B
@ ^ [X: A,Y: B] : ( Sa @ X @ ( G @ Y ) )
@ X3
@ Y3 ) ) ).
% option.rel_map(2)
thf(fact_8006_option_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X3: option @ A,Y3: option @ B] :
( ( rel_option @ C @ B @ Sb @ ( map_option @ A @ C @ I @ X3 ) @ Y3 )
= ( rel_option @ A @ B
@ ^ [X: A] : ( Sb @ ( I @ X ) )
@ X3
@ Y3 ) ) ).
% option.rel_map(1)
thf(fact_8007_option_Orel__eq,axiom,
! [A: $tType] :
( ( rel_option @ A @ A
@ ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [Y6: option @ A,Z2: option @ A] : Y6 = Z2 ) ) ).
% option.rel_eq
thf(fact_8008_option_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X3: option @ B] :
( ! [X4: B] : ( Ra @ X4 @ X4 )
=> ( rel_option @ B @ B @ Ra @ X3 @ X3 ) ) ).
% option.rel_refl
thf(fact_8009_combine__options__assoc,axiom,
! [A: $tType,F2: A > A > A,X3: option @ A,Y3: option @ A,Z: option @ A] :
( ! [X4: A,Y4: A,Z3: A] :
( ( F2 @ ( F2 @ X4 @ Y4 ) @ Z3 )
= ( F2 @ X4 @ ( F2 @ Y4 @ Z3 ) ) )
=> ( ( combine_options @ A @ F2 @ ( combine_options @ A @ F2 @ X3 @ Y3 ) @ Z )
= ( combine_options @ A @ F2 @ X3 @ ( combine_options @ A @ F2 @ Y3 @ Z ) ) ) ) ).
% combine_options_assoc
thf(fact_8010_combine__options__commute,axiom,
! [A: $tType,F2: A > A > A,X3: option @ A,Y3: option @ A] :
( ! [X4: A,Y4: A] :
( ( F2 @ X4 @ Y4 )
= ( F2 @ Y4 @ X4 ) )
=> ( ( combine_options @ A @ F2 @ X3 @ Y3 )
= ( combine_options @ A @ F2 @ Y3 @ X3 ) ) ) ).
% combine_options_commute
thf(fact_8011_combine__options__left__commute,axiom,
! [A: $tType,F2: A > A > A,Y3: option @ A,X3: option @ A,Z: option @ A] :
( ! [X4: A,Y4: A] :
( ( F2 @ X4 @ Y4 )
= ( F2 @ Y4 @ X4 ) )
=> ( ! [X4: A,Y4: A,Z3: A] :
( ( F2 @ ( F2 @ X4 @ Y4 ) @ Z3 )
= ( F2 @ X4 @ ( F2 @ Y4 @ Z3 ) ) )
=> ( ( combine_options @ A @ F2 @ Y3 @ ( combine_options @ A @ F2 @ X3 @ Z ) )
= ( combine_options @ A @ F2 @ X3 @ ( combine_options @ A @ F2 @ Y3 @ Z ) ) ) ) ) ).
% combine_options_left_commute
thf(fact_8012_option_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,Y2: A] :
~ ( rel_option @ A @ B @ R3 @ ( some @ A @ Y2 ) @ ( none @ B ) ) ).
% option.rel_distinct(2)
thf(fact_8013_option_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,Y2: B] :
~ ( rel_option @ A @ B @ R3 @ ( none @ A ) @ ( some @ B @ Y2 ) ) ).
% option.rel_distinct(1)
thf(fact_8014_option_Orel__cases,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,A2: option @ A,B2: option @ B] :
( ( rel_option @ A @ B @ R3 @ A2 @ B2 )
=> ( ( ( A2
= ( none @ A ) )
=> ( B2
!= ( none @ B ) ) )
=> ~ ! [X4: A] :
( ( A2
= ( some @ A @ X4 ) )
=> ! [Y4: B] :
( ( B2
= ( some @ B @ Y4 ) )
=> ~ ( R3 @ X4 @ Y4 ) ) ) ) ) ).
% option.rel_cases
thf(fact_8015_option_Orel__induct,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,X3: option @ A,Y3: option @ B,Q: ( option @ A ) > ( option @ B ) > $o] :
( ( rel_option @ A @ B @ R3 @ X3 @ Y3 )
=> ( ( Q @ ( none @ A ) @ ( none @ B ) )
=> ( ! [A24: A,B23: B] :
( ( R3 @ A24 @ B23 )
=> ( Q @ ( some @ A @ A24 ) @ ( some @ B @ B23 ) ) )
=> ( Q @ X3 @ Y3 ) ) ) ) ).
% option.rel_induct
thf(fact_8016_option__rel__Some2,axiom,
! [B: $tType,A: $tType,A4: A > B > $o,X3: option @ A,Y3: B] :
( ( rel_option @ A @ B @ A4 @ X3 @ ( some @ B @ Y3 ) )
= ( ? [X10: A] :
( ( X3
= ( some @ A @ X10 ) )
& ( A4 @ X10 @ Y3 ) ) ) ) ).
% option_rel_Some2
thf(fact_8017_option__rel__Some1,axiom,
! [A: $tType,B: $tType,A4: A > B > $o,X3: A,Y3: option @ B] :
( ( rel_option @ A @ B @ A4 @ ( some @ A @ X3 ) @ Y3 )
= ( ? [Y9: B] :
( ( Y3
= ( some @ B @ Y9 ) )
& ( A4 @ X3 @ Y9 ) ) ) ) ).
% option_rel_Some1
thf(fact_8018_option_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,X2: A,Y2: B] :
( ( R3 @ X2 @ Y2 )
=> ( rel_option @ A @ B @ R3 @ ( some @ A @ X2 ) @ ( some @ B @ Y2 ) ) ) ).
% option.rel_intros(2)
thf(fact_8019_option_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,X2: A,Y2: B] :
( ( rel_option @ A @ B @ R3 @ ( some @ A @ X2 ) @ ( some @ B @ Y2 ) )
= ( R3 @ X2 @ Y2 ) ) ).
% option.rel_inject(2)
thf(fact_8020_option_Octr__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R3: A > B > $o] : ( rel_option @ A @ B @ R3 @ ( none @ A ) @ ( none @ B ) ) ).
% option.ctr_transfer(1)
thf(fact_8021_option_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,A5: option @ A,B4: option @ B] :
( ( ( A5
= ( none @ A ) )
= ( B4
= ( none @ B ) ) )
& ( ( A5
!= ( none @ A ) )
=> ( ( B4
!= ( none @ B ) )
=> ( R6 @ ( the2 @ A @ A5 ) @ ( the2 @ B @ B4 ) ) ) ) ) ) ) ).
% option.rel_sel
thf(fact_8022_option_Orel__mono,axiom,
! [B: $tType,A: $tType,R3: A > B > $o,Ra: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R3 @ Ra )
=> ( ord_less_eq @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( rel_option @ A @ B @ R3 ) @ ( rel_option @ A @ B @ Ra ) ) ) ).
% option.rel_mono
thf(fact_8023_option_Omap__transfer,axiom,
! [A: $tType,B: $tType,F: $tType,E4: $tType,Rb: A > E4 > $o,Sd: B > F > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E4 > F ) @ ( ( option @ A ) > ( option @ B ) ) @ ( ( option @ E4 ) > ( option @ F ) ) @ ( bNF_rel_fun @ A @ E4 @ B @ F @ Rb @ Sd ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ E4 ) @ ( option @ B ) @ ( option @ F ) @ ( rel_option @ A @ E4 @ Rb ) @ ( rel_option @ B @ F @ Sd ) ) @ ( map_option @ A @ B ) @ ( map_option @ E4 @ F ) ) ).
% option.map_transfer
thf(fact_8024_option_Ocase__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S2: C > D > $o,R3: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > C ) > ( option @ A ) > C ) @ ( ( B > D ) > ( option @ B ) > D ) @ S2 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( option @ A ) > C ) @ ( ( option @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R3 @ S2 ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ C @ D @ ( rel_option @ A @ B @ R3 ) @ S2 ) ) @ ( case_option @ C @ A ) @ ( case_option @ D @ B ) ) ).
% option.case_transfer
thf(fact_8025_option_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( ( option @ C ) > ( option @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 ) )
@ ( bNF_rel_fun @ ( option @ A ) @ ( option @ C ) @ ( ( option @ B ) > $o ) @ ( ( option @ D ) > $o ) @ ( rel_option @ A @ C @ Sa )
@ ( bNF_rel_fun @ ( option @ B ) @ ( option @ D ) @ $o @ $o @ ( rel_option @ B @ D @ Sc )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 ) )
@ ( rel_option @ A @ B )
@ ( rel_option @ C @ D ) ) ).
% option.rel_transfer
thf(fact_8026_option_Octr__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R3: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( option @ A ) @ ( option @ B ) @ R3 @ ( rel_option @ A @ B @ R3 ) @ ( some @ A ) @ ( some @ B ) ) ).
% option.ctr_transfer(2)
thf(fact_8027_option_Odisc__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R3: A > B > $o] :
( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R3 )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ^ [Option3: option @ A] :
( Option3
!= ( none @ A ) )
@ ^ [Option3: option @ B] :
( Option3
!= ( none @ B ) ) ) ).
% option.disc_transfer(2)
thf(fact_8028_option_Odisc__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R3: A > B > $o] :
( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R3 )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ^ [Option3: option @ A] :
( Option3
= ( none @ A ) )
@ ^ [Option3: option @ B] :
( Option3
= ( none @ B ) ) ) ).
% option.disc_transfer(1)
thf(fact_8029_option_Orec__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S2: C > D > $o,R3: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > C ) > ( option @ A ) > C ) @ ( ( B > D ) > ( option @ B ) > D ) @ S2 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( option @ A ) > C ) @ ( ( option @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R3 @ S2 ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ C @ D @ ( rel_option @ A @ B @ R3 ) @ S2 ) ) @ ( rec_option @ C @ A ) @ ( rec_option @ D @ B ) ) ).
% option.rec_transfer
thf(fact_8030_option_OQuotient,axiom,
! [B: $tType,A: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A,T6: A > B > $o] :
( ( quotient @ A @ B @ R3 @ Abs @ Rep @ T6 )
=> ( quotient @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ A @ R3 ) @ ( map_option @ A @ B @ Abs ) @ ( map_option @ B @ A @ Rep ) @ ( rel_option @ A @ B @ T6 ) ) ) ).
% option.Quotient
thf(fact_8031_option__bind__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B6: C > D > $o] : ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ ( ( A > ( option @ C ) ) > ( option @ C ) ) @ ( ( B > ( option @ D ) ) > ( option @ D ) ) @ ( rel_option @ A @ B @ A4 ) @ ( bNF_rel_fun @ ( A > ( option @ C ) ) @ ( B > ( option @ D ) ) @ ( option @ C ) @ ( option @ D ) @ ( bNF_rel_fun @ A @ B @ ( option @ C ) @ ( option @ D ) @ A4 @ ( rel_option @ C @ D @ B6 ) ) @ ( rel_option @ C @ D @ B6 ) ) @ ( bind @ A @ C ) @ ( bind @ B @ D ) ) ).
% option_bind_transfer
thf(fact_8032_min__ext__compat,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R3 @ S2 ) @ R3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( min_ext @ A @ R3 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( min_ext @ A @ S2 ) @ ( insert @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( min_ext @ A @ R3 ) ) ) ).
% min_ext_compat
thf(fact_8033_length__removeAll__less,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X3 @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_removeAll_less
thf(fact_8034_union__comp__emptyR,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A ),C5: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A4 @ B6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ A4 @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ A4 @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ B6 @ C5 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyR
thf(fact_8035_union__comp__emptyL,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),C5: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A4 @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ B6 @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B6 ) @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyL
thf(fact_8036_length__removeAll__less__eq,axiom,
! [A: $tType,X3: A,Xs2: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X3 @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_removeAll_less_eq
thf(fact_8037_max__ext__compat,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R3 @ S2 ) @ R3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( max_ext @ A @ R3 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( max_ext @ A @ S2 ) @ ( insert @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( max_ext @ A @ R3 ) ) ) ).
% max_ext_compat
thf(fact_8038_reduction__pairI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R3 @ S2 ) @ R3 )
=> ( fun_reduction_pair @ A @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 ) ) ) ) ).
% reduction_pairI
thf(fact_8039_option_Opred__transfer,axiom,
! [A: $tType,B: $tType,R3: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( option @ A ) > $o ) @ ( ( option @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R3
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R3 )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( pred_option @ A )
@ ( pred_option @ B ) ) ).
% option.pred_transfer
thf(fact_8040_option_Opred__inject_I2_J,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( pred_option @ A @ P @ ( some @ A @ A2 ) )
= ( P @ A2 ) ) ).
% option.pred_inject(2)
thf(fact_8041_wf__less,axiom,
wf @ nat @ ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ).
% wf_less
thf(fact_8042_wf,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( wf @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ).
% wf
thf(fact_8043_wf__if__measure,axiom,
! [A: $tType,P: A > $o,F2: A > nat,G: A > A] :
( ! [X4: A] :
( ( P @ X4 )
=> ( ord_less @ nat @ ( F2 @ ( G @ X4 ) ) @ ( F2 @ X4 ) ) )
=> ( wf @ A
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [Y: A,X: A] :
( ( P @ X )
& ( Y
= ( G @ X ) ) ) ) ) ) ) ).
% wf_if_measure
thf(fact_8044_wf__no__loop,axiom,
! [B: $tType,R3: set @ ( product_prod @ B @ B )] :
( ( ( relcomp @ B @ B @ B @ R3 @ R3 )
= ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
=> ( wf @ B @ R3 ) ) ).
% wf_no_loop
thf(fact_8045_reduction__pair__def,axiom,
! [A: $tType] :
( ( fun_reduction_pair @ A )
= ( ^ [P3: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )] :
( ( wf @ A @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P3 ) )
& ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P3 ) @ ( product_snd @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P3 ) ) @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P3 ) ) ) ) ) ).
% reduction_pair_def
thf(fact_8046_option_Opred__mono,axiom,
! [A: $tType,P: A > $o,Pa: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ P @ Pa )
=> ( ord_less_eq @ ( ( option @ A ) > $o ) @ ( pred_option @ A @ P ) @ ( pred_option @ A @ Pa ) ) ) ).
% option.pred_mono
thf(fact_8047_wf__bounded__measure,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Ub: A > nat,F2: A > nat] :
( ! [A6: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A6 ) @ R2 )
=> ( ( ord_less_eq @ nat @ ( Ub @ B5 ) @ ( Ub @ A6 ) )
& ( ord_less_eq @ nat @ ( F2 @ B5 ) @ ( Ub @ A6 ) )
& ( ord_less @ nat @ ( F2 @ A6 ) @ ( F2 @ B5 ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wf_bounded_measure
thf(fact_8048_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P: B > $o,K: B,M: B > A] :
( ( wf @ A @ R2 )
=> ( ! [X4: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ ( transitive_trancl @ A @ R2 ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) )
=> ( ( P @ K )
=> ? [X4: B] :
( ( P @ X4 )
& ! [Y5: B] :
( ( P @ Y5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M @ X4 ) @ ( M @ Y5 ) ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ).
% wf_linord_ex_has_least
thf(fact_8049_wf__pair__less,axiom,
wf @ ( product_prod @ nat @ nat ) @ fun_pair_less ).
% wf_pair_less
thf(fact_8050_wf__int__ge__less__than2,axiom,
! [D3: int] : ( wf @ int @ ( int_ge_less_than2 @ D3 ) ) ).
% wf_int_ge_less_than2
thf(fact_8051_wf__int__ge__less__than,axiom,
! [D3: int] : ( wf @ int @ ( int_ge_less_than @ D3 ) ) ).
% wf_int_ge_less_than
thf(fact_8052_option_Opred__True,axiom,
! [A: $tType] :
( ( pred_option @ A
@ ^ [Uu3: A] : $true )
= ( ^ [Uu3: option @ A] : $true ) ) ).
% option.pred_True
thf(fact_8053_option_Opred__inject_I1_J,axiom,
! [A: $tType,P: A > $o] : ( pred_option @ A @ P @ ( none @ A ) ) ).
% option.pred_inject(1)
thf(fact_8054_option_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X3: option @ A,Ya: option @ A,F2: A > B,G: A > B] :
( ( X3 = Ya )
=> ( ( pred_option @ A
@ ^ [Z6: A] :
( ( F2 @ Z6 )
= ( G @ Z6 ) )
@ Ya )
=> ( ( map_option @ A @ B @ F2 @ X3 )
= ( map_option @ A @ B @ G @ Ya ) ) ) ) ).
% option.map_cong_pred
thf(fact_8055_option_Opred__set,axiom,
! [A: $tType] :
( ( pred_option @ A )
= ( ^ [P3: A > $o,X: option @ A] :
! [Y: A] :
( ( member @ A @ Y @ ( set_option @ A @ X ) )
=> ( P3 @ Y ) ) ) ) ).
% option.pred_set
thf(fact_8056_option_Opred__cong,axiom,
! [A: $tType,X3: option @ A,Ya: option @ A,P: A > $o,Pa: A > $o] :
( ( X3 = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set_option @ A @ Ya ) )
=> ( ( P @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( pred_option @ A @ P @ X3 )
= ( pred_option @ A @ Pa @ Ya ) ) ) ) ).
% option.pred_cong
thf(fact_8057_option_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X3: option @ A,Pa: A > $o] :
( ( pred_option @ A @ P @ X3 )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set_option @ A @ X3 ) )
=> ( ( P @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( pred_option @ A @ Pa @ X3 ) ) ) ).
% option.pred_mono_strong
thf(fact_8058_option_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F2: A > B,X3: option @ A] :
( ( pred_option @ B @ Q @ ( map_option @ A @ B @ F2 @ X3 ) )
= ( pred_option @ A @ ( comp @ B @ $o @ A @ Q @ F2 ) @ X3 ) ) ).
% option.pred_map
thf(fact_8059_reduction__pair__lemma,axiom,
! [A: $tType,P: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( fun_reduction_pair @ A @ P )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S2 @ ( product_snd @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P ) )
=> ( ( wf @ A @ S2 )
=> ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 ) ) ) ) ) ) ).
% reduction_pair_lemma
thf(fact_8060_pred__option__parametric,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( option @ A ) > $o ) @ ( ( option @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ A4 )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( pred_option @ A )
@ ( pred_option @ B ) ) ).
% pred_option_parametric
thf(fact_8061_and_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( comm_monoid @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% and.comm_monoid_axioms
thf(fact_8062_prod__decode__def,axiom,
( nat_prod_decode
= ( nat_prod_decode_aux @ ( zero_zero @ nat ) ) ) ).
% prod_decode_def
thf(fact_8063_mult_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( comm_monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% mult.comm_monoid_axioms
thf(fact_8064_comm__monoid_Ocomm__neutral,axiom,
! [A: $tType,F2: A > A > A,Z: A,A2: A] :
( ( comm_monoid @ A @ F2 @ Z )
=> ( ( F2 @ A2 @ Z )
= A2 ) ) ).
% comm_monoid.comm_neutral
thf(fact_8065_max__nat_Ocomm__monoid__axioms,axiom,
comm_monoid @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% max_nat.comm_monoid_axioms
thf(fact_8066_add_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( comm_monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% add.comm_monoid_axioms
thf(fact_8067_or_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( comm_monoid @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% or.comm_monoid_axioms
thf(fact_8068_gcd__nat_Ocomm__monoid__axioms,axiom,
comm_monoid @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% gcd_nat.comm_monoid_axioms
thf(fact_8069_xor_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( comm_monoid @ A @ ( bit_se5824344971392196577ns_xor @ A ) @ ( zero_zero @ A ) ) ) ).
% xor.comm_monoid_axioms
thf(fact_8070_list__decode_Opinduct,axiom,
! [A0: nat,P: nat > $o] :
( ( accp @ nat @ nat_list_decode_rel @ A0 )
=> ( ( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
=> ( ! [N2: nat] :
( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N2 ) )
=> ( ! [X6: nat,Y5: nat] :
( ( ( product_Pair @ nat @ nat @ X6 @ Y5 )
= ( nat_prod_decode @ N2 ) )
=> ( P @ Y5 ) )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% list_decode.pinduct
thf(fact_8071_list__decode_Oelims,axiom,
! [X3: nat,Y3: list @ nat] :
( ( ( nat_list_decode @ X3 )
= Y3 )
=> ( ( ( X3
= ( zero_zero @ nat ) )
=> ( Y3
!= ( nil @ nat ) ) )
=> ~ ! [N2: nat] :
( ( X3
= ( suc @ N2 ) )
=> ( Y3
!= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X: nat,Y: nat] : ( cons @ nat @ X @ ( nat_list_decode @ Y ) )
@ ( nat_prod_decode @ N2 ) ) ) ) ) ) ).
% list_decode.elims
thf(fact_8072_list__decode_Opsimps_I1_J,axiom,
( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
=> ( ( nat_list_decode @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ) ).
% list_decode.psimps(1)
thf(fact_8073_list__decode_Osimps_I1_J,axiom,
( ( nat_list_decode @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% list_decode.simps(1)
thf(fact_8074_list__decode_Opelims,axiom,
! [X3: nat,Y3: list @ nat] :
( ( ( nat_list_decode @ X3 )
= Y3 )
=> ( ( accp @ nat @ nat_list_decode_rel @ X3 )
=> ( ( ( X3
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( nil @ nat ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N2: nat] :
( ( X3
= ( suc @ N2 ) )
=> ( ( Y3
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X: nat,Y: nat] : ( cons @ nat @ X @ ( nat_list_decode @ Y ) )
@ ( nat_prod_decode @ N2 ) ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( suc @ N2 ) ) ) ) ) ) ) ).
% list_decode.pelims
thf(fact_8075_card__Plus__conv__if,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
( ( ( ( finite_finite @ A @ A4 )
& ( finite_finite @ B @ B6 ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B6 ) ) ) )
& ( ~ ( ( finite_finite @ A @ A4 )
& ( finite_finite @ B @ B6 ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) )
= ( zero_zero @ nat ) ) ) ) ).
% card_Plus_conv_if
thf(fact_8076_map__le__imp__upd__le,axiom,
! [A: $tType,B: $tType,M12: A > ( option @ B ),M23: A > ( option @ B ),X3: A,Y3: B] :
( ( map_le @ A @ B @ M12 @ M23 )
=> ( map_le @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M12 @ X3 @ ( none @ B ) ) @ ( fun_upd @ A @ ( option @ B ) @ M23 @ X3 @ ( some @ B @ Y3 ) ) ) ) ).
% map_le_imp_upd_le
thf(fact_8077_map__comp__None__iff,axiom,
! [A: $tType,C: $tType,B: $tType,M12: B > ( option @ A ),M23: C > ( option @ B ),K: C] :
( ( ( map_comp @ B @ A @ C @ M12 @ M23 @ K )
= ( none @ A ) )
= ( ( ( M23 @ K )
= ( none @ B ) )
| ? [K10: B] :
( ( ( M23 @ K )
= ( some @ B @ K10 ) )
& ( ( M12 @ K10 )
= ( none @ A ) ) ) ) ) ).
% map_comp_None_iff
thf(fact_8078_times__num__def,axiom,
( ( times_times @ num )
= ( ^ [M3: num,N3: num] : ( num_of_nat @ ( times_times @ nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N3 ) ) ) ) ) ).
% times_num_def
thf(fact_8079_map__comp__simps_I2_J,axiom,
! [B: $tType,C: $tType,A: $tType,M23: B > ( option @ A ),K: B,K7: A,M12: A > ( option @ C )] :
( ( ( M23 @ K )
= ( some @ A @ K7 ) )
=> ( ( map_comp @ A @ C @ B @ M12 @ M23 @ K )
= ( M12 @ K7 ) ) ) ).
% map_comp_simps(2)
thf(fact_8080_less__eq__num__def,axiom,
( ( ord_less_eq @ num )
= ( ^ [M3: num,N3: num] : ( ord_less_eq @ nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N3 ) ) ) ) ).
% less_eq_num_def
thf(fact_8081_nat__of__num_Osimps_I3_J,axiom,
! [X3: num] :
( ( nat_of_num @ ( bit1 @ X3 ) )
= ( suc @ ( plus_plus @ nat @ ( nat_of_num @ X3 ) @ ( nat_of_num @ X3 ) ) ) ) ).
% nat_of_num.simps(3)
thf(fact_8082_nat__of__num__pos,axiom,
! [X3: num] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat_of_num @ X3 ) ) ).
% nat_of_num_pos
thf(fact_8083_nat__of__num__neq__0,axiom,
! [X3: num] :
( ( nat_of_num @ X3 )
!= ( zero_zero @ nat ) ) ).
% nat_of_num_neq_0
thf(fact_8084_map__comp__Some__iff,axiom,
! [A: $tType,C: $tType,B: $tType,M12: B > ( option @ A ),M23: C > ( option @ B ),K: C,V2: A] :
( ( ( map_comp @ B @ A @ C @ M12 @ M23 @ K )
= ( some @ A @ V2 ) )
= ( ? [K10: B] :
( ( ( M23 @ K )
= ( some @ B @ K10 ) )
& ( ( M12 @ K10 )
= ( some @ A @ V2 ) ) ) ) ) ).
% map_comp_Some_iff
thf(fact_8085_nat__of__num__add,axiom,
! [X3: num,Y3: num] :
( ( nat_of_num @ ( plus_plus @ num @ X3 @ Y3 ) )
= ( plus_plus @ nat @ ( nat_of_num @ X3 ) @ ( nat_of_num @ Y3 ) ) ) ).
% nat_of_num_add
thf(fact_8086_nat__of__num__numeral,axiom,
( nat_of_num
= ( numeral_numeral @ nat ) ) ).
% nat_of_num_numeral
thf(fact_8087_nat__of__num__inverse,axiom,
! [X3: num] :
( ( num_of_nat @ ( nat_of_num @ X3 ) )
= X3 ) ).
% nat_of_num_inverse
thf(fact_8088_num__eq__iff,axiom,
( ( ^ [Y6: num,Z2: num] : Y6 = Z2 )
= ( ^ [X: num,Y: num] :
( ( nat_of_num @ X )
= ( nat_of_num @ Y ) ) ) ) ).
% num_eq_iff
thf(fact_8089_nat__of__num__code_I1_J,axiom,
( ( nat_of_num @ one2 )
= ( one_one @ nat ) ) ).
% nat_of_num_code(1)
thf(fact_8090_nat__of__num_Osimps_I2_J,axiom,
! [X3: num] :
( ( nat_of_num @ ( bit0 @ X3 ) )
= ( plus_plus @ nat @ ( nat_of_num @ X3 ) @ ( nat_of_num @ X3 ) ) ) ).
% nat_of_num.simps(2)
thf(fact_8091_nat__of__num__inc,axiom,
! [X3: num] :
( ( nat_of_num @ ( inc @ X3 ) )
= ( suc @ ( nat_of_num @ X3 ) ) ) ).
% nat_of_num_inc
thf(fact_8092_less__num__def,axiom,
( ( ord_less @ num )
= ( ^ [M3: num,N3: num] : ( ord_less @ nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N3 ) ) ) ) ).
% less_num_def
thf(fact_8093_nat__of__num__code_I2_J,axiom,
! [N: num] :
( ( nat_of_num @ ( bit0 @ N ) )
= ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ).
% nat_of_num_code(2)
thf(fact_8094_nat__of__num__mult,axiom,
! [X3: num,Y3: num] :
( ( nat_of_num @ ( times_times @ num @ X3 @ Y3 ) )
= ( times_times @ nat @ ( nat_of_num @ X3 ) @ ( nat_of_num @ Y3 ) ) ) ).
% nat_of_num_mult
thf(fact_8095_nat__of__num__sqr,axiom,
! [X3: num] :
( ( nat_of_num @ ( sqr @ X3 ) )
= ( times_times @ nat @ ( nat_of_num @ X3 ) @ ( nat_of_num @ X3 ) ) ) ).
% nat_of_num_sqr
thf(fact_8096_nat__of__num_Osimps_I1_J,axiom,
( ( nat_of_num @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_of_num.simps(1)
thf(fact_8097_num__of__nat__inverse,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nat_of_num @ ( num_of_nat @ N ) )
= N ) ) ).
% num_of_nat_inverse
thf(fact_8098_nat__of__num__code_I3_J,axiom,
! [N: num] :
( ( nat_of_num @ ( bit1 @ N ) )
= ( suc @ ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ) ).
% nat_of_num_code(3)
thf(fact_8099_plus__num__def,axiom,
( ( plus_plus @ num )
= ( ^ [M3: num,N3: num] : ( num_of_nat @ ( plus_plus @ nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N3 ) ) ) ) ) ).
% plus_num_def
thf(fact_8100_is__num_Ocases,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A2: A] :
( ( neg_numeral_is_num @ A @ A2 )
=> ( ( A2
!= ( one_one @ A ) )
=> ( ! [X4: A] :
( ( A2
= ( uminus_uminus @ A @ X4 ) )
=> ~ ( neg_numeral_is_num @ A @ X4 ) )
=> ~ ! [X4: A,Y4: A] :
( ( A2
= ( plus_plus @ A @ X4 @ Y4 ) )
=> ( ( neg_numeral_is_num @ A @ X4 )
=> ~ ( neg_numeral_is_num @ A @ Y4 ) ) ) ) ) ) ) ).
% is_num.cases
thf(fact_8101_is__num_Osimps,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_is_num @ A )
= ( ^ [A5: A] :
( ( A5
= ( one_one @ A ) )
| ? [X: A] :
( ( A5
= ( uminus_uminus @ A @ X ) )
& ( neg_numeral_is_num @ A @ X ) )
| ? [X: A,Y: A] :
( ( A5
= ( plus_plus @ A @ X @ Y ) )
& ( neg_numeral_is_num @ A @ X )
& ( neg_numeral_is_num @ A @ Y ) ) ) ) ) ) ).
% is_num.simps
thf(fact_8102_is__num__normalize_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( neg_numeral_is_num @ A @ ( one_one @ A ) ) ) ).
% is_num_normalize(4)
thf(fact_8103_is__num__normalize_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [X3: A,Y3: A] :
( ( neg_numeral_is_num @ A @ X3 )
=> ( ( neg_numeral_is_num @ A @ Y3 )
=> ( neg_numeral_is_num @ A @ ( plus_plus @ A @ X3 @ Y3 ) ) ) ) ) ).
% is_num_normalize(6)
thf(fact_8104_is__num__add__commute,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [X3: A,Y3: A] :
( ( neg_numeral_is_num @ A @ X3 )
=> ( ( neg_numeral_is_num @ A @ Y3 )
=> ( ( plus_plus @ A @ X3 @ Y3 )
= ( plus_plus @ A @ Y3 @ X3 ) ) ) ) ) ).
% is_num_add_commute
thf(fact_8105_is__num__add__left__commute,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [X3: A,Y3: A,Z: A] :
( ( neg_numeral_is_num @ A @ X3 )
=> ( ( neg_numeral_is_num @ A @ Y3 )
=> ( ( plus_plus @ A @ X3 @ ( plus_plus @ A @ Y3 @ Z ) )
= ( plus_plus @ A @ Y3 @ ( plus_plus @ A @ X3 @ Z ) ) ) ) ) ) ).
% is_num_add_left_commute
thf(fact_8106_is__num__numeral,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] : ( neg_numeral_is_num @ A @ ( numeral_numeral @ A @ K ) ) ) ).
% is_num_numeral
thf(fact_8107_is__num__normalize_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [X3: A] :
( ( neg_numeral_is_num @ A @ X3 )
=> ( neg_numeral_is_num @ A @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% is_num_normalize(5)
thf(fact_8108_arg__max__nat__le,axiom,
! [A: $tType,P: A > $o,X3: A,F2: A > nat,B2: nat] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less @ nat @ ( F2 @ Y4 ) @ B2 ) )
=> ( ord_less_eq @ nat @ ( F2 @ X3 ) @ ( F2 @ ( lattices_ord_arg_max @ A @ nat @ F2 @ P ) ) ) ) ) ).
% arg_max_nat_le
thf(fact_8109_arg__max__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less @ nat @ ( F2 @ Y4 ) @ B2 ) )
=> ( ( P @ ( lattices_ord_arg_max @ A @ nat @ F2 @ P ) )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( F2 @ Y5 ) @ ( F2 @ ( lattices_ord_arg_max @ A @ nat @ F2 @ P ) ) ) ) ) ) ) ).
% arg_max_nat_lemma
thf(fact_8110_arg__max__equality,axiom,
! [A: $tType,C: $tType] :
( ( order @ A )
=> ! [P: C > $o,K: C,F2: C > A] :
( ( P @ K )
=> ( ! [X4: C] :
( ( P @ X4 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( F2 @ K ) ) )
=> ( ( F2 @ ( lattices_ord_arg_max @ C @ A @ F2 @ P ) )
= ( F2 @ K ) ) ) ) ) ).
% arg_max_equality
thf(fact_8111_arg__max__natI,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less @ nat @ ( F2 @ Y4 ) @ B2 ) )
=> ( P @ ( lattices_ord_arg_max @ A @ nat @ F2 @ P ) ) ) ) ).
% arg_max_natI
thf(fact_8112_arg__maxI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [P: A > $o,X3: A,F2: A > B,Q: A > $o] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ~ ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ~ ( ord_less @ B @ ( F2 @ X4 ) @ ( F2 @ Y5 ) ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( lattices_ord_arg_max @ A @ B @ F2 @ P ) ) ) ) ) ) ).
% arg_maxI
thf(fact_8113_arg__max__on__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic1883929316492267755max_on @ B @ A )
= ( ^ [F3: B > A,S8: set @ B] :
( lattices_ord_arg_max @ B @ A @ F3
@ ^ [X: B] : ( member @ B @ X @ S8 ) ) ) ) ) ).
% arg_max_on_def
thf(fact_8114_arg__max__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattices_ord_arg_max @ B @ A )
= ( ^ [F3: B > A,P3: B > $o] : ( fChoice @ B @ ( lattic501386751176901750rg_max @ B @ A @ F3 @ P3 ) ) ) ) ) ).
% arg_max_def
thf(fact_8115_is__arg__max__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ( ( lattic501386751176901750rg_max @ A @ B )
= ( ^ [F3: A > B,P3: A > $o,X: A] :
( ( P3 @ X )
& ! [Y: A] :
( ( P3 @ Y )
=> ( ord_less_eq @ B @ ( F3 @ Y ) @ ( F3 @ X ) ) ) ) ) ) ) ).
% is_arg_max_linorder
thf(fact_8116_is__arg__max__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic501386751176901750rg_max @ B @ A )
= ( ^ [F3: B > A,P3: B > $o,X: B] :
( ( P3 @ X )
& ~ ? [Y: B] :
( ( P3 @ Y )
& ( ord_less @ A @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ) ) ).
% is_arg_max_def
thf(fact_8117_or__num_Oelims,axiom,
! [X3: num,Xa: num,Y3: num] :
( ( ( bit_un6697907153464112080or_num @ X3 @ Xa )
= Y3 )
=> ( ( ( X3 = one2 )
=> ( ( Xa = one2 )
=> ( Y3 != one2 ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y3
!= ( bit1 @ N2 ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y3
!= ( bit1 @ N2 ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ( ( Xa = one2 )
=> ( Y3
!= ( bit1 @ M4 ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y3
!= ( bit0 @ ( bit_un6697907153464112080or_num @ M4 @ N2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y3
!= ( bit1 @ ( bit_un6697907153464112080or_num @ M4 @ N2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ( ( Xa = one2 )
=> ( Y3
!= ( bit1 @ M4 ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y3
!= ( bit1 @ ( bit_un6697907153464112080or_num @ M4 @ N2 ) ) ) ) )
=> ~ ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y3
!= ( bit1 @ ( bit_un6697907153464112080or_num @ M4 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_num.elims
thf(fact_8118_or__num_Osimps_I2_J,axiom,
! [N: num] :
( ( bit_un6697907153464112080or_num @ one2 @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% or_num.simps(2)
thf(fact_8119_or__num_Osimps_I6_J,axiom,
! [M: num,N: num] :
( ( bit_un6697907153464112080or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( bit_un6697907153464112080or_num @ M @ N ) ) ) ).
% or_num.simps(6)
thf(fact_8120_or__num_Osimps_I8_J,axiom,
! [M: num,N: num] :
( ( bit_un6697907153464112080or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( bit_un6697907153464112080or_num @ M @ N ) ) ) ).
% or_num.simps(8)
thf(fact_8121_or__num_Osimps_I3_J,axiom,
! [N: num] :
( ( bit_un6697907153464112080or_num @ one2 @ ( bit1 @ N ) )
= ( bit1 @ N ) ) ).
% or_num.simps(3)
thf(fact_8122_or__num_Osimps_I7_J,axiom,
! [M: num] :
( ( bit_un6697907153464112080or_num @ ( bit1 @ M ) @ one2 )
= ( bit1 @ M ) ) ).
% or_num.simps(7)
thf(fact_8123_or__num_Osimps_I9_J,axiom,
! [M: num,N: num] :
( ( bit_un6697907153464112080or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( bit_un6697907153464112080or_num @ M @ N ) ) ) ).
% or_num.simps(9)
thf(fact_8124_or__num_Osimps_I1_J,axiom,
( ( bit_un6697907153464112080or_num @ one2 @ one2 )
= one2 ) ).
% or_num.simps(1)
thf(fact_8125_or__num_Osimps_I5_J,axiom,
! [M: num,N: num] :
( ( bit_un6697907153464112080or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit_un6697907153464112080or_num @ M @ N ) ) ) ).
% or_num.simps(5)
thf(fact_8126_numeral__or__num__eq,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: num] :
( ( numeral_numeral @ A @ ( bit_un6697907153464112080or_num @ M @ N ) )
= ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_or_num_eq
thf(fact_8127_or__num_Osimps_I4_J,axiom,
! [M: num] :
( ( bit_un6697907153464112080or_num @ ( bit0 @ M ) @ one2 )
= ( bit1 @ M ) ) ).
% or_num.simps(4)
thf(fact_8128_or__num_Opelims,axiom,
! [X3: num,Xa: num,Y3: num] :
( ( ( bit_un6697907153464112080or_num @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ X3 @ Xa ) )
=> ( ( ( X3 = one2 )
=> ( ( Xa = one2 )
=> ( ( Y3 = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y3
= ( bit1 @ N2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ( ( X3 = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y3
= ( bit1 @ N2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ( ( Xa = one2 )
=> ( ( Y3
= ( bit1 @ M4 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ one2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y3
= ( bit0 @ ( bit_un6697907153464112080or_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y3
= ( bit1 @ ( bit_un6697907153464112080or_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ( ( Xa = one2 )
=> ( ( Y3
= ( bit1 @ M4 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ one2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y3
= ( bit1 @ ( bit_un6697907153464112080or_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ~ ! [M4: num] :
( ( X3
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y3
= ( bit1 @ ( bit_un6697907153464112080or_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_num.pelims
thf(fact_8129_or__num__dict,axiom,
bit_un6697907153464112080or_num = bit_un2785000775030745342or_num ).
% or_num_dict
thf(fact_8130_or__num__rel__dict,axiom,
bit_un4773296044027857193um_rel = bit_un6909899581280750971um_rel ).
% or_num_rel_dict
thf(fact_8131_inv__unit__factor__eq__0__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A2: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A2 ) )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% inv_unit_factor_eq_0_iff
thf(fact_8132_unit__factor__simps_I1_J,axiom,
( ( unit_f5069060285200089521factor @ nat @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% unit_factor_simps(1)
thf(fact_8133_unit__factor__eq__0__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A2: A] :
( ( ( unit_f5069060285200089521factor @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% unit_factor_eq_0_iff
thf(fact_8134_unit__factor__0,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ( ( unit_f5069060285200089521factor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% unit_factor_0
thf(fact_8135_unit__factor__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( ( ( A2
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ( A2
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A2 @ B2 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_gcd
thf(fact_8136_unit__factor__nat__def,axiom,
( ( unit_f5069060285200089521factor @ nat )
= ( ^ [N3: nat] :
( if @ nat
@ ( N3
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( one_one @ nat ) ) ) ) ).
% unit_factor_nat_def
thf(fact_8137_unit__factor__dvd,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A2 ) @ B2 ) ) ) ).
% unit_factor_dvd
thf(fact_8138_unit__factor__power,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [A2: A,N: nat] :
( ( unit_f5069060285200089521factor @ A @ ( power_power @ A @ A2 @ N ) )
= ( power_power @ A @ ( unit_f5069060285200089521factor @ A @ A2 ) @ N ) ) ) ).
% unit_factor_power
thf(fact_8139_unit__factor__is__unit,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A2 ) @ ( one_one @ A ) ) ) ) ).
% unit_factor_is_unit
thf(fact_8140_coprime__crossproduct_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [B2: A,D3: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( unit_f5069060285200089521factor @ A @ B2 )
= ( unit_f5069060285200089521factor @ A @ D3 ) )
=> ( ( algebr8660921524188924756oprime @ A @ A2 @ B2 )
=> ( ( algebr8660921524188924756oprime @ A @ C2 @ D3 )
=> ( ( ( times_times @ A @ A2 @ D3 )
= ( times_times @ A @ B2 @ C2 ) )
= ( ( A2 = C2 )
& ( B2 = D3 ) ) ) ) ) ) ) ) ).
% coprime_crossproduct'
thf(fact_8141_unit__factor__Gcd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( ( ( gcd_Gcd @ A @ A4 )
= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A4 ) )
= ( zero_zero @ A ) ) )
& ( ( ( gcd_Gcd @ A @ A4 )
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A4 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_Gcd
thf(fact_8142_unit__factor__Gcd__fin,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( unit_f5069060285200089521factor @ A @ ( semiring_gcd_Gcd_fin @ A @ A4 ) )
= ( zero_neq_one_of_bool @ A
@ ( ( semiring_gcd_Gcd_fin @ A @ A4 )
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_factor_Gcd_fin
thf(fact_8143_frequently__at,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A2: A,S2: set @ A] :
( ( frequently @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ S2 ) )
= ( ! [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
=> ? [X: A] :
( ( member @ A @ X @ S2 )
& ( X != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ A2 ) @ D4 )
& ( P @ X ) ) ) ) ) ) ).
% frequently_at
thf(fact_8144_VEBT__internal_Ooption__comp__shift_Opelims_I1_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: option @ A,Xb: option @ A,Y3: $o] :
( ( ( vEBT_V6923181176774028177_shift @ A @ X3 @ Xa @ Xb )
= Y3 )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X3 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( none @ A ) )
=> ( ~ Y3
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X3 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Xb ) ) ) ) )
=> ( ! [V4: A] :
( ( Xa
= ( some @ A @ V4 ) )
=> ( ( Xb
= ( none @ A ) )
=> ( ~ Y3
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X3 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V4 ) @ ( none @ A ) ) ) ) ) ) )
=> ~ ! [X4: A] :
( ( Xa
= ( some @ A @ X4 ) )
=> ! [Y4: A] :
( ( Xb
= ( some @ A @ Y4 ) )
=> ( ( Y3
= ( X3 @ X4 @ Y4 ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X3 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ X4 ) @ ( some @ A @ Y4 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.pelims(1)
thf(fact_8145_VEBT__internal_Ogreater_Osimps,axiom,
( vEBT_VEBT_greater
= ( vEBT_V6923181176774028177_shift @ nat
@ ^ [X: nat,Y: nat] : ( ord_less @ nat @ Y @ X ) ) ) ).
% VEBT_internal.greater.simps
thf(fact_8146_VEBT__internal_Ogreater_Oelims_I1_J,axiom,
! [X3: option @ nat,Xa: option @ nat,Y3: $o] :
( ( ( vEBT_VEBT_greater @ X3 @ Xa )
= Y3 )
=> ( Y3
= ( vEBT_V6923181176774028177_shift @ nat
@ ^ [X: nat,Y: nat] : ( ord_less @ nat @ Y @ X )
@ X3
@ Xa ) ) ) ).
% VEBT_internal.greater.elims(1)
thf(fact_8147_VEBT__internal_Ogreater_Oelims_I2_J,axiom,
! [X3: option @ nat,Xa: option @ nat] :
( ( vEBT_VEBT_greater @ X3 @ Xa )
=> ( vEBT_V6923181176774028177_shift @ nat
@ ^ [X: nat,Y: nat] : ( ord_less @ nat @ Y @ X )
@ X3
@ Xa ) ) ).
% VEBT_internal.greater.elims(2)
thf(fact_8148_VEBT__internal_Ogreater_Oelims_I3_J,axiom,
! [X3: option @ nat,Xa: option @ nat] :
( ~ ( vEBT_VEBT_greater @ X3 @ Xa )
=> ~ ( vEBT_V6923181176774028177_shift @ nat
@ ^ [X: nat,Y: nat] : ( ord_less @ nat @ Y @ X )
@ X3
@ Xa ) ) ).
% VEBT_internal.greater.elims(3)
thf(fact_8149_VEBT__internal_Oless_Osimps,axiom,
( vEBT_VEBT_less
= ( vEBT_V6923181176774028177_shift @ nat @ ( ord_less @ nat ) ) ) ).
% VEBT_internal.less.simps
thf(fact_8150_VEBT__internal_Oless_Oelims_I1_J,axiom,
! [X3: option @ nat,Xa: option @ nat,Y3: $o] :
( ( ( vEBT_VEBT_less @ X3 @ Xa )
= Y3 )
=> ( Y3
= ( vEBT_V6923181176774028177_shift @ nat @ ( ord_less @ nat ) @ X3 @ Xa ) ) ) ).
% VEBT_internal.less.elims(1)
thf(fact_8151_VEBT__internal_Oless_Oelims_I2_J,axiom,
! [X3: option @ nat,Xa: option @ nat] :
( ( vEBT_VEBT_less @ X3 @ Xa )
=> ( vEBT_V6923181176774028177_shift @ nat @ ( ord_less @ nat ) @ X3 @ Xa ) ) ).
% VEBT_internal.less.elims(2)
thf(fact_8152_VEBT__internal_Oless_Oelims_I3_J,axiom,
! [X3: option @ nat,Xa: option @ nat] :
( ~ ( vEBT_VEBT_less @ X3 @ Xa )
=> ~ ( vEBT_V6923181176774028177_shift @ nat @ ( ord_less @ nat ) @ X3 @ Xa ) ) ).
% VEBT_internal.less.elims(3)
thf(fact_8153_VEBT__internal_Ooption__comp__shift_Osimps_I1_J,axiom,
! [A: $tType,Uu2: A > A > $o,Uv2: option @ A] :
~ ( vEBT_V6923181176774028177_shift @ A @ Uu2 @ ( none @ A ) @ Uv2 ) ).
% VEBT_internal.option_comp_shift.simps(1)
thf(fact_8154_VEBT__internal_Ooption__comp__shift_Oelims_I2_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: option @ A,Xb: option @ A] :
( ( vEBT_V6923181176774028177_shift @ A @ X3 @ Xa @ Xb )
=> ~ ! [X4: A] :
( ( Xa
= ( some @ A @ X4 ) )
=> ! [Y4: A] :
( ( Xb
= ( some @ A @ Y4 ) )
=> ~ ( X3 @ X4 @ Y4 ) ) ) ) ).
% VEBT_internal.option_comp_shift.elims(2)
thf(fact_8155_VEBT__internal_Ooption__comp__shift_Osimps_I3_J,axiom,
! [A: $tType,F2: A > A > $o,X3: A,Y3: A] :
( ( vEBT_V6923181176774028177_shift @ A @ F2 @ ( some @ A @ X3 ) @ ( some @ A @ Y3 ) )
= ( F2 @ X3 @ Y3 ) ) ).
% VEBT_internal.option_comp_shift.simps(3)
thf(fact_8156_VEBT__internal_Ooption__comp__shift_Opelims_I2_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: option @ A,Xb: option @ A] :
( ( vEBT_V6923181176774028177_shift @ A @ X3 @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X3 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ Xa @ Xb ) ) )
=> ~ ! [X4: A] :
( ( Xa
= ( some @ A @ X4 ) )
=> ! [Y4: A] :
( ( Xb
= ( some @ A @ Y4 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X3 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ X4 ) @ ( some @ A @ Y4 ) ) ) )
=> ~ ( X3 @ X4 @ Y4 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.pelims(2)
thf(fact_8157_VEBT__internal_Olesseq_Oelims_I3_J,axiom,
! [X3: option @ nat,Xa: option @ nat] :
( ~ ( vEBT_VEBT_lesseq @ X3 @ Xa )
=> ~ ( vEBT_V6923181176774028177_shift @ nat @ ( ord_less_eq @ nat ) @ X3 @ Xa ) ) ).
% VEBT_internal.lesseq.elims(3)
thf(fact_8158_VEBT__internal_Olesseq_Oelims_I2_J,axiom,
! [X3: option @ nat,Xa: option @ nat] :
( ( vEBT_VEBT_lesseq @ X3 @ Xa )
=> ( vEBT_V6923181176774028177_shift @ nat @ ( ord_less_eq @ nat ) @ X3 @ Xa ) ) ).
% VEBT_internal.lesseq.elims(2)
thf(fact_8159_VEBT__internal_Olesseq_Oelims_I1_J,axiom,
! [X3: option @ nat,Xa: option @ nat,Y3: $o] :
( ( ( vEBT_VEBT_lesseq @ X3 @ Xa )
= Y3 )
=> ( Y3
= ( vEBT_V6923181176774028177_shift @ nat @ ( ord_less_eq @ nat ) @ X3 @ Xa ) ) ) ).
% VEBT_internal.lesseq.elims(1)
thf(fact_8160_VEBT__internal_Olesseq_Osimps,axiom,
( vEBT_VEBT_lesseq
= ( vEBT_V6923181176774028177_shift @ nat @ ( ord_less_eq @ nat ) ) ) ).
% VEBT_internal.lesseq.simps
thf(fact_8161_VEBT__internal_Ooption__comp__shift_Osimps_I2_J,axiom,
! [A: $tType,Uw2: A > A > $o,V2: A] :
~ ( vEBT_V6923181176774028177_shift @ A @ Uw2 @ ( some @ A @ V2 ) @ ( none @ A ) ) ).
% VEBT_internal.option_comp_shift.simps(2)
thf(fact_8162_VEBT__internal_Ooption__comp__shift_Oelims_I1_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: option @ A,Xb: option @ A,Y3: $o] :
( ( ( vEBT_V6923181176774028177_shift @ A @ X3 @ Xa @ Xb )
= Y3 )
=> ( ( ( Xa
= ( none @ A ) )
=> Y3 )
=> ( ( ? [V4: A] :
( Xa
= ( some @ A @ V4 ) )
=> ( ( Xb
= ( none @ A ) )
=> Y3 ) )
=> ~ ! [X4: A] :
( ( Xa
= ( some @ A @ X4 ) )
=> ! [Y4: A] :
( ( Xb
= ( some @ A @ Y4 ) )
=> ( Y3
= ( ~ ( X3 @ X4 @ Y4 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.elims(1)
thf(fact_8163_VEBT__internal_Ooption__comp__shift_Oelims_I3_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: option @ A,Xb: option @ A] :
( ~ ( vEBT_V6923181176774028177_shift @ A @ X3 @ Xa @ Xb )
=> ( ( Xa
!= ( none @ A ) )
=> ( ( ? [V4: A] :
( Xa
= ( some @ A @ V4 ) )
=> ( Xb
!= ( none @ A ) ) )
=> ~ ! [X4: A] :
( ( Xa
= ( some @ A @ X4 ) )
=> ! [Y4: A] :
( ( Xb
= ( some @ A @ Y4 ) )
=> ( X3 @ X4 @ Y4 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.elims(3)
thf(fact_8164_VEBT__internal_Ooption__comp__shift_Opelims_I3_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: option @ A,Xb: option @ A] :
( ~ ( vEBT_V6923181176774028177_shift @ A @ X3 @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X3 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( none @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X3 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Xb ) ) ) )
=> ( ! [V4: A] :
( ( Xa
= ( some @ A @ V4 ) )
=> ( ( Xb
= ( none @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X3 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V4 ) @ ( none @ A ) ) ) ) ) )
=> ~ ! [X4: A] :
( ( Xa
= ( some @ A @ X4 ) )
=> ! [Y4: A] :
( ( Xb
= ( some @ A @ Y4 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X3 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ X4 ) @ ( some @ A @ Y4 ) ) ) )
=> ( X3 @ X4 @ Y4 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.pelims(3)
thf(fact_8165_unit__factor__Lcm__fin,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( unit_f5069060285200089521factor @ A @ ( semiring_gcd_Lcm_fin @ A @ A4 ) )
= ( zero_neq_one_of_bool @ A
@ ( ( semiring_gcd_Lcm_fin @ A @ A4 )
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_factor_Lcm_fin
thf(fact_8166_sum_Ocomm__monoid__list__set__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( groups4802862169904069756st_set @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum.comm_monoid_list_set_axioms
thf(fact_8167_Lcm__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite @ A @ A4 )
=> ( ( semiring_gcd_Lcm_fin @ A @ A4 )
= ( zero_zero @ A ) ) ) ) ).
% Lcm_fin.infinite
thf(fact_8168_Lcm__fin__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( ( semiring_gcd_Lcm_fin @ A @ A4 )
= ( zero_zero @ A ) )
= ( member @ A @ ( zero_zero @ A ) @ A4 ) ) ) ) ).
% Lcm_fin_0_iff
thf(fact_8169_Lcm__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A )
= ( ^ [A9: set @ A] : ( if @ A @ ( finite_finite @ A @ A9 ) @ ( finite_fold @ A @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ A9 ) @ ( zero_zero @ A ) ) ) ) ) ).
% Lcm_fin.eq_fold
thf(fact_8170_Lcm__fin__def,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A )
= ( bounde2362111253966948842tice_F @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ).
% Lcm_fin_def
thf(fact_8171_lcm__0__iff__nat,axiom,
! [M: nat,N: nat] :
( ( ( gcd_lcm @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% lcm_0_iff_nat
thf(fact_8172_lcm__0__iff__int,axiom,
! [M: int,N: int] :
( ( ( gcd_lcm @ int @ M @ N )
= ( zero_zero @ int ) )
= ( ( M
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ int ) ) ) ) ).
% lcm_0_iff_int
thf(fact_8173_lcm__int__int__eq,axiom,
! [M: nat,N: nat] :
( ( gcd_lcm @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
= ( semiring_1_of_nat @ int @ ( gcd_lcm @ nat @ M @ N ) ) ) ).
% lcm_int_int_eq
thf(fact_8174_lcm__1__iff__nat,axiom,
! [M: nat,N: nat] :
( ( ( gcd_lcm @ nat @ M @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% lcm_1_iff_nat
thf(fact_8175_lcm_Obottom__right__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A] :
( ( gcd_lcm @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% lcm.bottom_right_bottom
thf(fact_8176_lcm_Obottom__left__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A] :
( ( gcd_lcm @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% lcm.bottom_left_bottom
thf(fact_8177_lcm__neg__numeral__1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [N: num,A2: A] :
( ( gcd_lcm @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ A2 )
= ( gcd_lcm @ A @ ( numeral_numeral @ A @ N ) @ A2 ) ) ) ).
% lcm_neg_numeral_1
thf(fact_8178_lcm__neg__numeral__2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A2: A,N: num] :
( ( gcd_lcm @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( gcd_lcm @ A @ A2 @ ( numeral_numeral @ A @ N ) ) ) ) ).
% lcm_neg_numeral_2
thf(fact_8179_lcm__nat__abs__left__eq,axiom,
! [K: int,N: nat] :
( ( gcd_lcm @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
= ( nat2 @ ( gcd_lcm @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ) ).
% lcm_nat_abs_left_eq
% Type constructors (770)
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice,axiom,
bounded_lattice @ extended_enat ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_1,axiom,
! [A10: $tType] : ( bounded_lattice @ ( filter @ A10 ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_2,axiom,
bounded_lattice @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_3,axiom,
! [A10: $tType] : ( bounded_lattice @ ( set @ A10 ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_4,axiom,
! [A10: $tType,A16: $tType] :
( ( bounded_lattice @ A16 )
=> ( bounded_lattice @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,axiom,
! [A10: $tType,A16: $tType] :
( ( comple592849572758109894attice @ A16 )
=> ( counta4013691401010221786attice @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A10: $tType,A16: $tType] :
( ( comple6319245703460814977attice @ A16 )
=> ( condit1219197933456340205attice @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
! [A10: $tType,A16: $tType] :
( ( counta3822494911875563373attice @ A16 )
=> ( counta3822494911875563373attice @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A10: $tType,A16: $tType] :
( ( comple592849572758109894attice @ A16 )
=> ( comple592849572758109894attice @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A10: $tType,A16: $tType] :
( ( bounded_lattice @ A16 )
=> ( bounde4967611905675639751up_bot @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A10: $tType,A16: $tType] :
( ( bounded_lattice @ A16 )
=> ( bounde4346867609351753570nf_top @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A10: $tType,A16: $tType] :
( ( comple6319245703460814977attice @ A16 )
=> ( comple6319245703460814977attice @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A10: $tType,A16: $tType] :
( ( boolea8198339166811842893lgebra @ A16 )
=> ( boolea8198339166811842893lgebra @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A10: $tType,A16: $tType] :
( ( comple6319245703460814977attice @ A16 )
=> ( comple9053668089753744459l_ccpo @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A10: $tType,A16: $tType] :
( ( semilattice_sup @ A16 )
=> ( semilattice_sup @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A10: $tType,A16: $tType] :
( ( semilattice_inf @ A16 )
=> ( semilattice_inf @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A10: $tType,A16: $tType] :
( ( distrib_lattice @ A16 )
=> ( distrib_lattice @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A10: $tType,A16: $tType] :
( ( order_top @ A16 )
=> ( order_top @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A10: $tType,A16: $tType] :
( ( order_bot @ A16 )
=> ( order_bot @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A10: $tType,A16: $tType] :
( ( preorder @ A16 )
=> ( preorder @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A10: $tType,A16: $tType] :
( ( lattice @ A16 )
=> ( lattice @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A10: $tType,A16: $tType] :
( ( order @ A16 )
=> ( order @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A10: $tType,A16: $tType] :
( ( top @ A16 )
=> ( top @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A10: $tType,A16: $tType] :
( ( ord @ A16 )
=> ( ord @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A10: $tType,A16: $tType] :
( ( bot @ A16 )
=> ( bot @ ( A10 > A16 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A10: $tType,A16: $tType] :
( ( uminus @ A16 )
=> ( uminus @ ( A10 > A16 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_5,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___Rings_Onormalization__semidom__multiplicative,axiom,
normal6328177297339901930cative @ int ).
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri6575147826004484403cancel @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict9044650504122735259up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere580206878836729694up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere2412721322843649153imp_le @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
bit_se359711467146920520ations @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord2810124833399127020strict @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
topolo4958980785337419405_space @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
topolo1944317154257567458pology @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord715952674999750819strict @ int ).
thf(tcon_Int_Oint___Limits_Otopological__comm__monoid__add,axiom,
topolo5987344860129210374id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
linord4140545234300271783up_add @ int ).
thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
topolo2564578578187576103pology @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
semiri2026040879449505780visors @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring @ int ).
thf(tcon_Int_Oint___Limits_Otopological__semigroup__mult,axiom,
topolo4211221413907600880p_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide__unit__factor,axiom,
semido2269285787275462019factor @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord8928482502909563296strict @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
semiri3467727345109120633visors @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere6658533253407199908up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere166539214618696060dd_abs @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize,axiom,
semiri6843258321239162965malize @ int ).
thf(tcon_Int_Oint___Limits_Otopological__monoid__mult,axiom,
topolo1898628316856586783d_mult @ int ).
thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere6911136660526730532id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord5086331880401160121up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel2418104881723323429up_add @ int ).
thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_15535105094025558882visors @ int ).
thf(tcon_Int_Oint___Limits_Otopological__monoid__add,axiom,
topolo6943815403480290642id_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1802427076303600483id_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord4710134922213307826strict @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
bit_semiring_bits @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
topological_t2_space @ int ).
thf(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere2520102378445227354miring @ int ).
thf(tcon_Int_Oint___Rings_Onormalization__semidom,axiom,
normal8620421768224518004emidom @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord6961819062388156250ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__sup_6,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_7,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Lattices_Odistrib__lattice_8,axiom,
distrib_lattice @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
semiring_1_cancel @ int ).
thf(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
algebraic_semidom @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
ordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs @ int ).
thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
semiring_parity @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
semiring_modulo @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0 @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
semidom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide @ int ).
thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
semiring_numeral @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one @ int ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int ).
thf(tcon_Int_Oint___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_9,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__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_10,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_11,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_12,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_13,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_14,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_15,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_16,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_17,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_18,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_19,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_20,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_21,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_22,axiom,
normal6328177297339901930cative @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_23,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_24,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_25,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_26,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_27,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_28,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_29,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_30,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_31,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_32,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_33,axiom,
topolo4958980785337419405_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_34,axiom,
topolo1944317154257567458pology @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_35,axiom,
topolo5987344860129210374id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_36,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_37,axiom,
topolo2564578578187576103pology @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_38,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_39,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_40,axiom,
topolo4211221413907600880p_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_41,axiom,
semido2269285787275462019factor @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_42,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_43,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_44,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_45,axiom,
semiri6843258321239162965malize @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__mult_46,axiom,
topolo1898628316856586783d_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_47,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_48,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_49,axiom,
topolo6943815403480290642id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_50,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_51,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_52,axiom,
topological_t2_space @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_53,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom_54,axiom,
normal8620421768224518004emidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_55,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_56,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_57,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_58,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_59,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_60,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_61,axiom,
distrib_lattice @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_62,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_63,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_64,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_65,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_66,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_67,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_68,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_69,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_70,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_71,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_72,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_73,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_74,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_75,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_76,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_77,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_78,axiom,
zero_less_one @ 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_Oordered__comm__semiring_133,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_134,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_135,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_136,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_137,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_138,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_139,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_140,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_141,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_142,axiom,
distrib_lattice @ 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___Nat_Osemiring__char__0_159,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_160,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_161,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_162,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_163,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_164,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_165,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_166,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__divide_167,axiom,
idom_divide @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_168,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_169,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_170,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_171,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_172,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_173,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_174,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_175,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_176,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_177,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_178,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_179,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_180,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_181,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom_182,axiom,
semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_183,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_184,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_185,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_186,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_187,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_188,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_189,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_190,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_191,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_192,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_193,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_194,axiom,
dvd @ rat ).
thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_195,axiom,
! [A10: $tType] : ( counta4013691401010221786attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_196,axiom,
! [A10: $tType] : ( condit1219197933456340205attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_197,axiom,
! [A10: $tType] : ( counta3822494911875563373attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_198,axiom,
! [A10: $tType] : ( comple592849572758109894attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_199,axiom,
! [A10: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_200,axiom,
! [A10: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_201,axiom,
! [A10: $tType] : ( comple6319245703460814977attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_202,axiom,
! [A10: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_203,axiom,
! [A10: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_204,axiom,
! [A10: $tType] : ( semilattice_sup @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_205,axiom,
! [A10: $tType] : ( semilattice_inf @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Odistrib__lattice_206,axiom,
! [A10: $tType] : ( distrib_lattice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_207,axiom,
! [A10: $tType] : ( order_top @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_208,axiom,
! [A10: $tType] : ( order_bot @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_209,axiom,
! [A10: $tType] : ( preorder @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_210,axiom,
! [A10: $tType] : ( lattice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_211,axiom,
! [A10: $tType] : ( order @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_212,axiom,
! [A10: $tType] : ( top @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_213,axiom,
! [A10: $tType] : ( ord @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_214,axiom,
! [A10: $tType] : ( bot @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_215,axiom,
! [A10: $tType] : ( uminus @ ( set @ A10 ) ) ).
thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_216,axiom,
counta4013691401010221786attice @ $o ).
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___Topological__Spaces_Otopological__space_220,axiom,
topolo4958980785337419405_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_221,axiom,
topolo1944317154257567458pology @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_222,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_223,axiom,
bounde4346867609351753570nf_top @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_224,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_225,axiom,
topolo2564578578187576103pology @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_226,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_227,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_228,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_229,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_230,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_231,axiom,
distrib_lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_232,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_233,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_234,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_235,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_236,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_237,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_238,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_239,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_240,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_241,axiom,
uminus @ $o ).
thf(tcon_List_Olist___Nat_Osize_242,axiom,
! [A10: $tType] : ( size @ ( list @ A10 ) ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_243,axiom,
condit6923001295902523014norder @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_244,axiom,
condit1219197933456340205attice @ real ).
thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_245,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_246,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_247,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_248,axiom,
strict9044650504122735259up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_249,axiom,
ordere580206878836729694up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_250,axiom,
ordere2412721322843649153imp_le @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_251,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_252,axiom,
strict7427464778891057005id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_253,axiom,
ordere8940638589300402666id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_254,axiom,
topolo4958980785337419405_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_255,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_256,axiom,
archim462609752435547400_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_257,axiom,
linord715952674999750819strict @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ouniformity__dist,axiom,
real_V768167426530841204y_dist @ real ).
thf(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_258,axiom,
unboun7993243217541854897norder @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_259,axiom,
topolo5987344860129210374id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_260,axiom,
linord4140545234300271783up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_261,axiom,
topolo2564578578187576103pology @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_262,axiom,
semiri2026040879449505780visors @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_263,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_264,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_265,axiom,
linord8928482502909563296strict @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_266,axiom,
semiri3467727345109120633visors @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
real_V7819770556892013058_space @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_267,axiom,
ordere6658533253407199908up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_268,axiom,
ordere166539214618696060dd_abs @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_269,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_270,axiom,
ordere6911136660526730532id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_271,axiom,
linord5086331880401160121up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_272,axiom,
cancel2418104881723323429up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ouniformity,axiom,
topolo4638772830378233104ormity @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_273,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_274,axiom,
topolo6943815403480290642id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_275,axiom,
cancel1802427076303600483id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_276,axiom,
linord4710134922213307826strict @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_277,axiom,
topological_t2_space @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_278,axiom,
ordere2520102378445227354miring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_279,axiom,
linord6961819062388156250ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_280,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_281,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring_282,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_283,axiom,
ordered_semiring_0 @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_284,axiom,
linordered_semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder_285,axiom,
dense_linorder @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_286,axiom,
semilattice_sup @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__inf_287,axiom,
semilattice_inf @ real ).
thf(tcon_Real_Oreal___Lattices_Odistrib__lattice_288,axiom,
distrib_lattice @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_289,axiom,
ab_semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__cancel_290,axiom,
semiring_1_cancel @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_291,axiom,
comm_monoid_mult @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_292,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field_293,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring_294,axiom,
ordered_semiring @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring__abs_295,axiom,
ordered_ring_abs @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_296,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring_297,axiom,
linordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_298,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_299,axiom,
comm_semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_300,axiom,
comm_semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__order_301,axiom,
dense_order @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_302,axiom,
semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide_303,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Num_Osemiring__numeral_304,axiom,
semiring_numeral @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_305,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring_306,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one_307,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_308,axiom,
semiring_char_0 @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_309,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__char__0_310,axiom,
field_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_311,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring_312,axiom,
ordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_313,axiom,
idom_abs_sgn @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_314,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_315,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_316,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__divide_317,axiom,
idom_divide @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_318,axiom,
comm_ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_319,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_320,axiom,
semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__0_321,axiom,
semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_322,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot_323,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Lattices_Olattice_324,axiom,
lattice @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_325,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Rings_Omult__zero_326,axiom,
mult_zero @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring_327,axiom,
comm_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_328,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_329,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_330,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring_331,axiom,
semiring @ real ).
thf(tcon_Real_Oreal___Fields_Oinverse_332,axiom,
inverse @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom_333,axiom,
semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_334,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_335,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_336,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Oabs__if_337,axiom,
abs_if @ real ).
thf(tcon_Real_Oreal___Fields_Ofield_338,axiom,
field @ real ).
thf(tcon_Real_Oreal___Power_Opower_339,axiom,
power @ real ).
thf(tcon_Real_Oreal___Num_Onumeral_340,axiom,
numeral @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_341,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oplus_342,axiom,
plus @ real ).
thf(tcon_Real_Oreal___Rings_Oring_343,axiom,
ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom_344,axiom,
idom @ real ).
thf(tcon_Real_Oreal___Groups_Oone_345,axiom,
one @ real ).
thf(tcon_Real_Oreal___Rings_Odvd_346,axiom,
dvd @ real ).
thf(tcon_String_Ochar___Nat_Osize_347,axiom,
size @ char ).
thf(tcon_Sum__Type_Osum___Nat_Osize_348,axiom,
! [A10: $tType,A16: $tType] : ( size @ ( sum_sum @ A10 @ A16 ) ) ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_349,axiom,
! [A10: $tType] : ( condit1219197933456340205attice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_350,axiom,
! [A10: $tType] : ( counta3822494911875563373attice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_351,axiom,
! [A10: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_352,axiom,
! [A10: $tType] : ( bounde4346867609351753570nf_top @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_353,axiom,
! [A10: $tType] : ( comple6319245703460814977attice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_354,axiom,
! [A10: $tType] : ( comple9053668089753744459l_ccpo @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_355,axiom,
! [A10: $tType] : ( semilattice_sup @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_356,axiom,
! [A10: $tType] : ( semilattice_inf @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_357,axiom,
! [A10: $tType] : ( distrib_lattice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_358,axiom,
! [A10: $tType] : ( order_top @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_359,axiom,
! [A10: $tType] : ( order_bot @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_360,axiom,
! [A10: $tType] : ( preorder @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_361,axiom,
! [A10: $tType] : ( lattice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_362,axiom,
! [A10: $tType] : ( order @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Otop_363,axiom,
! [A10: $tType] : ( top @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_364,axiom,
! [A10: $tType] : ( ord @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Obot_365,axiom,
! [A10: $tType] : ( bot @ ( filter @ A10 ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_366,axiom,
! [A10: $tType] : ( size @ ( option @ A10 ) ) ).
thf(tcon_String_Oliteral___Groups_Osemigroup__add_367,axiom,
semigroup_add @ literal ).
thf(tcon_String_Oliteral___Orderings_Opreorder_368,axiom,
preorder @ literal ).
thf(tcon_String_Oliteral___Orderings_Olinorder_369,axiom,
linorder @ literal ).
thf(tcon_String_Oliteral___Groups_Omonoid__add_370,axiom,
monoid_add @ literal ).
thf(tcon_String_Oliteral___Orderings_Oorder_371,axiom,
order @ literal ).
thf(tcon_String_Oliteral___Orderings_Oord_372,axiom,
ord @ literal ).
thf(tcon_String_Oliteral___Groups_Ozero_373,axiom,
zero @ literal ).
thf(tcon_String_Oliteral___Groups_Oplus_374,axiom,
plus @ literal ).
thf(tcon_String_Oliteral___Nat_Osize_375,axiom,
size @ literal ).
thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_376,axiom,
semiri1453513574482234551roduct @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_377,axiom,
topolo3112930676232923870pology @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_378,axiom,
real_V8999393235501362500lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_379,axiom,
real_V2822296259951069270ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_380,axiom,
semiri6575147826004484403cancel @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_381,axiom,
real_V4412858255891104859lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_382,axiom,
real_V822414075346904944vector @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_383,axiom,
topolo4958980785337419405_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_384,axiom,
real_V3459762299906320749_field @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_385,axiom,
real_V5047593784448816457lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_386,axiom,
real_V768167426530841204y_dist @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_387,axiom,
topolo5987344860129210374id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_388,axiom,
semiri2026040879449505780visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_389,axiom,
real_V2191834092415804123ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_390,axiom,
real_V8037385150606011577_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_391,axiom,
topolo4211221413907600880p_mult @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_392,axiom,
topolo7287701948861334536_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_393,axiom,
topolo8386298272705272623_space @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_394,axiom,
semiri3467727345109120633visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_395,axiom,
real_V7819770556892013058_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_396,axiom,
real_V4867850818363320053vector @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_397,axiom,
cancel2418104881723323429up_add @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ouniformity_398,axiom,
topolo4638772830378233104ormity @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_399,axiom,
ring_15535105094025558882visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_400,axiom,
real_V7773925162809079976_field @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_401,axiom,
topolo6943815403480290642id_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_402,axiom,
cancel1802427076303600483id_add @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_403,axiom,
topological_t2_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___Nat_Osemiring__char__0_418,axiom,
semiring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_419,axiom,
ab_group_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_420,axiom,
field_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_421,axiom,
zero_neq_one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_422,axiom,
idom_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_423,axiom,
monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__divide_424,axiom,
idom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_425,axiom,
comm_ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_426,axiom,
monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_427,axiom,
semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_428,axiom,
semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_429,axiom,
group_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Omult__zero_430,axiom,
mult_zero @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_431,axiom,
comm_ring @ complex ).
thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_432,axiom,
neg_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_433,axiom,
ring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring_434,axiom,
semiring @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Oinverse_435,axiom,
inverse @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom_436,axiom,
semidom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ouminus_437,axiom,
uminus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1_438,axiom,
ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield_439,axiom,
field @ complex ).
thf(tcon_Complex_Ocomplex___Power_Opower_440,axiom,
power @ complex ).
thf(tcon_Complex_Ocomplex___Num_Onumeral_441,axiom,
numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ozero_442,axiom,
zero @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oplus_443,axiom,
plus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring_444,axiom,
ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom_445,axiom,
idom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oone_446,axiom,
one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Odvd_447,axiom,
dvd @ complex ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_448,axiom,
condit6923001295902523014norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_449,axiom,
counta4013691401010221786attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_450,axiom,
condit1219197933456340205attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_451,axiom,
counta3822494911875563373attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_452,axiom,
comple592849572758109894attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_453,axiom,
strict9044650504122735259up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_454,axiom,
strict7427464778891057005id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_455,axiom,
canoni5634975068530333245id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_456,axiom,
bounde4967611905675639751up_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_457,axiom,
bounde4346867609351753570nf_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder,axiom,
comple5582772986160207858norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_458,axiom,
linord4140545234300271783up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_459,axiom,
comple6319245703460814977attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_460,axiom,
linord181362715937106298miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_461,axiom,
semiri3467727345109120633visors @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_462,axiom,
ordere6658533253407199908up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_463,axiom,
ordere6911136660526730532id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_464,axiom,
ordere2520102378445227354miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_465,axiom,
comple9053668089753744459l_ccpo @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_466,axiom,
semilattice_sup @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_467,axiom,
semilattice_inf @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_468,axiom,
distrib_lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_469,axiom,
ab_semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_470,axiom,
comm_monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_471,axiom,
ab_semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_472,axiom,
ordered_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_473,axiom,
comm_monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_474,axiom,
comm_semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_475,axiom,
comm_semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_476,axiom,
semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_477,axiom,
semiring_numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_478,axiom,
semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_479,axiom,
zero_less_one @ 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,
! [A10: $tType,A16: $tType] :
( ( ( topolo4958980785337419405_space @ A10 )
& ( topolo4958980785337419405_space @ A16 ) )
=> ( topolo4958980785337419405_space @ ( product_prod @ A10 @ A16 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_505,axiom,
! [A10: $tType,A16: $tType] :
( ( ( topological_t2_space @ A10 )
& ( topological_t2_space @ A16 ) )
=> ( topological_t2_space @ ( product_prod @ A10 @ A16 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_506,axiom,
! [A10: $tType,A16: $tType] : ( size @ ( product_prod @ A10 @ A16 ) ) ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_507,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_508,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_509,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_510,axiom,
euclid8789492081693882211th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_511,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_512,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_513,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_514,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_515,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_516,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_517,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_518,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_519,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_520,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_521,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_522,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_523,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_524,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_525,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_526,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_527,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_528,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_529,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_530,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_531,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_532,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_533,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_534,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_535,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_536,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_537,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_538,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_539,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_540,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_541,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_542,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_543,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_544,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_545,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_546,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_547,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_548,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_549,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_550,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_551,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_552,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_553,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_554,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_555,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_556,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_557,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_558,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_559,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_560,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_561,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_562,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_563,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_564,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_565,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_566,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_567,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_568,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_569,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_570,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_571,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_572,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_573,axiom,
ring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_574,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_575,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_576,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_577,axiom,
idom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_578,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_579,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_580,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_581,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_582,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_583,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_584,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_585,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_586,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_587,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_588,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom_589,axiom,
semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_590,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_591,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_592,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_593,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_594,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_595,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_596,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_597,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_598,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_599,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_600,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_601,axiom,
dvd @ code_integer ).
thf(tcon_VEBT__Definitions_OVEBT___Nat_Osize_602,axiom,
size @ vEBT_VEBT ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X3: A,Y3: A] :
( ( if @ A @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y3: A] :
( ( if @ A @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X9: A] : ( P @ X9 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less @ nat @ x @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ deg ) ).
%------------------------------------------------------------------------------