TPTP Problem File: ITP292^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP292^2 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_Intf_Imperative 00161_004904
% 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 : 0096_VEBT_Intf_Imperative_00161_004904 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 10129 (2952 unt; 745 typ; 0 def)
% Number of atoms : 32012 (10664 equ; 2 cnn)
% Maximal formula atoms : 71 ( 3 avg)
% Number of connectives : 212844 (2873 ~; 354 |;2533 &;192215 @)
% ( 0 <=>;14869 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 8 avg)
% Number of types : 18 ( 17 usr)
% Number of type conns : 4384 (4384 >; 0 *; 0 +; 0 <<)
% Number of symbols : 733 ( 728 usr; 14 con; 0-9 aty)
% Number of variables : 31719 (2683 ^;27386 !;1032 ?;31719 :)
% ( 618 !>; 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 23:43:10.845
%------------------------------------------------------------------------------
% Could-be-implicit typings (32)
thf(ty_t_VEBT__BuildupMemImp_OVEBTi,type,
vEBT_VEBTi: $tType ).
thf(ty_t_VEBT__Definitions_OVEBT,type,
vEBT_VEBT: $tType ).
thf(ty_t_Heap__Time__Monad_OHeap,type,
heap_Time_Heap: $tType > $tType ).
thf(ty_t_Code__Numeral_Ointeger,type,
code_integer: $tType ).
thf(ty_t_Heap_Oheap_Oheap__ext,type,
heap_ext: $tType > $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_Numeral__Type_Onum1,type,
numeral_num1: $tType ).
thf(ty_t_Numeral__Type_Onum0,type,
numeral_num0: $tType ).
thf(ty_t_Numeral__Type_Obit1,type,
numeral_bit1: $tType > $tType ).
thf(ty_t_Numeral__Type_Obit0,type,
numeral_bit0: $tType > $tType ).
thf(ty_t_Extended__Nat_Oenat,type,
extended_enat: $tType ).
thf(ty_t_Multiset_Omultiset,type,
multiset: $tType > $tType ).
thf(ty_t_Complex_Ocomplex,type,
complex: $tType ).
thf(ty_t_Assertions_Oassn,type,
assn: $tType ).
thf(ty_t_String_Oliteral,type,
literal: $tType ).
thf(ty_t_Uint32_Ouint32,type,
uint32: $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_Heap_Oarray,type,
array: $tType > $tType ).
thf(ty_t_Word_Oword,type,
word: $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Heap_Oref,type,
ref: $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 (713)
thf(sy_cl_Type__Length_Olen0,type,
type_len0:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : $o ).
thf(sy_cl_Typerep_Otyperep,type,
typerep:
!>[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_Heap_Oheap,type,
heap:
!>[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_Groups_Oplus,type,
plus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Onumeral,type,
numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Otimes,type,
times:
!>[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_Rings_Osemidom,type,
semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Oinverse,type,
inverse:
!>[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_Type__Length_Olen,type,
type_len:
!>[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_Cardinality_Ocard2,type,
card2:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Transcendental_Oln,type,
ln:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__lattice,type,
finite_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Osemiring__numeral,type,
semiring_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__modulo,type,
semidom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__modulo,type,
semiring_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__parity,type,
semiring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring__abs,type,
ordered_ring_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Code__Evaluation_Oterm__of,type,
code_term_of:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Generic__set__bit_Oset__bit,type,
generic_set_set_bit:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oalgebraic__semidom,type,
algebraic_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__cancel,type,
semiring_1_cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Odistrib__lattice,type,
distrib_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Obanach,type,
real_Vector_banach:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring,type,
linordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Partial__Order_Occpo,type,
comple9053668089753744459l_ccpo:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__distrib__lattice,type,
finite8700451911770168679attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_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_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[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_Rings_Oring__1__no__zero__divisors,type,
ring_15535105094025558882visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel2418104881723323429up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord5086331880401160121up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere6911136660526730532id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
real_V4867850818363320053vector:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim2362893244070406136eiling:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere166539214618696060dd_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere6658533253407199908up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__ab__group__add,type,
topolo1287966508704411220up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
real_V7819770556892013058_space:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Operfect__space,type,
topolo8386298272705272623_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ouniform__space,type,
topolo7287701948861334536_space:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__semigroup__mult,type,
topolo4211221413907600880p_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ocomplete__space,type,
real_V8037385150606011577_space:
!>[A: $tType] : $o ).
thf(sy_cl_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_Bit__Comprehension_Obit__comprehension,type,
bit_bi6583157726757044596ension:
!>[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_Quickcheck__Narrowing_Opartial__term__of,type,
quickc6926020345158392990erm_of:
!>[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_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_Topological__Spaces_Olinear__continuum__topology,type,
topolo8458572112393995274pology:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere1937475149494474687imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit5016429287641298734tinuum:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
euclid5411537665997757685th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Countable__Complete__Lattices_Ocountable__complete__lattice,type,
counta3822494911875563373attice:
!>[A: $tType] : $o ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri1453513574482234551roduct:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit1219197933456340205attice:
!>[A: $tType] : $o ).
thf(sy_cl_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_Array__Time_Olen,type,
array_len:
!>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ nat ) ) ).
thf(sy_c_Array__Time_Omake,type,
array_make:
!>[A: $tType] : ( nat > ( nat > A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Omap__entry,type,
array_map_entry:
!>[A: $tType] : ( nat > ( A > A ) > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Onth,type,
array_nth:
!>[A: $tType] : ( ( array @ A ) > nat > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Array__Time_Oswap,type,
array_swap:
!>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Array__Time_Oupd,type,
array_upd:
!>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Assertions_Oassn_ORep__assn,type,
rep_assn: assn > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oentails,type,
entails: assn > assn > $o ).
thf(sy_c_Assertions_Oex__assn,type,
ex_assn:
!>[A: $tType] : ( ( A > assn ) > assn ) ).
thf(sy_c_Assertions_Oprecise,type,
precise:
!>[A: $tType,B: $tType] : ( ( A > B > assn ) > $o ) ).
thf(sy_c_Assertions_Opure__assn,type,
pure_assn: $o > assn ).
thf(sy_c_Assertions_Osnga__assn,type,
snga_assn:
!>[A: $tType] : ( ( array @ A ) > ( list @ A ) > assn ) ).
thf(sy_c_Assertions_Osngr__assn,type,
sngr_assn:
!>[A: $tType] : ( ( ref @ A ) > A > assn ) ).
thf(sy_c_Assertions_Otimes__assn__raw,type,
times_assn_raw: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Otimes__assn__raw__rel,type,
times_assn_raw_rel: ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $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_BNF__Wellorder__Constructions_OFunc,type,
bNF_Wellorder_Func:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( A > B ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc__map,type,
bNF_We4925052301507509544nc_map:
!>[B: $tType,C: $tType,A: $tType,D: $tType] : ( ( set @ B ) > ( C > A ) > ( B > D ) > ( D > C ) > B > A ) ).
thf(sy_c_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__Comprehension_Obit__comprehension__class_Oset__bits,type,
bit_bi4170147762399595738t_bits:
!>[A: $tType] : ( ( nat > $o ) > A ) ).
thf(sy_c_Bit__Comprehension_Owf__set__bits__int,type,
bit_wf_set_bits_int: ( nat > $o ) > $o ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : ( A > nat > $o ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
thf(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftl,type,
bit_Sh4282982442137083160shiftl:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftr,type,
bit_Sh4282982442137083166shiftr:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Shifts__Infix__Syntax_Osshiftr,type,
bit_Sh8784991116023147202shiftr:
!>[A: $tType] : ( ( word @ A ) > nat > ( word @ A ) ) ).
thf(sy_c_Bits__Integer_Ointeger__set__bit,type,
bits_integer_set_bit: code_integer > code_integer > $o > code_integer ).
thf(sy_c_Bits__Integer_Ointeger__shiftl,type,
bits_integer_shiftl: code_integer > code_integer > code_integer ).
thf(sy_c_Bits__Integer_Ointeger__shiftr,type,
bits_integer_shiftr: code_integer > code_integer > code_integer ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
boolea3799213064322606851m_diff:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
thf(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ).
thf(sy_c_Code__Numeral_Odivmod__abs,type,
code_divmod_abs: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Odivmod__integer,type,
code_divmod_integer: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
code_int_of_integer: code_integer > int ).
thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: int > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
code_integer_of_nat: nat > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__num,type,
code_integer_of_num: num > code_integer ).
thf(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
thf(sy_c_Code__Numeral_Onegative,type,
code_negative: num > code_integer ).
thf(sy_c_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: code_integer > num ).
thf(sy_c_Code__Numeral_Opositive,type,
code_positive: num > code_integer ).
thf(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Onarrowing__samples,type,
code_T4080844693773952564amples:
!>[A: $tType] : ( ( code_integer > ( product_prod @ A @ A ) ) > A > code_integer > ( list @ A ) ) ).
thf(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Onarrowing__samples__rel,type,
code_T1710151556404007877es_rel:
!>[A: $tType] : ( ( code_integer > ( product_prod @ A @ A ) ) > code_integer > code_integer > $o ) ).
thf(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Opartial__term__of__sample,type,
code_T4081349890594273596sample:
!>[A: $tType] : ( ( code_integer > ( product_prod @ A @ A ) ) > A > code_integer > A ) ).
thf(sy_c_Code__Target__Word__Base_Oset__bits__aux,type,
code_T2661198915054445665ts_aux:
!>[A: $tType] : ( ( nat > $o ) > nat > ( word @ A ) > ( word @ A ) ) ).
thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
complete_Inf_Inf:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complex_Ocomplex_OComplex,type,
complex2: real > real > complex ).
thf(sy_c_Deriv_Ohas__derivative,type,
has_derivative:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Deriv_Ohas__field__derivative,type,
has_field_derivative:
!>[A: $tType] : ( ( A > A ) > A > ( filter @ A ) > $o ) ).
thf(sy_c_Divides_Oadjust__div,type,
adjust_div: ( product_prod @ int @ int ) > int ).
thf(sy_c_Divides_Oadjust__mod,type,
adjust_mod: int > int > int ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Filter_Oat__bot,type,
at_bot:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oat__top,type,
at_top:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oeventually,type,
eventually:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofiltercomap,type,
filtercomap:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Omap__filter__on,type,
map_filter_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute,type,
finite6289374366891150609ommute:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
finite4664212375090638736ute_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
finite673082921795544331dem_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem__on__axioms,type,
finite4980608107308702382axioms:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_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_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Ostrict__mono__on,type,
strict_mono_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_GCD_OGcd__class_OGcd,type,
gcd_Gcd:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > ( product_prod @ int @ int ) ).
thf(sy_c_GCD_Obezw__rel,type,
bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Ogcd__class_Ogcd,type,
gcd_gcd:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__nat__rel,type,
gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_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_Generic__set__bit_Oset__bit__class_Oset__bit,type,
generi7602027413899671122et_bit:
!>[A: $tType] : ( A > nat > $o > A ) ).
thf(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum,type,
groups3894954378712506084id_sum:
!>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
groups7311177749621191930dd_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
groups1027152243600224163dd_sum:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
groups7121269368397514597t_prod:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
groups1962203154675924110t_prod:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
groups4207007520872428315er_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ).
thf(sy_c_Groups__List_Omonoid__add_Osum__list,type,
groups4543113879258116180m_list:
!>[A: $tType] : ( ( A > A > A ) > A > ( list @ A ) > A ) ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
groups8242544230860333062m_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_HOL_ONO__MATCH,type,
nO_MATCH:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_Hash__Instances_Ohash__code__prod,type,
hash_hash_code_prod:
!>[A: $tType,B: $tType] : ( ( A > uint32 ) > ( B > uint32 ) > ( product_prod @ A @ B ) > uint32 ) ).
thf(sy_c_Heap_Oarray_Osize__array,type,
size_array:
!>[A: $tType] : ( ( A > nat ) > ( array @ A ) > nat ) ).
thf(sy_c_Heap__Time__Monad_Oreturn,type,
heap_Time_return:
!>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Hoare__Triple_Ohoare__triple,type,
hoare_hoare_triple:
!>[A: $tType] : ( assn > ( heap_Time_Heap @ A ) > ( A > assn ) > $o ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
infini527867602293511546merate:
!>[A: $tType] : ( ( set @ A ) > nat > A ) ).
thf(sy_c_Int_Oint__ge__less__than,type,
int_ge_less_than: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osemilattice__neutr__order,type,
semila1105856199041335345_order:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices__Big_Olinorder_OMax,type,
lattices_Max:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder_OMin,type,
lattices_Min:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
lattic643756798350308766er_Min:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Limits_OBfun,type,
bfun:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Limits_Oat__infinity,type,
at_infinity:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_List_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_Oenumerate,type,
enumerate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
thf(sy_c_List_Ofilter,type,
filter2:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ofind,type,
find:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
thf(sy_c_List_Ofolding__insort__key,type,
folding_insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > ( set @ B ) > ( B > A ) > $o ) ).
thf(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( ( B > A > B ) > B > ( list @ A ) > B ) ).
thf(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexn,type,
lexn:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > nat > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olinorder_Oinsort__key,type,
insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
sorted8670434370408473282of_set:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( set @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__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_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist_Osize__list,type,
size_list:
!>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).
thf(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Onths,type,
nths:
!>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
thf(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_List_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_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_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_Ograph,type,
graph:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Map_Omap__add,type,
map_add:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Omap__of,type,
map_of:
!>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Omap__upds,type,
map_upds:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Oran,type,
ran:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Map_Orestrict__map,type,
restrict_map:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > A > ( option @ B ) ) ).
thf(sy_c_Misc_Obijective,type,
bijective:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
thf(sy_c_Misc_Obrk__rel,type,
brk_rel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) ) ) ) ).
thf(sy_c_Misc_Odflt__None__set,type,
dflt_None_set:
!>[A: $tType] : ( ( set @ A ) > ( option @ ( set @ A ) ) ) ).
thf(sy_c_Misc_Oinv__on,type,
inv_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > B > A ) ).
thf(sy_c_Misc_Olist__collect__set,type,
list_collect_set:
!>[B: $tType,A: $tType] : ( ( B > ( set @ A ) ) > ( list @ B ) > ( set @ A ) ) ).
thf(sy_c_Misc_Omap__to__set,type,
map_to_set:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Misc_Omerge__list,type,
merge_list:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omergesort,type,
mergesort:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omergesort__by__rel,type,
mergesort_by_rel:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omergesort__by__rel__merge,type,
merges9089515139780605204_merge:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omergesort__by__rel__merge__rel,type,
merges2244889521215249637ge_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > $o ) ).
thf(sy_c_Misc_Omergesort__by__rel__rel,type,
mergesort_by_rel_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_Misc_Omergesort__by__rel__split,type,
merges295452479951948502_split:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
thf(sy_c_Misc_Omergesort__by__rel__split__rel,type,
merges7066485432131860899it_rel:
!>[A: $tType] : ( ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) > ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_Misc_Omergesort__remdups,type,
mergesort_remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Opairself,type,
pairself:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( product_prod @ A @ A ) > ( product_prod @ B @ B ) ) ).
thf(sy_c_Misc_Opairself__rel,type,
pairself_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) > ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Misc_Orel__of,type,
rel_of:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( ( product_prod @ A @ B ) > $o ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Misc_Oremove__rev,type,
remove_rev:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Oset__to__map,type,
set_to_map:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( option @ A ) ) ).
thf(sy_c_Misc_Oslice,type,
slice:
!>[A: $tType] : ( nat > nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Othe__default,type,
the_default:
!>[A: $tType] : ( A > ( option @ A ) > A ) ).
thf(sy_c_Misc_Ouncurry,type,
uncurry:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Most__significant__bit_Omsb__class_Omsb,type,
most_s684356279273892711sb_msb:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Multiset_Oadd__mset,type,
add_mset:
!>[A: $tType] : ( A > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Olinorder__class_Opart,type,
linorder_part:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) ) ).
thf(sy_c_Multiset_Osubseteq__mset,type,
subseteq_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > $o ) ).
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_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri8178284476397505188at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > ( set @ nat ) ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: ( set @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_NthRoot_Oroot,type,
root: nat > real > real ).
thf(sy_c_NthRoot_Osqrt,type,
sqrt: real > real ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Numeral__Type_Omod__ring,type,
numeral_mod_ring:
!>[A: $tType] : ( int > ( A > int ) > ( int > A ) > $o ) ).
thf(sy_c_Numeral__Type_Omod__type,type,
numeral_mod_type:
!>[A: $tType] : ( int > ( A > int ) > ( int > A ) > $o ) ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
thf(sy_c_Option_Ooption_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_Order__Continuity_Ocountable__complete__lattice__class_Occlfp,type,
order_532582986084564980_cclfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : 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_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_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_Ointernal__case__prod,type,
produc5280177257484947105e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
product_rec_set_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Quicksort_Olinorder__class_Oquicksort,type,
linorder_quicksort:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Quicksort_Olinorder__class_Oquicksort__rel,type,
linord6200660962353139674rt_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_Rat_Onormalize,type,
normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
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_Odist__class_Odist,type,
real_V557655796197034286t_dist:
!>[A: $tType] : ( A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
real_V7770717601297561774m_norm:
!>[A: $tType] : ( A > real ) ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
real_V8093663219630862766scaleR:
!>[A: $tType] : ( real > A > A ) ).
thf(sy_c_Refine__Imp__Hol_Orefines,type,
refine_Imp_refines:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ A ) > $o ) ).
thf(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( ( set @ A ) > ( 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_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Series_Osuminf,type,
suminf:
!>[A: $tType] : ( ( nat > A ) > A ) ).
thf(sy_c_Series_Osummable,type,
summable:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Series_Osums,type,
sums:
!>[A: $tType] : ( ( nat > A ) > A > $o ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OBex,type,
bex:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Ofilter,type,
filter3:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Oremove,type,
remove:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
set_fo6178422350223883121st_nat:
!>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
set_fo1817059534552279752at_rel:
!>[A: $tType] : ( ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > $o ) ).
thf(sy_c_Set__Interval_Oord__class_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_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_Signed__Division_Osigned__division__class_Osigned__divide,type,
signed7115095781618012415divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Signed__Division_Osigned__division__class_Osigned__modulo,type,
signed6721504322012087516modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_Syntax__Match_Osyntax__fo__nomatch,type,
syntax7388354845996824322omatch:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_Time__Reasoning_OTBOUND,type,
time_TBOUND:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > nat > $o ) ).
thf(sy_c_Time__Reasoning_Ofails,type,
time_fails:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > $o ) ).
thf(sy_c_Time__Reasoning_Ohtt,type,
time_htt:
!>[A: $tType] : ( assn > ( heap_Time_Heap @ A ) > ( A > assn ) > nat > $o ) ).
thf(sy_c_Time__Reasoning_Otime,type,
time_time:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > nat ) ).
thf(sy_c_Topological__Spaces_Omonoseq,type,
topological_monoseq:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
topolo1002775350975398744n_open:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ot2__space__class_OLim,type,
topolo3827282254853284352ce_Lim:
!>[F: $tType,A: $tType] : ( ( filter @ F ) > ( F > A ) > A ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
topolo174197925503356063within:
!>[A: $tType] : ( A > ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type,
topolo7230453075368039082e_nhds:
!>[A: $tType] : ( A > ( filter @ A ) ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy,type,
topolo3814608138187158403Cauchy:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_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_Transcendental_Oarccos,type,
arccos: real > real ).
thf(sy_c_Transcendental_Oarcosh,type,
arcosh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oarcsin,type,
arcsin: real > real ).
thf(sy_c_Transcendental_Oarctan,type,
arctan: real > real ).
thf(sy_c_Transcendental_Oarsinh,type,
arsinh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oartanh,type,
artanh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Ocos,type,
cos:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Ocos__coeff,type,
cos_coeff: nat > real ).
thf(sy_c_Transcendental_Ocosh,type,
cosh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Ocot,type,
cot:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Odiffs,type,
diffs:
!>[A: $tType] : ( ( nat > A ) > nat > A ) ).
thf(sy_c_Transcendental_Oexp,type,
exp:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oln__class_Oln,type,
ln_ln:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Olog,type,
log: real > real > real ).
thf(sy_c_Transcendental_Opi,type,
pi: real ).
thf(sy_c_Transcendental_Opowr,type,
powr:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Transcendental_Osin,type,
sin:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Osin__coeff,type,
sin_coeff: nat > real ).
thf(sy_c_Transcendental_Osinh,type,
sinh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Otan,type,
tan:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Otanh,type,
tanh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Type__Length_Olen0__class_Olen__of,type,
type_len0_len_of:
!>[A: $tType] : ( ( itself @ A ) > nat ) ).
thf(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Uint32_Odiv0__uint32,type,
div0_uint32: uint32 > uint32 ).
thf(sy_c_Uint32_Omod0__uint32,type,
mod0_uint32: uint32 > uint32 ).
thf(sy_c_Uint32_Oset__bits__aux__uint32,type,
set_bits_aux_uint32: ( nat > $o ) > nat > uint32 > uint32 ).
thf(sy_c_Uint32_Osigned__drop__bit__uint32,type,
signed489701013188660438uint32: nat > uint32 > uint32 ).
thf(sy_c_Uint32_Ouint32_ORep__uint32,type,
rep_uint32: uint32 > ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) ).
thf(sy_c_Uint32_Ouint32__divmod,type,
uint32_divmod: uint32 > uint32 > ( product_prod @ uint32 @ uint32 ) ).
thf(sy_c_Uint32_Ouint32__sdiv,type,
uint32_sdiv: uint32 > uint32 > uint32 ).
thf(sy_c_Uint32_Ouint32__set__bit,type,
uint32_set_bit: uint32 > code_integer > $o > uint32 ).
thf(sy_c_Uint32_Ouint32__shiftl,type,
uint32_shiftl: uint32 > code_integer > uint32 ).
thf(sy_c_Uint32_Ouint32__shiftr,type,
uint32_shiftr: uint32 > code_integer > uint32 ).
thf(sy_c_Uint32_Ouint32__sshiftr,type,
uint32_sshiftr: uint32 > code_integer > uint32 ).
thf(sy_c_Uint32_Ouint32__test__bit,type,
uint32_test_bit: uint32 > code_integer > $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_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
vEBT_T_m_a_x_t_rel: vEBT_VEBT > vEBT_VEBT > $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,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_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
vEBT_T5462971552011256508_l_rel: vEBT_VEBT > vEBT_VEBT > $o ).
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__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
vEBT_V441764108873111860ildupi: nat > nat ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H,type,
vEBT_V9176841429113362141ildupi: nat > int ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
vEBT_V3352910403632780892pi_rel: nat > nat > $o ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
vEBT_V2957053500504383685pi_rel: nat > nat > $o ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
vEBT_VEBT_Tb: nat > int ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
vEBT_VEBT_Tb2: nat > nat ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
vEBT_VEBT_Tb_rel: nat > nat > $o ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
vEBT_VEBT_Tb_rel2: nat > nat > $o ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ohighi,type,
vEBT_VEBT_highi: nat > nat > ( heap_Time_Heap @ nat ) ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Olowi,type,
vEBT_VEBT_lowi: nat > nat > ( heap_Time_Heap @ nat ) ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli,type,
vEBT_VEBT_minNulli: vEBT_VEBTi > ( heap_Time_Heap @ $o ) ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei,type,
vEBT_VEBT_replicatei:
!>[A: $tType] : ( nat > ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ ( list @ A ) ) ) ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H,type,
vEBT_V739175172307565963ildupi: nat > ( heap_Time_Heap @ vEBT_VEBTi ) ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__inserti_H,type,
vEBT_V3964819847710782039nserti: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ vEBT_VEBTi ) ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__memberi_H,type,
vEBT_V854960066525838166emberi: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ $o ) ).
thf(sy_c_VEBT__BuildupMemImp_OVEBTi_OLeafi,type,
vEBT_Leafi: $o > $o > vEBT_VEBTi ).
thf(sy_c_VEBT__BuildupMemImp_OVEBTi_ONodei,type,
vEBT_Nodei: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > vEBT_VEBTi ).
thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi,type,
vEBT_case_VEBTi:
!>[A: $tType] : ( ( ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > A ) > ( $o > $o > A ) > vEBT_VEBTi > A ) ).
thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Osize__VEBTi,type,
vEBT_size_VEBTi: vEBT_VEBTi > nat ).
thf(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw,type,
vEBT_vebt_assn_raw: vEBT_VEBT > vEBT_VEBTi > assn ).
thf(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw__rel,type,
vEBT_v8524038756793281170aw_rel: ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) > ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) > $o ).
thf(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi,type,
vEBT_vebt_buildupi: nat > ( heap_Time_Heap @ vEBT_VEBTi ) ).
thf(sy_c_VEBT__BuildupMemImp_Ovebt__inserti,type,
vEBT_vebt_inserti: vEBT_VEBTi > nat > ( heap_Time_Heap @ vEBT_VEBTi ) ).
thf(sy_c_VEBT__BuildupMemImp_Ovebt__maxti,type,
vEBT_vebt_maxti: vEBT_VEBTi > ( heap_Time_Heap @ ( option @ nat ) ) ).
thf(sy_c_VEBT__BuildupMemImp_Ovebt__maxti__rel,type,
vEBT_vebt_maxti_rel: vEBT_VEBTi > vEBT_VEBTi > $o ).
thf(sy_c_VEBT__BuildupMemImp_Ovebt__memberi,type,
vEBT_vebt_memberi: vEBT_VEBTi > nat > ( heap_Time_Heap @ $o ) ).
thf(sy_c_VEBT__BuildupMemImp_Ovebt__minti,type,
vEBT_vebt_minti: vEBT_VEBTi > ( heap_Time_Heap @ ( option @ nat ) ) ).
thf(sy_c_VEBT__BuildupMemImp_Ovebt__minti__rel,type,
vEBT_vebt_minti_rel: vEBT_VEBTi > vEBT_VEBTi > $o ).
thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
vEBT_Node: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_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__DelImperative_OVEBT__internal_Ovebt__deletei_H,type,
vEBT_V1365221501068881998eletei: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ vEBT_VEBTi ) ).
thf(sy_c_VEBT__DelImperative_Ovebt__deletei,type,
vEBT_vebt_deletei: vEBT_VEBTi > nat > ( heap_Time_Heap @ vEBT_VEBTi ) ).
thf(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
vEBT_T_d_e_l_e_t_e: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e__rel,type,
vEBT_T8441311223069195367_e_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
vEBT_V1232361888498592333_e_t_e: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
vEBT_V6368547301243506412_e_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Delete_Ovebt__delete,type,
vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT ).
thf(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
vEBT_vebt_delete_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
vEBT_VEBT_height: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
vEBT_VEBT_height_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H,type,
vEBT_VEBT_insert: vEBT_VEBT > nat > vEBT_VEBT ).
thf(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H__rel,type,
vEBT_VEBT_insert_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Insert_Ovebt__insert,type,
vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
vEBT_vebt_insert_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Intf__Imperative_Ovebt__assn,type,
vEBT_Intf_vebt_assn: nat > ( set @ nat ) > vEBT_VEBTi > assn ).
thf(sy_c_VEBT__List__Assn_OlistI__assn,type,
vEBT_List_listI_assn:
!>[A: $tType,B: $tType] : ( ( set @ nat ) > ( A > B > assn ) > ( list @ A ) > ( list @ B ) > assn ) ).
thf(sy_c_VEBT__List__Assn_Olist__assn,type,
vEBT_List_list_assn:
!>[A: $tType,C: $tType] : ( ( A > C > assn ) > ( list @ A ) > ( list @ C ) > assn ) ).
thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ).
thf(sy_c_VEBT__Member_Ovebt__member,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
vEBT_vebt_member_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
vEBT_VEBT_add: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
vEBT_VEBT_greater: ( option @ nat ) > ( option @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
vEBT_VEBT_less: ( option @ nat ) > ( option @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
vEBT_VEBT_lesseq: ( option @ nat ) > ( option @ nat ) > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
vEBT_VEBT_max_in_set: ( set @ nat ) > nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
vEBT_VEBT_min_in_set: ( set @ nat ) > nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
vEBT_VEBT_mul: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift,type,
vEBT_V2048590022279873568_shift:
!>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
vEBT_VEBT_power: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
vEBT_vebt_maxt: vEBT_VEBT > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
vEBT_vebt_mint: vEBT_VEBT > ( option @ nat ) ).
thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
vEBT_is_pred_in_set: ( set @ nat ) > nat > nat > $o ).
thf(sy_c_VEBT__Pred_Ovebt__pred,type,
vEBT_vebt_pred: vEBT_VEBT > nat > ( option @ nat ) ).
thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
vEBT_vebt_pred_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
vEBT_V8646137997579335489_i_l_d: nat > nat ).
thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
vEBT_V8346862874174094_d_u_p: nat > nat ).
thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
vEBT_V1247956027447740395_p_rel: nat > nat > $o ).
thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
vEBT_V5144397997797733112_d_rel: nat > nat > $o ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
vEBT_VEBT_cnt: vEBT_VEBT > real ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
vEBT_VEBT_cnt2: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
vEBT_VEBT_cnt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
vEBT_VEBT_cnt_rel2: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
vEBT_VEBT_space: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
vEBT_VEBT_space2: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
vEBT_VEBT_space_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
vEBT_VEBT_space_rel2: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__predi_H,type,
vEBT_VEBT_vebt_predi: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) ) ).
thf(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__succi_H,type,
vEBT_VEBT_vebt_succi: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) ) ).
thf(sy_c_VEBT__SuccPredImperative_Ovebt__predi,type,
vEBT_vebt_predi: vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) ) ).
thf(sy_c_VEBT__SuccPredImperative_Ovebt__succi,type,
vEBT_vebt_succi: vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) ) ).
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_VEBT__Uniqueness_OVEBT__internal_OperInsTrans,type,
vEBT_V6289311342943941716sTrans: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Wellfounded_Ofinite__psubset,type,
finite_psubset:
!>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ).
thf(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__ext,type,
max_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__extp,type,
max_extp:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Omin__ext,type,
min_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omlex__prod,type,
mlex_prod:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Word_OWord,type,
word2:
!>[A: $tType] : ( int > ( word @ A ) ) ).
thf(sy_c_Word_Ois__down,type,
is_down:
!>[A: $tType,B: $tType] : ( ( ( word @ A ) > ( word @ B ) ) > $o ) ).
thf(sy_c_Word_Ois__up,type,
is_up:
!>[A: $tType,B: $tType] : ( ( ( word @ A ) > ( word @ B ) ) > $o ) ).
thf(sy_c_Word_Opcr__word,type,
pcr_word:
!>[B: $tType] : ( int > ( word @ B ) > $o ) ).
thf(sy_c_Word_Orevcast,type,
revcast:
!>[A: $tType,B: $tType] : ( ( word @ A ) > ( word @ B ) ) ).
thf(sy_c_Word_Oring__1__class_Osigned,type,
ring_1_signed:
!>[B: $tType,A: $tType] : ( ( word @ B ) > A ) ).
thf(sy_c_Word_Osemiring__1__class_Ounsigned,type,
semiring_1_unsigned:
!>[B: $tType,A: $tType] : ( ( word @ B ) > A ) ).
thf(sy_c_Word_Osigned__cast,type,
signed_cast:
!>[A: $tType,B: $tType] : ( ( word @ A ) > ( word @ B ) ) ).
thf(sy_c_Word_Osigned__drop__bit,type,
signed_drop_bit:
!>[A: $tType] : ( nat > ( word @ A ) > ( word @ A ) ) ).
thf(sy_c_Word_Oslice,type,
slice2:
!>[A: $tType,B: $tType] : ( nat > ( word @ A ) > ( word @ B ) ) ).
thf(sy_c_Word_Oslice1,type,
slice1:
!>[A: $tType,B: $tType] : ( nat > ( word @ A ) > ( word @ B ) ) ).
thf(sy_c_Word_Othe__signed__int,type,
the_signed_int:
!>[A: $tType] : ( ( word @ A ) > int ) ).
thf(sy_c_Word_Oudvd,type,
udvd:
!>[A: $tType] : ( ( word @ A ) > ( word @ A ) > $o ) ).
thf(sy_c_Word_Oword__cat,type,
word_cat:
!>[A: $tType,B: $tType,C: $tType] : ( ( word @ A ) > ( word @ B ) > ( word @ C ) ) ).
thf(sy_c_Word_Oword__int__case,type,
word_int_case:
!>[B: $tType,A: $tType] : ( ( int > B ) > ( word @ A ) > B ) ).
thf(sy_c_Word_Oword__rcat,type,
word_rcat:
!>[A: $tType,B: $tType] : ( ( list @ ( word @ A ) ) > ( word @ B ) ) ).
thf(sy_c_Word_Oword__reverse,type,
word_reverse:
!>[A: $tType] : ( ( word @ A ) > ( word @ A ) ) ).
thf(sy_c_Word_Oword__roti,type,
word_roti:
!>[A: $tType] : ( int > ( word @ A ) > ( word @ A ) ) ).
thf(sy_c_Word_Oword__rotl,type,
word_rotl:
!>[A: $tType] : ( nat > ( word @ A ) > ( word @ A ) ) ).
thf(sy_c_Word_Oword__rotr,type,
word_rotr:
!>[A: $tType] : ( nat > ( word @ A ) > ( word @ A ) ) ).
thf(sy_c_Word_Oword__sle,type,
word_sle:
!>[A: $tType] : ( ( word @ A ) > ( word @ A ) > $o ) ).
thf(sy_c_Word_Oword__sless,type,
word_sless:
!>[A: $tType] : ( ( word @ A ) > ( word @ A ) > $o ) ).
thf(sy_c_Word_Oword__split,type,
word_split:
!>[A: $tType,B: $tType,C: $tType] : ( ( word @ A ) > ( product_prod @ ( word @ B ) @ ( word @ C ) ) ) ).
thf(sy_c_Word_Oword__succ,type,
word_succ:
!>[A: $tType] : ( ( word @ A ) > ( word @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_a,type,
a: heap_ext @ product_unit ).
thf(sy_v_b,type,
b: set @ nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_r,type,
r: vEBT_VEBTi ).
thf(sy_v_s,type,
s: set @ nat ).
thf(sy_v_t,type,
t: vEBT_VEBT ).
thf(sy_v_x,type,
x: nat ).
thf(sy_v_xa,type,
xa: nat ).
% Relevant facts (8187)
thf(fact_0_invar__vebt__delete,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T2 @ X ) @ N ) ) ).
% invar_vebt_delete
thf(fact_1_set__vebt__equal,axiom,
! [T_1: vEBT_VEBT,N: nat,T_2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T_1 @ N )
=> ( ( vEBT_invar_vebt @ T_2 @ N )
=> ( ( T_1 = T_2 )
= ( ( vEBT_set_vebt @ T_1 )
= ( vEBT_set_vebt @ T_2 ) ) ) ) ) ).
% set_vebt_equal
thf(fact_2_unique__tree,axiom,
! [T2: vEBT_VEBT,N: nat,S: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_invar_vebt @ S @ N )
=> ( ( ( vEBT_set_vebt @ T2 )
= ( vEBT_set_vebt @ S ) )
=> ( S = T2 ) ) ) ) ).
% unique_tree
thf(fact_3_insert_H__pres__valid,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( vEBT_invar_vebt @ ( vEBT_VEBT_insert @ T2 @ X ) @ N ) ) ).
% insert'_pres_valid
thf(fact_4_perIT__concat,axiom,
! [S: vEBT_VEBT,T2: vEBT_VEBT,U: vEBT_VEBT] :
( ( vEBT_V6289311342943941716sTrans @ S @ T2 )
=> ( ( vEBT_V6289311342943941716sTrans @ T2 @ U )
=> ( vEBT_V6289311342943941716sTrans @ S @ U ) ) ) ).
% perIT_concat
thf(fact_5_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_6_uniquetree,axiom,
! [T2: vEBT_VEBT,N: nat,S: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_invar_vebt @ S @ N )
=> ( ( ( vEBT_VEBT_set_vebt @ T2 )
= ( vEBT_VEBT_set_vebt @ S ) )
=> ( S = T2 ) ) ) ) ).
% uniquetree
thf(fact_7_Rep__assn__inject,axiom,
! [X: assn,Y: assn] :
( ( ( rep_assn @ X )
= ( rep_assn @ Y ) )
= ( X = Y ) ) ).
% Rep_assn_inject
thf(fact_8_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X2 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_9_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_10_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_11_valid__eq1,axiom,
! [T2: vEBT_VEBT,D2: nat] :
( ( vEBT_invar_vebt @ T2 @ D2 )
=> ( vEBT_VEBT_valid @ T2 @ D2 ) ) ).
% valid_eq1
thf(fact_12_valid__eq2,axiom,
! [T2: vEBT_VEBT,D2: nat] :
( ( vEBT_VEBT_valid @ T2 @ D2 )
=> ( vEBT_invar_vebt @ T2 @ D2 ) ) ).
% valid_eq2
thf(fact_13_set__vebt__member,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_vebt_member @ T2 @ X )
= ( member @ nat @ X @ ( vEBT_set_vebt @ T2 ) ) ) ) ).
% set_vebt_member
thf(fact_14_dele__member__cont__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T2 @ X ) @ Y )
= ( ( X != Y )
& ( vEBT_vebt_member @ T2 @ Y ) ) ) ) ).
% dele_member_cont_corr
thf(fact_15_dele__bmo__cont__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T2 @ X ) @ Y )
= ( ( X != Y )
& ( vEBT_V8194947554948674370ptions @ T2 @ Y ) ) ) ) ).
% dele_bmo_cont_corr
thf(fact_16_valid__member__both__member__options,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X )
=> ( vEBT_vebt_member @ T2 @ X ) ) ) ).
% valid_member_both_member_options
thf(fact_17_both__member__options__equiv__member,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X )
= ( vEBT_vebt_member @ T2 @ X ) ) ) ).
% both_member_options_equiv_member
thf(fact_18_snga__assn__raw_Ocases,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
~ ! [R: array @ A,X3: list @ A,H: heap_ext @ product_unit,As: set @ nat] :
( X
!= ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) ) ) ).
% snga_assn_raw.cases
thf(fact_19_sngr__assn__raw_Ocases,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
~ ! [R: ref @ A,X3: A,H: heap_ext @ product_unit,As: set @ nat] :
( X
!= ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) ) ) ).
% sngr_assn_raw.cases
thf(fact_20_times__assn__raw_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
~ ! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H: heap_ext @ product_unit,As: set @ nat] :
( X
!= ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ P @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) ) ).
% times_assn_raw.cases
thf(fact_21_pure__assn__raw_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ $o @ ( product_prod @ A @ ( set @ B ) )] :
~ ! [B4: $o,H: A,As: set @ B] :
( X
!= ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ B4 @ ( product_Pair @ A @ ( set @ B ) @ H @ As ) ) ) ).
% pure_assn_raw.cases
thf(fact_22_prod__induct7,axiom,
! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
( ! [A4: A,B4: B,C2: C,D3: D,E2: E,F2: F,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct7
thf(fact_23_prod__induct6,axiom,
! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
( ! [A4: A,B4: B,C2: C,D3: D,E2: E,F2: F] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D3 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct6
thf(fact_24_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A4: A,B4: B,C2: C,D3: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D3 @ E2 ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct5
thf(fact_25_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A4: A,B4: B,C2: C,D3: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D3 ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct4
thf(fact_26_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A4: A,B4: B,C2: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
=> ( P2 @ X ) ) ).
% prod_induct3
thf(fact_27_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
~ ! [A4: A,B4: B,C2: C,D3: D,E2: E,F2: F,G2: G] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_28_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
~ ! [A4: A,B4: B,C2: C,D3: D,E2: E,F2: F] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D3 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_29_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A4: A,B4: B,C2: C,D3: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D3 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_30_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A4: A,B4: B,C2: C,D3: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D3 ) ) ) ) ).
% prod_cases4
thf(fact_31_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B4: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_32_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_33_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P3: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P2 @ P3 ) ) ).
% prod_cases
thf(fact_34_surj__pair,axiom,
! [A: $tType,B: $tType,P3: product_prod @ A @ B] :
? [X3: A,Y3: B] :
( P3
= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_35_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A4: A,B4: B] :
( Y
!= ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_36_one__assn__raw_Ocases,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( X
!= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ).
% one_assn_raw.cases
thf(fact_37_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_38_set__vebt__finite,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( finite_finite2 @ nat @ ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).
% set_vebt_finite
thf(fact_39_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_40_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_41_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q2 ) ) ) ).
% Collect_cong
thf(fact_42_HOL_Oext,axiom,
! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
( ! [X3: A] :
( ( F3 @ X3 )
= ( G3 @ X3 ) )
=> ( F3 = G3 ) ) ).
% HOL.ext
thf(fact_43_VEBT__internal_OperInsTrans_Ointros_I1_J,axiom,
! [T2: vEBT_VEBT] : ( vEBT_V6289311342943941716sTrans @ T2 @ T2 ) ).
% VEBT_internal.perInsTrans.intros(1)
thf(fact_44_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
( ( produc5280177257484947105e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C3 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_45_valid__tree__deg__neq__0,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_tree_deg_neq_0
thf(fact_46_valid__0__not,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_0_not
thf(fact_47_min__Null__member,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_minNull @ T2 )
=> ~ ( vEBT_vebt_member @ T2 @ X ) ) ).
% min_Null_member
thf(fact_48_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_49_deg__deg__n,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( Deg = N ) ) ).
% deg_deg_n
thf(fact_50_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S: B,R3: set @ ( product_prod @ A @ B ),S2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_51_bex2I,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,S3: set @ ( product_prod @ A @ B ),P2: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S3 )
=> ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S3 )
=> ( P2 @ A2 @ B2 ) )
=> ? [A4: A,B4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ S3 )
& ( P2 @ A4 @ B4 ) ) ) ) ).
% bex2I
thf(fact_52_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
~ ! [F2: A > B,A4: A,B4: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F2 @ ( product_Pair @ A @ A @ A4 @ B4 ) ) ) ).
% pairself.cases
thf(fact_53_set__vebt__minNull,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_minNull @ T2 )
= ( ( vEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% set_vebt_minNull
thf(fact_54_VEBT_Oinject_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT,Y11: option @ ( product_prod @ nat @ nat ),Y12: nat,Y13: list @ vEBT_VEBT,Y14: vEBT_VEBT] :
( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
= ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 )
& ( X13 = Y13 )
& ( X14 = Y14 ) ) ) ).
% VEBT.inject(1)
thf(fact_55_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_finite2 @ A )
= ( ^ [A6: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_56_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_57_deg__SUcn__Node,axiom,
! [Tree: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
=> ? [Info2: option @ ( product_prod @ nat @ nat ),TreeList2: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( Tree
= ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList2 @ S4 ) ) ) ).
% deg_SUcn_Node
thf(fact_58_Leaf__0__not,axiom,
! [A2: $o,B2: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) ) ).
% Leaf_0_not
thf(fact_59_uncurry__apply,axiom,
! [B: $tType,A: $tType,C: $tType,F3: B > C > A,A2: B,B2: C] :
( ( uncurry @ B @ C @ A @ F3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( F3 @ A2 @ B2 ) ) ).
% uncurry_apply
thf(fact_60_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_61_finite__set__choice,axiom,
! [B: $tType,A: $tType,A5: set @ A,P2: A > B > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ? [X_12: B] : ( P2 @ X3 @ X_12 ) )
=> ? [F2: A > B] :
! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( P2 @ X5 @ ( F2 @ X5 ) ) ) ) ) ).
% finite_set_choice
thf(fact_62_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A5: set @ A] : ( finite_finite2 @ A @ A5 ) ) ).
% finite
thf(fact_63_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_64_VEBT_Oinject_I2_J,axiom,
! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
( ( ( vEBT_Leaf @ X21 @ X22 )
= ( vEBT_Leaf @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% VEBT.inject(2)
thf(fact_65_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_66_succ__none__empty,axiom,
! [Xs: set @ nat,A2: nat] :
( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs @ A2 @ X_1 )
=> ( ( finite_finite2 @ nat @ Xs )
=> ~ ? [X5: nat] :
( ( member @ nat @ X5 @ Xs )
& ( ord_less @ nat @ A2 @ X5 ) ) ) ) ).
% succ_none_empty
thf(fact_67_pred__none__empty,axiom,
! [Xs: set @ nat,A2: nat] :
( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs @ A2 @ X_1 )
=> ( ( finite_finite2 @ nat @ Xs )
=> ~ ? [X5: nat] :
( ( member @ nat @ X5 @ Xs )
& ( ord_less @ nat @ X5 @ A2 ) ) ) ) ).
% pred_none_empty
thf(fact_68_memb__imp__not__empty,axiom,
! [A: $tType,X: A,S3: set @ A] :
( ( member @ A @ X @ S3 )
=> ( S3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% memb_imp_not_empty
thf(fact_69_set__notEmptyE,axiom,
! [A: $tType,S3: set @ A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
~ ( member @ A @ X3 @ S3 ) ) ).
% set_notEmptyE
thf(fact_70_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_71_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_72_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_73_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_74_VEBT_Odistinct_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT,X21: $o,X22: $o] :
( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
!= ( vEBT_Leaf @ X21 @ X22 ) ) ).
% VEBT.distinct(1)
thf(fact_75_VEBT_Oexhaust,axiom,
! [Y: vEBT_VEBT] :
( ! [X112: option @ ( product_prod @ nat @ nat ),X122: nat,X132: list @ vEBT_VEBT,X142: vEBT_VEBT] :
( Y
!= ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X212: $o,X222: $o] :
( Y
!= ( vEBT_Leaf @ X212 @ X222 ) ) ) ).
% VEBT.exhaust
thf(fact_76_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_77_infinite__imp__nonempty,axiom,
! [A: $tType,S3: set @ A] :
( ~ ( finite_finite2 @ A @ S3 )
=> ( S3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_78_VEBT__internal_Onaive__member_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A4: $o,B4: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X3 ) )
=> ( ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Ux ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V: nat,TreeList2: list @ vEBT_VEBT,S4: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) @ X3 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_79_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_80_mod__h__bot__indep,axiom,
! [P2: assn,H2: heap_ext @ product_unit,H3: heap_ext @ product_unit] :
( ( rep_assn @ P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( rep_assn @ P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_indep
thf(fact_81_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_82_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_83_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_84_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_85_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_86_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_87_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_88_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_89_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_90_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_91_invar__vebt__buildup,axiom,
! [N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% invar_vebt_buildup
thf(fact_92_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_93_T__vebt__buildupi__gq__0,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).
% T_vebt_buildupi_gq_0
thf(fact_94_set__vebt__buildup,axiom,
! [I: nat] :
( ( vEBT_set_vebt @ ( vEBT_vebt_buildup @ I ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_vebt_buildup
thf(fact_95_buildup__gives__empty,axiom,
! [N: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% buildup_gives_empty
thf(fact_96_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_97_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_98_obtain__set__pred,axiom,
! [Z: nat,X: nat,A5: set @ nat] :
( ( ord_less @ nat @ Z @ X )
=> ( ( vEBT_VEBT_min_in_set @ A5 @ Z )
=> ( ( finite_finite2 @ nat @ A5 )
=> ? [X_1: nat] : ( vEBT_is_pred_in_set @ A5 @ X @ X_1 ) ) ) ) ).
% obtain_set_pred
thf(fact_99_obtain__set__succ,axiom,
! [X: nat,Z: nat,A5: set @ nat,B5: set @ nat] :
( ( ord_less @ nat @ X @ Z )
=> ( ( vEBT_VEBT_max_in_set @ A5 @ Z )
=> ( ( finite_finite2 @ nat @ B5 )
=> ( ( A5 = B5 )
=> ? [X_1: nat] : ( vEBT_is_succ_in_set @ A5 @ X @ X_1 ) ) ) ) ) ).
% obtain_set_succ
thf(fact_100_finite__psubset__induct,axiom,
! [A: $tType,A5: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [A7: set @ A] :
( ( finite_finite2 @ A @ A7 )
=> ( ! [B6: set @ A] :
( ( ord_less @ ( set @ A ) @ B6 @ A7 )
=> ( P2 @ B6 ) )
=> ( P2 @ A7 ) ) )
=> ( P2 @ A5 ) ) ) ).
% finite_psubset_induct
thf(fact_101_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ ( zero_zero @ nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_102_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P2: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ B @ ( F3 @ Y4 ) @ ( F3 @ X3 ) )
=> ( P2 @ Y4 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A2 ) ) ) ).
% measure_induct_rule
thf(fact_103_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P2: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ B @ ( F3 @ Y4 ) @ ( F3 @ X3 ) )
=> ( P2 @ Y4 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A2 ) ) ) ).
% measure_induct
thf(fact_104_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_105_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_106_infinite__descent__measure,axiom,
! [A: $tType,P2: A > $o,V2: A > nat,X: A] :
( ! [X3: A] :
( ~ ( P2 @ X3 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V2 @ Y4 ) @ ( V2 @ X3 ) )
& ~ ( P2 @ Y4 ) ) )
=> ( P2 @ X ) ) ).
% infinite_descent_measure
thf(fact_107_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less @ nat @ X @ Y )
=> ( ord_less @ nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_108_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P2 @ N2 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ~ ( P2 @ M2 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_109_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( P2 @ M2 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_110_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_111_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_112_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_113_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_114_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_115_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_116_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_117_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_118_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_119_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_120_zero__induct,axiom,
! [P2: nat > $o,K: nat] :
( ( P2 @ K )
=> ( ! [N2: nat] :
( ( P2 @ ( suc @ N2 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_121_diff__induct,axiom,
! [P2: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P2 @ X3 @ ( zero_zero @ nat ) )
=> ( ! [Y3: nat] : ( P2 @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P2 @ X3 @ Y3 )
=> ( P2 @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P2 @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_122_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_123_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_124_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_125_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_126_old_Onat_Odistinct_I2_J,axiom,
! [Nat4: nat] :
( ( suc @ Nat4 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_127_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( ( zero_zero @ nat )
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_128_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_129_strict__inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P2 @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( P2 @ ( suc @ I2 ) )
=> ( P2 @ I2 ) ) )
=> ( P2 @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_130_less__Suc__induct,axiom,
! [I: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I2: nat] : ( P2 @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K2 )
=> ( ( P2 @ I2 @ J2 )
=> ( ( P2 @ J2 @ K2 )
=> ( P2 @ I2 @ K2 ) ) ) ) )
=> ( P2 @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_131_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_132_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_133_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_134_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less @ nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_135_Nat_OAll__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ N )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( P2 @ I3 ) ) ) ) ).
% Nat.All_less_Suc
thf(fact_136_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_137_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_138_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N ) )
& ( P2 @ I3 ) ) )
= ( ( P2 @ N )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
& ( P2 @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_139_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_140_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_141_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_142_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_143_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_144_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_145_infinite__descent0__measure,axiom,
! [A: $tType,V2: A > nat,P2: A > $o,X: A] :
( ! [X3: A] :
( ( ( V2 @ X3 )
= ( zero_zero @ nat ) )
=> ( P2 @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V2 @ X3 ) )
=> ( ~ ( P2 @ X3 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V2 @ Y4 ) @ ( V2 @ X3 ) )
& ~ ( P2 @ Y4 ) ) ) )
=> ( P2 @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_146_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( P2 @ N2 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ~ ( P2 @ M2 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_147_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_148_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_149_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_150_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_151_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_152_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_153_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_154_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N3 )
=> ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ N3 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_155_Ex__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N ) )
& ( P2 @ I3 ) ) )
= ( ( P2 @ ( zero_zero @ nat ) )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
& ( P2 @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_156_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_157_All__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ ( zero_zero @ nat ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( P2 @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_158_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_159_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_160_vebt__delete_Osimps_I3_J,axiom,
! [A2: $o,B2: $o,N: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ N ) ) )
= ( vEBT_Leaf @ A2 @ B2 ) ) ).
% vebt_delete.simps(3)
thf(fact_161_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_162_succ__member,axiom,
! [T2: vEBT_VEBT,X: nat,Y: nat] :
( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X @ Y )
= ( ( vEBT_vebt_member @ T2 @ Y )
& ( ord_less @ nat @ X @ Y )
& ! [Z2: nat] :
( ( ( vEBT_vebt_member @ T2 @ Z2 )
& ( ord_less @ nat @ X @ Z2 ) )
=> ( ord_less_eq @ nat @ Y @ Z2 ) ) ) ) ).
% succ_member
thf(fact_163_pred__member,axiom,
! [T2: vEBT_VEBT,X: nat,Y: nat] :
( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X @ Y )
= ( ( vEBT_vebt_member @ T2 @ Y )
& ( ord_less @ nat @ Y @ X )
& ! [Z2: nat] :
( ( ( vEBT_vebt_member @ T2 @ Z2 )
& ( ord_less @ nat @ Z2 @ X ) )
=> ( ord_less_eq @ nat @ Z2 @ Y ) ) ) ) ).
% pred_member
thf(fact_164_buildup__nothing__in__leaf,axiom,
! [N: nat,X: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_leaf
thf(fact_165_VEBT__internal_OT__vebt__buildupi_Osimps_I2_J,axiom,
( ( vEBT_V441764108873111860ildupi @ ( suc @ ( zero_zero @ nat ) ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% VEBT_internal.T_vebt_buildupi.simps(2)
thf(fact_166_VEBT__internal_OT__vebt__buildupi_Osimps_I1_J,axiom,
( ( vEBT_V441764108873111860ildupi @ ( zero_zero @ nat ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% VEBT_internal.T_vebt_buildupi.simps(1)
thf(fact_167_VEBT__internal_Oinsert_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A4: $o,B4: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X3 ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ X3 ) ) ) ).
% VEBT_internal.insert'.cases
thf(fact_168_forall__finite_I3_J,axiom,
! [X: nat,P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ ( suc @ X ) ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ ( zero_zero @ nat ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ X ) )
=> ( P2 @ ( suc @ I3 ) ) ) ) ) ).
% forall_finite(3)
thf(fact_169_forall__finite_I2_J,axiom,
! [P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ ( zero_zero @ nat ) ) )
=> ( P2 @ I3 ) ) )
= ( P2 @ ( zero_zero @ nat ) ) ) ).
% forall_finite(2)
thf(fact_170_Comparator__Generator_OAll__less__Suc,axiom,
! [X: nat,P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ X ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ ( zero_zero @ nat ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ X )
=> ( P2 @ ( suc @ I3 ) ) ) ) ) ).
% Comparator_Generator.All_less_Suc
thf(fact_171_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X6: set @ A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ X6 )
& ( ord_less @ A @ X3 @ Xa ) ) )
=> ~ ( finite_finite2 @ A @ X6 ) ) ) ) ).
% infinite_growing
thf(fact_172_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ S3 )
& ~ ? [Xa: A] :
( ( member @ A @ Xa @ S3 )
& ( ord_less @ A @ Xa @ X3 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_173_empty__Collect__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P2 ) )
= ( ! [X4: A] :
~ ( P2 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_174_max__in__set__def,axiom,
( vEBT_VEBT_max_in_set
= ( ^ [Xs2: set @ nat,X4: nat] :
( ( member @ nat @ X4 @ Xs2 )
& ! [Y5: nat] :
( ( member @ nat @ Y5 @ Xs2 )
=> ( ord_less_eq @ nat @ Y5 @ X4 ) ) ) ) ) ).
% max_in_set_def
thf(fact_175_min__in__set__def,axiom,
( vEBT_VEBT_min_in_set
= ( ^ [Xs2: set @ nat,X4: nat] :
( ( member @ nat @ X4 @ Xs2 )
& ! [Y5: nat] :
( ( member @ nat @ Y5 @ Xs2 )
=> ( ord_less_eq @ nat @ X4 @ Y5 ) ) ) ) ) ).
% min_in_set_def
thf(fact_176_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_177_all__not__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_178_Collect__empty__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
~ ( P2 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_179_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_180_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_181_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_182_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_183_psubsetD,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C3: A] :
( ( ord_less @ ( set @ A ) @ A5 @ B5 )
=> ( ( member @ A @ C3 @ A5 )
=> ( member @ A @ C3 @ B5 ) ) ) ).
% psubsetD
thf(fact_184_psubset__trans,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less @ ( set @ A ) @ B5 @ C4 )
=> ( ord_less @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% psubset_trans
thf(fact_185_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B2: nat] :
( ( P2 @ K )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y4: nat] :
( ( P2 @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_186_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_187_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_188_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_189_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_190_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_191_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ( C3 = D2 )
=> ( ord_less_eq @ A @ A2 @ D2 ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_192_ex__has__least__nat,axiom,
! [A: $tType,P2: A > $o,K: A,M: A > nat] :
( ( P2 @ K )
=> ? [X3: A] :
( ( P2 @ X3 )
& ! [Y4: A] :
( ( P2 @ Y4 )
=> ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y4 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_193_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
=> ( ( ord_less_eq @ nat @ N @ N3 )
=> ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( F3 @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_194_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N3 )
=> ( ord_less_eq @ A @ ( F3 @ N ) @ ( F3 @ N3 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_195_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P2: A > $o,K: A,F3: A > nat,B2: nat] :
( ( P2 @ K )
=> ( ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less @ nat @ ( F3 @ Y3 ) @ B2 ) )
=> ? [X3: A] :
( ( P2 @ X3 )
& ! [Y4: A] :
( ( P2 @ Y4 )
=> ( ord_less_eq @ nat @ ( F3 @ Y4 ) @ ( F3 @ X3 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_196_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_197_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( ord_less_eq @ A @ A2 @ X3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A5 )
=> ( ( ord_less_eq @ A @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_198_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( ord_less_eq @ A @ X3 @ A2 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A5 )
=> ( ( ord_less_eq @ A @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_199_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_200_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_201_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_202_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M6 )
=> ? [M3: nat] :
( M6
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_203_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_204_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_205_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_206_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N2 )
=> ( P2 @ M2 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_207_nat__induct__at__least,axiom,
! [M: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( P2 @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_208_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R3: nat > nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ! [X3: nat] : ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z3: nat] :
( ( R3 @ X3 @ Y3 )
=> ( ( R3 @ Y3 @ Z3 )
=> ( R3 @ X3 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R3 @ N2 @ ( suc @ N2 ) )
=> ( R3 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_209_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_210_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_211_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_212_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_213_exists__leI,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [N4: nat] :
( ( ord_less @ nat @ N4 @ N )
=> ~ ( P2 @ N4 ) )
=> ( P2 @ N ) )
=> ? [N5: nat] :
( ( ord_less_eq @ nat @ N5 @ N )
& ( P2 @ N5 ) ) ) ).
% exists_leI
thf(fact_214_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M5: nat,N6: nat] :
( ( ord_less_eq @ nat @ M5 @ N6 )
& ( M5 != N6 ) ) ) ) ).
% nat_less_le
thf(fact_215_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_216_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M5: nat,N6: nat] :
( ( ord_less @ nat @ M5 @ N6 )
| ( M5 = N6 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_217_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_218_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_219_less__mono__imp__le__mono,axiom,
! [F3: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ ( F3 @ I2 ) @ ( F3 @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F3 @ I ) @ ( F3 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_220_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A5 )
=> ( ( ord_less_eq @ A @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_221_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A5 )
=> ( ( ord_less_eq @ A @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_222_nat__compl__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N2 )
=> ( P2 @ Nn ) )
=> ( P2 @ ( suc @ N2 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_compl_induct
thf(fact_223_nat__compl__induct_H,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N2 )
=> ( P2 @ Nn ) )
=> ( P2 @ ( suc @ N2 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_compl_induct'
thf(fact_224_nat__in__between__eq_I1_J,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less @ nat @ A2 @ B2 )
& ( ord_less_eq @ nat @ B2 @ ( suc @ A2 ) ) )
= ( B2
= ( suc @ A2 ) ) ) ).
% nat_in_between_eq(1)
thf(fact_225_nat__in__between__eq_I2_J,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_eq @ nat @ A2 @ B2 )
& ( ord_less @ nat @ B2 @ ( suc @ A2 ) ) )
= ( B2 = A2 ) ) ).
% nat_in_between_eq(2)
thf(fact_226_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_227_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_228_dec__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P2 @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( ord_less @ nat @ N2 @ J )
=> ( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_229_inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P2 @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( ord_less @ nat @ N2 @ J )
=> ( ( P2 @ ( suc @ N2 ) )
=> ( P2 @ N2 ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% inc_induct
thf(fact_230_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_231_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_232_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_233_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N6: nat] : ( ord_less_eq @ nat @ ( suc @ N6 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_234_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_235_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K2 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_236_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Ux2 ) ).
% VEBT_internal.naive_member.simps(2)
thf(fact_237_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less @ nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ K2 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_238_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_239_equals0D,axiom,
! [A: $tType,A5: set @ A,A2: A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A5 ) ) ).
% equals0D
thf(fact_240_equals0I,axiom,
! [A: $tType,A5: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A5 )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_241_ex__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A5 ) )
= ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_242_not__psubset__empty,axiom,
! [A: $tType,A5: set @ A] :
~ ( ord_less @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_243_VEBT__internal_Oreplicatei_Ocases,axiom,
! [A: $tType,X: product_prod @ nat @ ( heap_Time_Heap @ A )] :
( ! [X3: heap_Time_Heap @ A] :
( X
!= ( product_Pair @ nat @ ( heap_Time_Heap @ A ) @ ( zero_zero @ nat ) @ X3 ) )
=> ~ ! [N2: nat,X3: heap_Time_Heap @ A] :
( X
!= ( product_Pair @ nat @ ( heap_Time_Heap @ A ) @ ( suc @ N2 ) @ X3 ) ) ) ).
% VEBT_internal.replicatei.cases
thf(fact_244_forall__finite_I1_J,axiom,
! [P2: nat > $o,I4: nat] :
( ( ord_less @ nat @ I4 @ ( zero_zero @ nat ) )
=> ( P2 @ I4 ) ) ).
% forall_finite(1)
thf(fact_245_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ Tree @ N )
=> ( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_V5719532721284313246member @ Tree @ X )
| ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).
% member_valid_both_member_options
thf(fact_246_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T3: vEBT_VEBT,X4: nat] :
( ( vEBT_V5719532721284313246member @ T3 @ X4 )
| ( vEBT_VEBT_membermima @ T3 @ X4 ) ) ) ) ).
% both_member_options_def
thf(fact_247_is__succ__in__set__def,axiom,
( vEBT_is_succ_in_set
= ( ^ [Xs2: set @ nat,X4: nat,Y5: nat] :
( ( member @ nat @ Y5 @ Xs2 )
& ( ord_less @ nat @ X4 @ Y5 )
& ! [Z2: nat] :
( ( member @ nat @ Z2 @ Xs2 )
=> ( ( ord_less @ nat @ X4 @ Z2 )
=> ( ord_less_eq @ nat @ Y5 @ Z2 ) ) ) ) ) ) ).
% is_succ_in_set_def
thf(fact_248_is__pred__in__set__def,axiom,
( vEBT_is_pred_in_set
= ( ^ [Xs2: set @ nat,X4: nat,Y5: nat] :
( ( member @ nat @ Y5 @ Xs2 )
& ( ord_less @ nat @ Y5 @ X4 )
& ! [Z2: nat] :
( ( member @ nat @ Z2 @ Xs2 )
=> ( ( ord_less @ nat @ Z2 @ X4 )
=> ( ord_less_eq @ nat @ Z2 @ Y5 ) ) ) ) ) ) ).
% is_pred_in_set_def
thf(fact_249_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_250_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_251_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_252_mod__h__bot__iff_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P3: ref @ A,X: A,H2: heap_ext @ product_unit] :
~ ( rep_assn @ ( sngr_assn @ A @ P3 @ X ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_iff(3)
thf(fact_253_buildup__nothing__in__min__max,axiom,
! [N: nat,X: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_min_max
thf(fact_254_VEBT__internal_OminNull_Osimps_I3_J,axiom,
! [Uu2: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu2 @ $true ) ) ).
% VEBT_internal.minNull.simps(3)
thf(fact_255_VEBT__internal_OminNull_Osimps_I2_J,axiom,
! [Uv2: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv2 ) ) ).
% VEBT_internal.minNull.simps(2)
thf(fact_256_empty__subsetI,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% empty_subsetI
thf(fact_257_subset__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_258_psubsetI,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( A5 != B5 )
=> ( ord_less @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% psubsetI
thf(fact_259_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_260_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_261_subset__emptyI,axiom,
! [A: $tType,A5: set @ A] :
( ! [X3: A] :
~ ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_262_rev__finite__subset,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% rev_finite_subset
thf(fact_263_infinite__super,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ T4 )
=> ( ~ ( finite_finite2 @ A @ S3 )
=> ~ ( finite_finite2 @ A @ T4 ) ) ) ).
% infinite_super
thf(fact_264_finite__subset,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_subset
thf(fact_265_mod__false,axiom,
! [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ ( bot_bot @ assn ) @ H2 ) ).
% mod_false
thf(fact_266_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B7 )
| ( A6 = B7 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_267_subset__psubset__trans,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less @ ( set @ A ) @ B5 @ C4 )
=> ( ord_less @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_268_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
& ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_269_psubset__subset__trans,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ord_less @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_270_psubset__imp__subset,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_271_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
& ( A6 != B7 ) ) ) ) ).
% psubset_eq
thf(fact_272_psubsetE,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% psubsetE
thf(fact_273_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
! [Uu2: $o,Uv2: $o,Uw2: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) ).
% VEBT_internal.membermima.simps(1)
thf(fact_274_order__antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% order_antisym_conv
thf(fact_275_linorder__le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_le_cases
thf(fact_276_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F3: A > B,C3: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F3 @ B2 )
= C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_277_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C3: B] :
( ( A2
= ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_278_linorder__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_linear
thf(fact_279_order__eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% order_eq_refl
thf(fact_280_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F3: A > C,C3: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_281_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C3: B] :
( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_282_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y6: A,Z4: A] : Y6 = Z4 )
= ( ^ [A8: A,B8: A] :
( ( ord_less_eq @ A @ A8 @ B8 )
& ( ord_less_eq @ A @ B8 @ A8 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_283_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_284_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G3 @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F3 @ G3 ) ) ) ).
% le_funI
thf(fact_285_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ).
% le_funE
thf(fact_286_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ).
% le_funD
thf(fact_287_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_288_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_289_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_290_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y6: A,Z4: A] : Y6 = Z4 )
= ( ^ [A8: A,B8: A] :
( ( ord_less_eq @ A @ B8 @ A8 )
& ( ord_less_eq @ A @ A8 @ B8 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_291_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: A,B4: A] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_292_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_293_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% order.trans
thf(fact_294_order__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% order_antisym
thf(fact_295_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_296_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_297_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y6: A,Z4: A] : Y6 = Z4 )
= ( ^ [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_298_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_299_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_300_order__less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_imp_not_less
thf(fact_301_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% order_less_imp_not_eq2
thf(fact_302_order__less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% order_less_imp_not_eq
thf(fact_303_linorder__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_less_linear
thf(fact_304_order__less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,P2: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P2 ) ) ) ).
% order_less_imp_triv
thf(fact_305_order__less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_not_sym
thf(fact_306_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F3: A > C,C3: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F3 @ B2 ) @ C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% order_less_subst2
thf(fact_307_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C3: B] :
( ( ord_less @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_308_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% order_less_irrefl
thf(fact_309_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F3: A > B,C3: B] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ( F3 @ B2 )
= C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ B @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_310_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C3: B] :
( ( A2
= ( F3 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_311_order__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% order_less_trans
thf(fact_312_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_313_linorder__neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neq_iff
thf(fact_314_order__less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_asym
thf(fact_315_linorder__neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE
thf(fact_316_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_317_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_318_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_319_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_320_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% order.strict_trans
thf(fact_321_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: A] : ( P2 @ A4 @ A4 )
=> ( ! [A4: A,B4: A] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A2 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_322_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P4: A > $o] :
? [X7: A] : ( P4 @ X7 ) )
= ( ^ [P5: A > $o] :
? [N6: A] :
( ( P5 @ N6 )
& ! [M5: A] :
( ( ord_less @ A @ M5 @ N6 )
=> ~ ( P5 @ M5 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_323_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_324_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_325_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_326_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_327_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P2: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X3 )
=> ( P2 @ Y4 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A2 ) ) ) ).
% less_induct
thf(fact_328_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_329_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2 = B2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_330_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_331_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_332_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z3: A] :
( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% dense
thf(fact_333_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% gt_ex
thf(fact_334_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% lt_ex
thf(fact_335_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_336_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_337_linorder__le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_le_less_linear
thf(fact_338_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F3: A > C,C3: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_339_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C3: B] :
( ( ord_less @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_340_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F3: A > C,C3: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F3 @ B2 ) @ C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_341_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C3: B] :
( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_342_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% order_less_le_trans
thf(fact_343_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% order_le_less_trans
thf(fact_344_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_345_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_346_order__less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% order_less_imp_le
thf(fact_347_linorder__not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_not_less
thf(fact_348_linorder__not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_not_le
thf(fact_349_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ) ).
% order_less_le
thf(fact_350_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y5: A] :
( ( ord_less @ A @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ) ).
% order_le_less
thf(fact_351_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_352_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_353_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B8: A,A8: A] :
( ( ord_less_eq @ A @ B8 @ A8 )
& ~ ( ord_less_eq @ A @ A8 @ B8 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_354_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_355_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_356_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B8: A,A8: A] :
( ( ord_less_eq @ A @ B8 @ A8 )
& ( A8 != B8 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_357_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B8: A,A8: A] :
( ( ord_less @ A @ B8 @ A8 )
| ( A8 = B8 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_358_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_359_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_360_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A8: A,B8: A] :
( ( ord_less_eq @ A @ A8 @ B8 )
& ~ ( ord_less_eq @ A @ B8 @ A8 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_361_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% order.strict_trans2
thf(fact_362_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% order.strict_trans1
thf(fact_363_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A8: A,B8: A] :
( ( ord_less_eq @ A @ A8 @ B8 )
& ( A8 != B8 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_364_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B8: A] :
( ( ord_less @ A @ A8 @ B8 )
| ( A8 = B8 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_365_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_366_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
& ~ ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% less_le_not_le
thf(fact_367_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_le
thf(fact_368_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,Y: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_ge
thf(fact_369_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_370_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_371_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_372_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_373_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_374_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_375_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_376_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_377_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_378_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_379_VEBT__internal_OminNull_Osimps_I1_J,axiom,
vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% VEBT_internal.minNull.simps(1)
thf(fact_380_deg1Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
= ( ? [A8: $o,B8: $o] :
( T2
= ( vEBT_Leaf @ A8 @ B8 ) ) ) ) ).
% deg1Leaf
thf(fact_381_deg__1__Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
=> ? [A4: $o,B4: $o] :
( T2
= ( vEBT_Leaf @ A4 @ B4 ) ) ) ).
% deg_1_Leaf
thf(fact_382_deg__1__Leafy,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( N
= ( one_one @ nat ) )
=> ? [A4: $o,B4: $o] :
( T2
= ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ).
% deg_1_Leafy
thf(fact_383_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N7: set @ nat] :
? [M5: nat] :
! [X4: nat] :
( ( member @ nat @ X4 @ N7 )
=> ( ord_less_eq @ nat @ X4 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_384_infinite__nat__iff__unbounded__le,axiom,
! [S3: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S3 ) )
= ( ! [M5: nat] :
? [N6: nat] :
( ( ord_less_eq @ nat @ M5 @ N6 )
& ( member @ nat @ N6 @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_385_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N7: set @ nat] :
? [M5: nat] :
! [X4: nat] :
( ( member @ nat @ X4 @ N7 )
=> ( ord_less @ nat @ X4 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_386_infinite__nat__iff__unbounded,axiom,
! [S3: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S3 ) )
= ( ! [M5: nat] :
? [N6: nat] :
( ( ord_less @ nat @ M5 @ N6 )
& ( member @ nat @ N6 @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_387_bounded__nat__set__is__finite,axiom,
! [N8: set @ nat,N: nat] :
( ! [X3: nat] :
( ( member @ nat @ X3 @ N8 )
=> ( ord_less @ nat @ X3 @ N ) )
=> ( finite_finite2 @ nat @ N8 ) ) ).
% bounded_nat_set_is_finite
thf(fact_388_unbounded__k__infinite,axiom,
! [K: nat,S3: set @ nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ K @ M3 )
=> ? [N9: nat] :
( ( ord_less @ nat @ M3 @ N9 )
& ( member @ nat @ N9 @ S3 ) ) )
=> ~ ( finite_finite2 @ nat @ S3 ) ) ).
% unbounded_k_infinite
thf(fact_389_VEBT__internal_Oheight_Ocases,axiom,
! [X: vEBT_VEBT] :
( ! [A4: $o,B4: $o] :
( X
!= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ! [Uu: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ Uu @ Deg2 @ TreeList2 @ Summary2 ) ) ) ).
% VEBT_internal.height.cases
thf(fact_390_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S3: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ? [X5: A] :
( ( member @ A @ X5 @ S3 )
& ( ord_less @ B @ ( F3 @ X5 ) @ ( F3 @ ( lattic7623131987881927897min_on @ A @ B @ F3 @ S3 ) ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_391_subsetI,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ A @ X3 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% subsetI
thf(fact_392_subset__antisym,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( A5 = B5 ) ) ) ).
% subset_antisym
thf(fact_393_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_394_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
( ( ord_less_eq @ ( A > B ) @ F4 @ G4 )
& ~ ( ord_less_eq @ ( A > B ) @ G4 @ F4 ) ) ) ) ) ).
% less_fun_def
thf(fact_395_subset__Collect__conv,axiom,
! [A: $tType,S3: set @ A,P2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ ( collect @ A @ P2 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ S3 )
=> ( P2 @ X4 ) ) ) ) ).
% subset_Collect_conv
thf(fact_396_in__mono,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( member @ A @ X @ A5 )
=> ( member @ A @ X @ B5 ) ) ) ).
% in_mono
thf(fact_397_subsetD,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( member @ A @ C3 @ A5 )
=> ( member @ A @ C3 @ B5 ) ) ) ).
% subsetD
thf(fact_398_equalityE,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% equalityE
thf(fact_399_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( member @ A @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_400_equalityD1,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% equalityD1
thf(fact_401_Set_OequalityD2,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% Set.equalityD2
thf(fact_402_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A6 )
=> ( member @ A @ T3 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_403_subset__refl,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% subset_refl
thf(fact_404_Collect__mono,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_mono
thf(fact_405_subset__trans,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% subset_trans
thf(fact_406_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y6: set @ A,Z4: set @ A] : Y6 = Z4 )
= ( ^ [A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_407_Collect__mono__iff,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q2 ) )
= ( ! [X4: A] :
( ( P2 @ X4 )
=> ( Q2 @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_408_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_409_one__natural_Orsp,axiom,
( ( one_one @ nat )
= ( one_one @ nat ) ) ).
% one_natural.rsp
thf(fact_410_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_411_nat__geq__1__eq__neqz,axiom,
! [X: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ X )
= ( X
!= ( zero_zero @ nat ) ) ) ).
% nat_geq_1_eq_neqz
thf(fact_412_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
! [Uu2: $o,Uv2: $o,D2: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D2 )
= ( D2
= ( one_one @ nat ) ) ) ).
% VEBT_internal.valid'.simps(1)
thf(fact_413_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( P2 @ ( one_one @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_414_bounded__Max__nat,axiom,
! [P2: nat > $o,X: nat,M7: nat] :
( ( P2 @ X )
=> ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( ord_less_eq @ nat @ X3 @ M7 ) )
=> ~ ! [M3: nat] :
( ( P2 @ M3 )
=> ~ ! [X5: nat] :
( ( P2 @ X5 )
=> ( ord_less_eq @ nat @ X5 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_415_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( ( ( X
= ( zero_zero @ nat ) )
=> A2 )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> B2 )
& ( X
= ( one_one @ nat ) ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
thf(fact_416_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S3: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic7623131987881927897min_on @ A @ B @ F3 @ S3 ) @ S3 ) ) ) ) ).
% arg_min_if_finite(1)
thf(fact_417_finite__transitivity__chain,axiom,
! [A: $tType,A5: set @ A,R3: A > A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [X3: A] :
~ ( R3 @ X3 @ X3 )
=> ( ! [X3: A,Y3: A,Z3: A] :
( ( R3 @ X3 @ Y3 )
=> ( ( R3 @ Y3 @ Z3 )
=> ( R3 @ X3 @ Z3 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ A5 )
& ( R3 @ X3 @ Y4 ) ) )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_418_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S3: set @ A,Y: A,F3: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ Y @ S3 )
=> ( ord_less_eq @ B @ ( F3 @ ( lattic7623131987881927897min_on @ A @ B @ F3 @ S3 ) ) @ ( F3 @ Y ) ) ) ) ) ) ).
% arg_min_least
thf(fact_419_minNull__delete__time__bound_H,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T2 @ X ) )
=> ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X ) @ ( one_one @ nat ) ) ) ) ).
% minNull_delete_time_bound'
thf(fact_420_vebt__member__code_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( ( ( X
= ( zero_zero @ nat ) )
=> A2 )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> B2 )
& ( X
= ( one_one @ nat ) ) ) ) ) ) ).
% vebt_member_code(1)
thf(fact_421_vebt__delete__code_I1_J,axiom,
! [X: nat,A2: $o,B2: $o] :
( ( ( X
= ( zero_zero @ nat ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_Leaf @ $false @ B2 ) ) )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_Leaf @ A2 @ $false ) ) )
& ( ( X
!= ( one_one @ nat ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_Leaf @ A2 @ B2 ) ) ) ) ) ) ).
% vebt_delete_code(1)
thf(fact_422_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_423_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_424_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_425_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_426_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_427_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_428_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_429_Collect__empty__eq__bot,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( P2
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_430_assn__basic__inequalities_I3_J,axiom,
( ( bot_bot @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(3)
thf(fact_431_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_432_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ).
% subrelI
thf(fact_433_mod__emp__simp,axiom,
! [H2: heap_ext @ product_unit] : ( rep_assn @ ( one_one @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% mod_emp_simp
thf(fact_434_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_435_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_436_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_437_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_438_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_439_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I2_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(2)
thf(fact_440_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(1)
thf(fact_441_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I3_J,axiom,
! [A2: $o,B2: $o,N: nat] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ N ) ) )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
thf(fact_442_nat__descend__induct,axiom,
! [N: nat,P2: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less @ nat @ N @ K2 )
=> ( P2 @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ K2 @ I4 )
=> ( P2 @ I4 ) )
=> ( P2 @ K2 ) ) )
=> ( P2 @ M ) ) ) ).
% nat_descend_induct
thf(fact_443_exists__least__lemma,axiom,
! [P2: nat > $o] :
( ~ ( P2 @ ( zero_zero @ nat ) )
=> ( ? [X_12: nat] : ( P2 @ X_12 )
=> ? [N2: nat] :
( ~ ( P2 @ N2 )
& ( P2 @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_444_dependent__nat__choice,axiom,
! [A: $tType,P2: nat > A > $o,Q2: nat > A > A > $o] :
( ? [X_12: A] : ( P2 @ ( zero_zero @ nat ) @ X_12 )
=> ( ! [X3: A,N2: nat] :
( ( P2 @ N2 @ X3 )
=> ? [Y4: A] :
( ( P2 @ ( suc @ N2 ) @ Y4 )
& ( Q2 @ N2 @ X3 @ Y4 ) ) )
=> ? [F2: nat > A] :
! [N9: nat] :
( ( P2 @ N9 @ ( F2 @ N9 ) )
& ( Q2 @ N9 @ ( F2 @ N9 ) @ ( F2 @ ( suc @ N9 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_445_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ( ( X
!= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ! [Va: nat] :
( X
!= ( suc @ ( suc @ Va ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
thf(fact_446_list__decode_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_447_mod__h__bot__iff_I4_J,axiom,
! [B: $tType] :
( ( heap @ B )
=> ! [Q3: array @ B,Y: list @ B,H2: heap_ext @ product_unit] :
~ ( rep_assn @ ( snga_assn @ B @ Q3 @ Y ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_iff(4)
thf(fact_448_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A2: A,B2: A,P2: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( P2 @ A2 )
=> ( ~ ( P2 @ B2 )
=> ? [C2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
& ( ord_less_eq @ A @ C2 @ B2 )
& ! [X5: A] :
( ( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less @ A @ X5 @ C2 ) )
=> ( P2 @ X5 ) )
& ! [D4: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less @ A @ X3 @ D4 ) )
=> ( P2 @ X3 ) )
=> ( ord_less_eq @ A @ D4 @ C2 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_449_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_450_less__by__empty,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A ),B5: set @ ( product_prod @ A @ A )] :
( ( A5
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ B5 ) ) ).
% less_by_empty
thf(fact_451_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(2)
thf(fact_452_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_453_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(1)
thf(fact_454_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit5016429287641298734tinuum @ A )
=> ! [A2: A] :
? [B4: A] :
( ( ord_less @ A @ A2 @ B4 )
| ( ord_less @ A @ B4 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_455_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ~ ( ord_less_eq @ A @ T2 @ X5 ) ) ) ).
% minf(8)
thf(fact_456_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ord_less_eq @ A @ X5 @ T2 ) ) ) ).
% minf(6)
thf(fact_457_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ord_less_eq @ A @ T2 @ X5 ) ) ) ).
% pinf(8)
thf(fact_458_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ~ ( ord_less_eq @ A @ X5 @ T2 ) ) ) ).
% pinf(6)
thf(fact_459_field__lbound__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D1: A,D22: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ D22 )
=> ? [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
& ( ord_less @ A @ E2 @ D1 )
& ( ord_less @ A @ E2 @ D22 ) ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_460_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_461_VEBT_Osize__gen_I2_J,axiom,
! [X21: $o,X22: $o] :
( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
= ( zero_zero @ nat ) ) ).
% VEBT.size_gen(2)
thf(fact_462_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_463_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
thf(fact_464_frac__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( archimedean_frac @ A @ X )
= X )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% frac_eq
thf(fact_465_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_466_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_467_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_468_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_469_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_470_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_471_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_472_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_473_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_474_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_475_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_476_zero__comp__diff__simps_I1_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% zero_comp_diff_simps(1)
thf(fact_477_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_478_zero__comp__diff__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% zero_comp_diff_simps(2)
thf(fact_479_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_480_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_481_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_482_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_483_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_484_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_485_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_486_Suc__diff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( suc @ ( minus_minus @ nat @ N @ M ) )
= ( minus_minus @ nat @ N @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_diff
thf(fact_487_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C3 @ D2 ) )
=> ( ( A2 = B2 )
= ( C3 = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_488_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C3 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_489_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_490_le__some__optE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M: A,X: option @ A] :
( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ M ) @ X )
=> ~ ! [M8: A] :
( ( X
= ( some @ A @ M8 ) )
=> ~ ( ord_less_eq @ A @ M @ M8 ) ) ) ) ).
% le_some_optE
thf(fact_491_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y6: A,Z4: A] : Y6 = Z4 )
= ( ^ [A8: A,B8: A] :
( ( minus_minus @ A @ A8 @ B8 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_492_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ D2 @ C3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_493_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C3 @ A2 ) @ ( minus_minus @ A @ C3 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_494_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ C3 ) ) ) ) ).
% diff_right_mono
thf(fact_495_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C3 @ D2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
= ( ord_less_eq @ A @ C3 @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_496_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ C3 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_497_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C3 @ A2 ) @ ( minus_minus @ A @ C3 @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_498_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C3 @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
= ( ord_less @ A @ C3 @ D2 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_499_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ D2 @ C3 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_500_zero__induct__lemma,axiom,
! [P2: nat > $o,K: nat,I: nat] :
( ( P2 @ K )
=> ( ! [N2: nat] :
( ( P2 @ ( suc @ N2 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_501_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_502_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_503_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_504_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_505_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_506_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_507_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_508_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_509_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_510_le__diff__iff_H,axiom,
! [A2: nat,C3: nat,B2: nat] :
( ( ord_less_eq @ nat @ A2 @ C3 )
=> ( ( ord_less_eq @ nat @ B2 @ C3 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C3 @ A2 ) @ ( minus_minus @ nat @ C3 @ B2 ) )
= ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_511_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_512_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
thf(fact_513_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B8: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A8 @ B8 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_514_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A8: A,B8: A] : ( ord_less @ A @ ( minus_minus @ A @ A8 @ B8 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_515_Suc__to__right,axiom,
! [N: nat,M: nat] :
( ( ( suc @ N )
= M )
=> ( N
= ( minus_minus @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% Suc_to_right
thf(fact_516_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_517_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_518_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_519_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_520_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_521_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_522_diff__less__mono,axiom,
! [A2: nat,B2: nat,C3: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( ( ord_less_eq @ nat @ C3 @ A2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A2 @ C3 ) @ ( minus_minus @ nat @ B2 @ C3 ) ) ) ) ).
% diff_less_mono
thf(fact_523_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
thf(fact_524_VEBT__internal_OminNull_Osimps_I5_J,axiom,
! [Uz: product_prod @ nat @ nat,Va2: nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va2 @ Vb @ Vc ) ) ).
% VEBT_internal.minNull.simps(5)
thf(fact_525_frac__ge__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X ) ) ) ).
% frac_ge_0
thf(fact_526_frac__lt__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X ) @ ( one_one @ A ) ) ) ).
% frac_lt_1
thf(fact_527_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_528_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o,P6: A > $o,Q2: A > $o,Q4: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( P2 @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ( ( P2 @ X5 )
& ( Q2 @ X5 ) )
= ( ( P6 @ X5 )
& ( Q4 @ X5 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_529_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o,P6: A > $o,Q2: A > $o,Q4: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( P2 @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ( ( P2 @ X5 )
| ( Q2 @ X5 ) )
= ( ( P6 @ X5 )
| ( Q4 @ X5 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_530_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( X5 != T2 ) ) ) ).
% pinf(3)
thf(fact_531_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( X5 != T2 ) ) ) ).
% pinf(4)
thf(fact_532_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ~ ( ord_less @ A @ X5 @ T2 ) ) ) ).
% pinf(5)
thf(fact_533_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ord_less @ A @ T2 @ X5 ) ) ) ).
% pinf(7)
thf(fact_534_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F5: D] :
? [Z3: C] :
! [X5: C] :
( ( ord_less @ C @ Z3 @ X5 )
=> ( F5 = F5 ) ) ) ).
% pinf(11)
thf(fact_535_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o,P6: A > $o,Q2: A > $o,Q4: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( P2 @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( Q2 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ( ( P2 @ X5 )
& ( Q2 @ X5 ) )
= ( ( P6 @ X5 )
& ( Q4 @ X5 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_536_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o,P6: A > $o,Q2: A > $o,Q4: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( P2 @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( Q2 @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ( ( P2 @ X5 )
| ( Q2 @ X5 ) )
= ( ( P6 @ X5 )
| ( Q4 @ X5 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_537_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( X5 != T2 ) ) ) ).
% minf(3)
thf(fact_538_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( X5 != T2 ) ) ) ).
% minf(4)
thf(fact_539_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ord_less @ A @ X5 @ T2 ) ) ) ).
% minf(5)
thf(fact_540_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ~ ( ord_less @ A @ T2 @ X5 ) ) ) ).
% minf(7)
thf(fact_541_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F5: D] :
? [Z3: C] :
! [X5: C] :
( ( ord_less @ C @ X5 @ Z3 )
=> ( F5 = F5 ) ) ) ).
% minf(11)
thf(fact_542_vebt__delete_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,TrLst: list @ vEBT_VEBT,Smry: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) ) ).
% vebt_delete.simps(5)
thf(fact_543_vebt__member_Osimps_I3_J,axiom,
! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz ) @ X ) ).
% vebt_member.simps(3)
thf(fact_544_VEBT__internal_OminNull_Oelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ! [Uv: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv ) )
=> ( ! [Uu: $o] :
( X
!= ( vEBT_Leaf @ Uu @ $true ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).
% VEBT_internal.minNull.elims(3)
thf(fact_545_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Va2: list @ vEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) @ X )
= ( ( X = Mi )
| ( X = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_546_nz__le__conv__less,axiom,
! [K: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ K @ M )
=> ( ord_less @ nat @ ( minus_minus @ nat @ K @ ( suc @ ( zero_zero @ nat ) ) ) @ M ) ) ) ).
% nz_le_conv_less
thf(fact_547_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_548_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_549_Suc__n__minus__m__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ M )
=> ( ( suc @ ( minus_minus @ nat @ N @ M ) )
= ( minus_minus @ nat @ N @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_n_minus_m_eq
thf(fact_550_vebt__delete_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,Tr: list @ vEBT_VEBT,Sm: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) ) ).
% vebt_delete.simps(6)
thf(fact_551_vebt__member_Osimps_I4_J,axiom,
! [V3: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X ) ).
% vebt_member.simps(4)
thf(fact_552_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
thf(fact_553_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_554_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_555_less__option__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
= ( ord_less @ A @ X @ Y ) ) ) ).
% less_option_Some
thf(fact_556_less__eq__option__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_eq_option_Some
thf(fact_557_Suc__if__eq,axiom,
! [A: $tType,F3: nat > A,H2: nat > A,G3: A,N: nat] :
( ! [N2: nat] :
( ( F3 @ ( suc @ N2 ) )
= ( H2 @ N2 ) )
=> ( ( ( F3 @ ( zero_zero @ nat ) )
= G3 )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( ( F3 @ N )
= G3 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( F3 @ N )
= ( H2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% Suc_if_eq
thf(fact_558_mi__eq__ma__no__ch,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
=> ( ( Mi = Ma )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
& ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% mi_eq_ma_no_ch
thf(fact_559_diff__shunt__var,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( minus_minus @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% diff_shunt_var
thf(fact_560_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K3: nat,M5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M5 @ K3 ) @ ( product_Pair @ nat @ nat @ M5 @ ( minus_minus @ nat @ K3 @ M5 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus @ nat @ M5 @ ( suc @ K3 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_561_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y )
=> ( ( ( ord_less_eq @ nat @ Xa2 @ X )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa2 @ ( minus_minus @ nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa2 @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_562_VEBT__internal_Omembermima_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu: $o,Uv: $o,Uw: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) )
=> ( ! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT,Uz2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) @ Uz2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ X3 ) )
=> ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) @ X3 ) )
=> ~ ! [V: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd ) @ X3 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_563_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A4: $o,B4: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( zero_zero @ nat ) ) )
=> ( ! [A4: $o,B4: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [A4: $o,B4: $o,N2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ N2 ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Uu: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Uu ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ X3 ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
thf(fact_564_diff__diff__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ M @ ( minus_minus @ nat @ M @ N ) ) )
= ( ( ord_less @ nat @ I @ M )
& ( ord_less @ nat @ I @ N ) ) ) ).
% diff_diff_less
thf(fact_565_bijective__Empty,axiom,
! [B: $tType,A: $tType] : ( bijective @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% bijective_Empty
thf(fact_566_Diff__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= A5 ) ).
% Diff_empty
thf(fact_567_empty__Diff,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_568_Diff__cancel,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_569_finite__Diff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% finite_Diff
thf(fact_570_finite__Diff2,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
= ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_Diff2
thf(fact_571_Diff__eq__empty__iff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_572_lesseq__shift,axiom,
( ( ord_less_eq @ nat )
= ( ^ [X4: nat,Y5: nat] : ( vEBT_VEBT_lesseq @ ( some @ nat @ X4 ) @ ( some @ nat @ Y5 ) ) ) ) ).
% lesseq_shift
thf(fact_573_less__eq__option__None__code,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X ) ) ).
% less_eq_option_None_code
thf(fact_574_less__option__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] :
~ ( ord_less @ ( option @ A ) @ X @ ( none @ A ) ) ) ).
% less_option_None
thf(fact_575_not__Some__eq2,axiom,
! [B: $tType,A: $tType,V3: option @ ( product_prod @ A @ B )] :
( ( ! [X4: A,Y5: B] :
( V3
!= ( some @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) ) ) )
= ( V3
= ( none @ ( product_prod @ A @ B ) ) ) ) ).
% not_Some_eq2
thf(fact_576_less__eq__option__Some__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X ) @ ( none @ A ) ) ) ).
% less_eq_option_Some_None
thf(fact_577_less__option__None__Some__code,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less @ ( option @ A ) @ ( none @ A ) @ ( some @ A @ X ) ) ) ).
% less_option_None_Some_code
thf(fact_578_less__eq__option__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X ) ) ).
% less_eq_option_None
thf(fact_579_less__eq__option__None__is__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] :
( ( ord_less_eq @ ( option @ A ) @ X @ ( none @ A ) )
=> ( X
= ( none @ A ) ) ) ) ).
% less_eq_option_None_is_None
thf(fact_580_less__option__None__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less @ ( option @ A ) @ ( none @ A ) @ ( some @ A @ X ) ) ) ).
% less_option_None_Some
thf(fact_581_less__option__None__is__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] :
( ( ord_less @ ( option @ A ) @ ( none @ A ) @ X )
=> ? [Z3: A] :
( X
= ( some @ A @ Z3 ) ) ) ) ).
% less_option_None_is_Some
thf(fact_582_Diff__infinite__finite,axiom,
! [A: $tType,T4: set @ A,S3: set @ A] :
( ( finite_finite2 @ A @ T4 )
=> ( ~ ( finite_finite2 @ A @ S3 )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S3 @ T4 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_583_Diff__mono,axiom,
! [A: $tType,A5: set @ A,C4: set @ A,D5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ D5 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D5 ) ) ) ) ).
% Diff_mono
thf(fact_584_Diff__subset,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ A5 ) ).
% Diff_subset
thf(fact_585_double__diff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ( minus_minus @ ( set @ A ) @ B5 @ ( minus_minus @ ( set @ A ) @ C4 @ A5 ) )
= A5 ) ) ) ).
% double_diff
thf(fact_586_psubset__imp__ex__mem,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B5 )
=> ? [B4: A] : ( member @ A @ B4 @ ( minus_minus @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_587_subset__minus__empty,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_minus_empty
thf(fact_588_vebt__delete_Osimps_I4_J,axiom,
! [Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Uu2 )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) ) ).
% vebt_delete.simps(4)
thf(fact_589_VEBT__internal_OminNull_Osimps_I4_J,axiom,
! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ).
% VEBT_internal.minNull.simps(4)
thf(fact_590_vebt__member_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X ) ).
% vebt_member.simps(2)
thf(fact_591_vebt__member__code_I2_J,axiom,
! [T2: nat,R2: list @ vEBT_VEBT,E3: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ T2 @ R2 @ E3 ) @ X ) ).
% vebt_member_code(2)
thf(fact_592_bijective__def,axiom,
! [B: $tType,A: $tType] :
( ( bijective @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
( ! [X4: A,Y5: B,Z2: B] :
( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Z2 ) @ R4 ) )
=> ( Y5 = Z2 ) )
& ! [X4: A,Y5: A,Z2: B] :
( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Z2 ) @ R4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y5 @ Z2 ) @ R4 ) )
=> ( X4 = Y5 ) ) ) ) ) ).
% bijective_def
thf(fact_593_VEBT__internal_OminNull_Oelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) ) ) ) ).
% VEBT_internal.minNull.elims(2)
thf(fact_594_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT,Uz: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Uz ) ).
% VEBT_internal.membermima.simps(2)
thf(fact_595_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,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Uu2 )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
thf(fact_596_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I4_J,axiom,
! [Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Uu2 )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
thf(fact_597_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ Y )
=> ( ( ? [Uv: $o] :
( X
= ( vEBT_Leaf @ $true @ Uv ) )
=> Y )
=> ( ( ? [Uu: $o] :
( X
= ( vEBT_Leaf @ Uu @ $true ) )
=> Y )
=> ( ( ? [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) )
=> ~ Y )
=> ~ ( ? [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> Y ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
thf(fact_598_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu: $o,Uv: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) ) )
=> ( ! [A4: $o,Uw: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [A4: $o,B4: $o,Va: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ Va ) ) ) )
=> ( ! [Uy: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) @ Vb2 ) )
=> ( ! [V: product_prod @ nat @ nat,Vd: list @ vEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd @ Ve ) @ Vf ) )
=> ( ! [V: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) @ Vj ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
thf(fact_599_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu: $o,B4: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ B4 ) @ ( zero_zero @ nat ) ) )
=> ( ! [Uv: $o,Uw: $o,N2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) )
=> ( ! [Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) @ Va3 ) )
=> ( ! [V: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc2 @ Vd ) @ Ve ) )
=> ( ! [V: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) @ Vi ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
thf(fact_600_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A4: $o,B4: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X3 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S4 ) @ X3 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S4 ) @ X3 ) )
=> ( ! [V: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
thf(fact_601_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A4: $o,B4: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X3 ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X3 ) )
=> ( ! [V: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) @ X3 ) )
=> ( ! [V: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
thf(fact_602_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
! [X: vEBT_VEBT] :
( ! [A4: $o,B4: $o] :
( X
!= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
thf(fact_603_List_Ofinite__set,axiom,
! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% List.finite_set
thf(fact_604_delete__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% delete_correct
thf(fact_605_T_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_Ocases,axiom,
! [X: vEBT_VEBT] :
( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ( ! [Uv: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv ) )
=> ( ! [Uu: $o] :
( X
!= ( vEBT_Leaf @ Uu @ $true ) )
=> ( ! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
thf(fact_606_insertCI,axiom,
! [A: $tType,A2: A,B5: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B5 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% insertCI
thf(fact_607_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A5: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A5 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A5 ) ) ) ).
% insert_iff
thf(fact_608_insert__absorb2,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A5 ) )
= ( insert @ A @ X @ A5 ) ) ).
% insert_absorb2
thf(fact_609_DiffI,axiom,
! [A: $tType,C3: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ A5 )
=> ( ~ ( member @ A @ C3 @ B5 )
=> ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% DiffI
thf(fact_610_Diff__iff,axiom,
! [A: $tType,C3: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
= ( ( member @ A @ C3 @ A5 )
& ~ ( member @ A @ C3 @ B5 ) ) ) ).
% Diff_iff
thf(fact_611_Diff__idemp,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ B5 )
= ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ).
% Diff_idemp
thf(fact_612_set__vebt__delete,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% set_vebt_delete
thf(fact_613_delete__correct_H,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% delete_correct'
thf(fact_614_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_615_finite__insert,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A5 ) )
= ( finite_finite2 @ A @ A5 ) ) ).
% finite_insert
thf(fact_616_insert__subset,axiom,
! [A: $tType,X: A,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ B5 )
= ( ( member @ A @ X @ B5 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% insert_subset
thf(fact_617_Diff__insert0,axiom,
! [A: $tType,X: A,A5: set @ A,B5: set @ A] :
( ~ ( member @ A @ X @ A5 )
=> ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B5 ) )
= ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_618_insert__Diff1,axiom,
! [A: $tType,X: A,B5: set @ A,A5: set @ A] :
( ( member @ A @ X @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ B5 )
= ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_619_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A5: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A5 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_620_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A5: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A5 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_621_insert__Diff__single,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A2 @ A5 ) ) ).
% insert_Diff_single
thf(fact_622_finite__Diff__insert,axiom,
! [A: $tType,A5: set @ A,A2: A,B5: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ B5 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% finite_Diff_insert
thf(fact_623_DiffE,axiom,
! [A: $tType,C3: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
=> ~ ( ( member @ A @ C3 @ A5 )
=> ( member @ A @ C3 @ B5 ) ) ) ).
% DiffE
thf(fact_624_DiffD1,axiom,
! [A: $tType,C3: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
=> ( member @ A @ C3 @ A5 ) ) ).
% DiffD1
thf(fact_625_DiffD2,axiom,
! [A: $tType,C3: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
=> ~ ( member @ A @ C3 @ B5 ) ) ).
% DiffD2
thf(fact_626_insert__Diff__if,axiom,
! [A: $tType,X: A,B5: set @ A,A5: set @ A] :
( ( ( member @ A @ X @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ B5 )
= ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) )
& ( ~ ( member @ A @ X @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ B5 )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_627_insertE,axiom,
! [A: $tType,A2: A,B2: A,A5: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A5 ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ A5 ) ) ) ).
% insertE
thf(fact_628_insertI1,axiom,
! [A: $tType,A2: A,B5: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B5 ) ) ).
% insertI1
thf(fact_629_insertI2,axiom,
! [A: $tType,A2: A,B5: set @ A,B2: A] :
( ( member @ A @ A2 @ B5 )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% insertI2
thf(fact_630_Set_Oset__insert,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( member @ A @ X @ A5 )
=> ~ ! [B9: set @ A] :
( ( A5
= ( insert @ A @ X @ B9 ) )
=> ( member @ A @ X @ B9 ) ) ) ).
% Set.set_insert
thf(fact_631_insert__ident,axiom,
! [A: $tType,X: A,A5: set @ A,B5: set @ A] :
( ~ ( member @ A @ X @ A5 )
=> ( ~ ( member @ A @ X @ B5 )
=> ( ( ( insert @ A @ X @ A5 )
= ( insert @ A @ X @ B5 ) )
= ( A5 = B5 ) ) ) ) ).
% insert_ident
thf(fact_632_insert__absorb,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( ( insert @ A @ A2 @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_633_insert__eq__iff,axiom,
! [A: $tType,A2: A,A5: set @ A,B2: A,B5: set @ A] :
( ~ ( member @ A @ A2 @ A5 )
=> ( ~ ( member @ A @ B2 @ B5 )
=> ( ( ( insert @ A @ A2 @ A5 )
= ( insert @ A @ B2 @ B5 ) )
= ( ( ( A2 = B2 )
=> ( A5 = B5 ) )
& ( ( A2 != B2 )
=> ? [C5: set @ A] :
( ( A5
= ( insert @ A @ B2 @ C5 ) )
& ~ ( member @ A @ B2 @ C5 )
& ( B5
= ( insert @ A @ A2 @ C5 ) )
& ~ ( member @ A @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_634_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A5: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A5 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A5 ) ) ) ).
% insert_commute
thf(fact_635_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ? [B9: set @ A] :
( ( A5
= ( insert @ A @ A2 @ B9 ) )
& ~ ( member @ A @ A2 @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_636_singleton__inject,axiom,
! [A: $tType,A2: A,B2: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_637_insert__not__empty,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( insert @ A @ A2 @ A5 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_638_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C3: A,D2: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C3 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C3 )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_639_singleton__iff,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_640_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_641_insert__minus__eq,axiom,
! [A: $tType,X: A,Y: A,A5: set @ A] :
( ( X != Y )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% insert_minus_eq
thf(fact_642_set__minus__singleton__eq,axiom,
! [A: $tType,X: A,X6: set @ A] :
( ~ ( member @ A @ X @ X6 )
=> ( ( minus_minus @ ( set @ A ) @ X6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X6 ) ) ).
% set_minus_singleton_eq
thf(fact_643_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ~ ( member @ A @ X @ A5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_644_Diff__insert2,axiom,
! [A: $tType,A5: set @ A,A2: A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_645_insert__Diff,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_646_Diff__insert,axiom,
! [A: $tType,A5: set @ A,A2: A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_647_finite_OinsertI,axiom,
! [A: $tType,A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ A @ ( insert @ A @ A2 @ A5 ) ) ) ).
% finite.insertI
thf(fact_648_subset__insertI2,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% subset_insertI2
thf(fact_649_subset__insertI,axiom,
! [A: $tType,B5: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A2 @ B5 ) ) ).
% subset_insertI
thf(fact_650_subset__insert,axiom,
! [A: $tType,X: A,A5: set @ A,B5: set @ A] :
( ~ ( member @ A @ X @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% subset_insert
thf(fact_651_insert__mono,axiom,
! [A: $tType,C4: set @ A,D5: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ D5 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C4 ) @ ( insert @ A @ A2 @ D5 ) ) ) ).
% insert_mono
thf(fact_652_insert__subsetI,axiom,
! [A: $tType,X: A,A5: set @ A,X6: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X6 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_653_subset__Diff__insert,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,X: A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B5 @ ( insert @ A @ X @ C4 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B5 @ C4 ) )
& ~ ( member @ A @ X @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_654_finite_Ocases,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A7: set @ A] :
( ? [A4: A] :
( A2
= ( insert @ A @ A4 @ A7 ) )
=> ~ ( finite_finite2 @ A @ A7 ) ) ) ) ).
% finite.cases
thf(fact_655_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A8: set @ A] :
( ( A8
= ( bot_bot @ ( set @ A ) ) )
| ? [A6: set @ A,B8: A] :
( ( A8
= ( insert @ A @ B8 @ A6 ) )
& ( finite_finite2 @ A @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_656_finite__induct,axiom,
! [A: $tType,F5: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( P2 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P2 @ F6 )
=> ( P2 @ ( insert @ A @ X3 @ F6 ) ) ) ) )
=> ( P2 @ F5 ) ) ) ) ).
% finite_induct
thf(fact_657_finite__ne__induct,axiom,
! [A: $tType,F5: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( F5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] : ( P2 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( F6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P2 @ F6 )
=> ( P2 @ ( insert @ A @ X3 @ F6 ) ) ) ) ) )
=> ( P2 @ F5 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_658_infinite__finite__induct,axiom,
! [A: $tType,P2: ( set @ A ) > $o,A5: set @ A] :
( ! [A7: set @ A] :
( ~ ( finite_finite2 @ A @ A7 )
=> ( P2 @ A7 ) )
=> ( ( P2 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P2 @ F6 )
=> ( P2 @ ( insert @ A @ X3 @ F6 ) ) ) ) )
=> ( P2 @ A5 ) ) ) ) ).
% infinite_finite_induct
thf(fact_659_infinite__remove,axiom,
! [A: $tType,S3: set @ A,A2: A] :
( ~ ( finite_finite2 @ A @ S3 )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_660_infinite__coinduct,axiom,
! [A: $tType,X6: ( set @ A ) > $o,A5: set @ A] :
( ( X6 @ A5 )
=> ( ! [A7: set @ A] :
( ( X6 @ A7 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A7 )
& ( ( X6 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A5 ) ) ) ).
% infinite_coinduct
thf(fact_661_finite__empty__induct,axiom,
! [A: $tType,A5: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( P2 @ A5 )
=> ( ! [A4: A,A7: set @ A] :
( ( finite_finite2 @ A @ A7 )
=> ( ( member @ A @ A4 @ A7 )
=> ( ( P2 @ A7 )
=> ( P2 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_662_subset__singleton__iff,axiom,
! [A: $tType,X6: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X6 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X6
= ( bot_bot @ ( set @ A ) ) )
| ( X6
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_663_subset__singletonD,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( A5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_664_subset__insert__iff,axiom,
! [A: $tType,A5: set @ A,X: A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B5 ) )
= ( ( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) )
& ( ~ ( member @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_665_Diff__single__insert,axiom,
! [A: $tType,A5: set @ A,X: A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_666_remove__subset,axiom,
! [A: $tType,X: A,S3: set @ A] :
( ( member @ A @ X @ S3 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ S3 ) ) ).
% remove_subset
thf(fact_667_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S3: set @ B,P2: ( set @ B ) > $o,F3: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( P2 @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B,S5: set @ B] :
( ( finite_finite2 @ B @ S5 )
=> ( ! [Y4: B] :
( ( member @ B @ Y4 @ S5 )
=> ( ord_less_eq @ A @ ( F3 @ Y4 ) @ ( F3 @ X3 ) ) )
=> ( ( P2 @ S5 )
=> ( P2 @ ( insert @ B @ X3 @ S5 ) ) ) ) )
=> ( P2 @ S3 ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_668_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( P2 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B4: A,A7: set @ A] :
( ( finite_finite2 @ A @ A7 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A7 )
=> ( ord_less @ A @ X5 @ B4 ) )
=> ( ( P2 @ A7 )
=> ( P2 @ ( insert @ A @ B4 @ A7 ) ) ) ) )
=> ( P2 @ A5 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_669_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( P2 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B4: A,A7: set @ A] :
( ( finite_finite2 @ A @ A7 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A7 )
=> ( ord_less @ A @ B4 @ X5 ) )
=> ( ( P2 @ A7 )
=> ( P2 @ ( insert @ A @ B4 @ A7 ) ) ) ) )
=> ( P2 @ A5 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_670_finite__subset__induct,axiom,
! [A: $tType,F5: set @ A,A5: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F5 @ A5 )
=> ( ( P2 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A4 @ A5 )
=> ( ~ ( member @ A @ A4 @ F6 )
=> ( ( P2 @ F6 )
=> ( P2 @ ( insert @ A @ A4 @ F6 ) ) ) ) ) )
=> ( P2 @ F5 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_671_finite__subset__induct_H,axiom,
! [A: $tType,F5: set @ A,A5: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F5 @ A5 )
=> ( ( P2 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A4 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F6 @ A5 )
=> ( ~ ( member @ A @ A4 @ F6 )
=> ( ( P2 @ F6 )
=> ( P2 @ ( insert @ A @ A4 @ F6 ) ) ) ) ) ) )
=> ( P2 @ F5 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_672_remove__induct,axiom,
! [A: $tType,P2: ( set @ A ) > $o,B5: set @ A] :
( ( P2 @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B5 )
=> ( P2 @ B5 ) )
=> ( ! [A7: set @ A] :
( ( finite_finite2 @ A @ A7 )
=> ( ( A7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A7 @ B5 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A7 )
=> ( P2 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P2 @ A7 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% remove_induct
thf(fact_673_finite__remove__induct,axiom,
! [A: $tType,B5: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B5 )
=> ( ( P2 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A7: set @ A] :
( ( finite_finite2 @ A @ A7 )
=> ( ( A7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A7 @ B5 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A7 )
=> ( P2 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P2 @ A7 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_674_finite__induct__select,axiom,
! [A: $tType,S3: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ S3 )
=> ( ( P2 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T5: set @ A] :
( ( ord_less @ ( set @ A ) @ T5 @ S3 )
=> ( ( P2 @ T5 )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( minus_minus @ ( set @ A ) @ S3 @ T5 ) )
& ( P2 @ ( insert @ A @ X5 @ T5 ) ) ) ) )
=> ( P2 @ S3 ) ) ) ) ).
% finite_induct_select
thf(fact_675_psubset__insert__iff,axiom,
! [A: $tType,A5: set @ A,X: A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B5 ) )
= ( ( ( member @ A @ X @ B5 )
=> ( ord_less @ ( set @ A ) @ A5 @ B5 ) )
& ( ~ ( member @ A @ X @ B5 )
=> ( ( ( member @ A @ X @ A5 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) )
& ( ~ ( member @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_676_finite__list,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [Xs3: list @ A] :
( ( set2 @ A @ Xs3 )
= A5 ) ) ).
% finite_list
thf(fact_677_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B5 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X4 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_678_geqmaxNone,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less_eq @ nat @ Ma @ X )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( none @ nat ) ) ) ) ).
% geqmaxNone
thf(fact_679_pred__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Px: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X )
= ( some @ nat @ Px ) )
= ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X @ Px ) ) ) ).
% pred_corr
thf(fact_680_pred__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X )
= ( some @ nat @ Sx ) )
= ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T2 ) @ X @ Sx ) ) ) ).
% pred_correct
thf(fact_681_set__vebt__pred,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Px: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X )
= ( some @ nat @ Px ) )
= ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T2 ) @ X @ Px ) ) ) ).
% set_vebt_pred
thf(fact_682_succ__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X )
= ( some @ nat @ Sx ) )
= ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X @ Sx ) ) ) ).
% succ_corr
thf(fact_683_succ__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X )
= ( some @ nat @ Sx ) )
= ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T2 ) @ X @ Sx ) ) ) ).
% succ_correct
thf(fact_684_maxt__sound,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X )
=> ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X ) ) ) ) ).
% maxt_sound
thf(fact_685_maxt__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X ) )
=> ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X ) ) ) ).
% maxt_corr
thf(fact_686_set__vebt__maxt,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X ) )
= ( vEBT_VEBT_max_in_set @ ( vEBT_set_vebt @ T2 ) @ X ) ) ) ).
% set_vebt_maxt
thf(fact_687_mint__sound,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X )
=> ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X ) ) ) ) ).
% mint_sound
thf(fact_688_maxbmo,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X ) )
=> ( vEBT_V8194947554948674370ptions @ T2 @ X ) ) ).
% maxbmo
thf(fact_689_minNullmin,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ T2 )
=> ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) ) ) ).
% minNullmin
thf(fact_690_minminNull,axiom,
! [T2: vEBT_VEBT] :
( ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) )
=> ( vEBT_VEBT_minNull @ T2 ) ) ).
% minminNull
thf(fact_691_vebt__minNull__mint,axiom,
( vEBT_VEBT_minNull
= ( ^ [T3: vEBT_VEBT] :
( ( vEBT_vebt_mint @ T3 )
= ( none @ nat ) ) ) ) ).
% vebt_minNull_mint
thf(fact_692_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_693_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_694_mint__corr__help,axiom,
! [T2: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ Mini ) )
=> ( ( vEBT_vebt_member @ T2 @ X )
=> ( ord_less_eq @ nat @ Mini @ X ) ) ) ) ).
% mint_corr_help
thf(fact_695_maxt__corr__help,axiom,
! [T2: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ Maxi ) )
=> ( ( vEBT_vebt_member @ T2 @ X )
=> ( ord_less_eq @ nat @ X @ Maxi ) ) ) ) ).
% maxt_corr_help
thf(fact_696_set__vebt__mint_H,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X ) )
= ( ( member @ nat @ X @ ( vEBT_set_vebt @ T2 ) )
& ! [X4: nat] :
( ( member @ nat @ X4 @ ( vEBT_set_vebt @ T2 ) )
=> ( ord_less_eq @ nat @ X @ X4 ) ) ) ) ) ).
% set_vebt_mint'
thf(fact_697_set__vebt__maxt_H,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X ) )
= ( ( member @ nat @ X @ ( vEBT_set_vebt @ T2 ) )
& ! [X4: nat] :
( ( member @ nat @ X4 @ ( vEBT_set_vebt @ T2 ) )
=> ( ord_less_eq @ nat @ X4 @ X ) ) ) ) ) ).
% set_vebt_maxt'
thf(fact_698_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_699_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_700_set__vebt__mint,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X ) )
= ( vEBT_VEBT_min_in_set @ ( vEBT_set_vebt @ T2 ) @ X ) ) ) ).
% set_vebt_mint
thf(fact_701_mint__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X ) )
=> ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X ) ) ) ).
% mint_corr
thf(fact_702_vebt__succ_Osimps_I2_J,axiom,
! [Uv2: $o,Uw2: $o,N: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N ) )
= ( none @ nat ) ) ).
% vebt_succ.simps(2)
thf(fact_703_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_704_vebt__succ_Osimps_I3_J,axiom,
! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz ) @ Va2 )
= ( none @ nat ) ) ).
% vebt_succ.simps(3)
thf(fact_705_vebt__pred_Osimps_I4_J,axiom,
! [Uy2: nat,Uz: list @ vEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz @ Va2 ) @ Vb )
= ( none @ nat ) ) ).
% vebt_pred.simps(4)
thf(fact_706_vebt__succ_Osimps_I4_J,axiom,
! [V3: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) @ Ve2 )
= ( none @ nat ) ) ).
% vebt_succ.simps(4)
thf(fact_707_vebt__pred_Osimps_I5_J,axiom,
! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Vf2 )
= ( none @ nat ) ) ).
% vebt_pred.simps(5)
thf(fact_708_vebt__succ_Osimps_I5_J,axiom,
! [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Vi2 )
= ( none @ nat ) ) ).
% vebt_succ.simps(5)
thf(fact_709_vebt__pred_Osimps_I6_J,axiom,
! [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Vj2 )
= ( none @ nat ) ) ).
% vebt_pred.simps(6)
thf(fact_710_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_711_vebt__succ__code_I1_J,axiom,
! [B2: $o,X: nat,A2: $o] :
( ( ( B2
& ( X
= ( zero_zero @ nat ) ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ ( B2
& ( X
= ( zero_zero @ nat ) ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( none @ nat ) ) ) ) ).
% vebt_succ_code(1)
thf(fact_712_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_713_vebt__pred__code,axiom,
! [X: nat,A2: $o,B2: $o] :
( ( ( X
= ( zero_zero @ nat ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( none @ nat ) ) )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> ( ( A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( none @ nat ) ) ) ) )
& ( ( X
!= ( one_one @ nat ) )
=> ( ( B2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( none @ nat ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred_code
thf(fact_714_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_715_vebt__maxt_Oelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_maxt @ X )
= Y )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( ( A4
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( Y
= ( none @ nat ) ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat] :
( ? [Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
=> ( Y
!= ( some @ nat @ Ma2 ) ) ) ) ) ) ).
% vebt_maxt.elims
thf(fact_716_vebt__mint_Oelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_mint @ X )
= Y )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ( ( A4
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( Y
= ( none @ nat ) ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat] :
( ? [Ma2: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
=> ( Y
!= ( some @ nat @ Mi2 ) ) ) ) ) ) ).
% vebt_mint.elims
thf(fact_717_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_718_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_719_vebt__maxt_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz ) )
= ( some @ nat @ Ma ) ) ).
% vebt_maxt.simps(3)
thf(fact_720_vebt__mint_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz ) )
= ( some @ nat @ Mi ) ) ).
% vebt_mint.simps(3)
thf(fact_721_set__vebt__mint_H_H,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( ( vEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) ) )
& ( ( ( vEBT_set_vebt @ T2 )
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ ( lattic643756798350308766er_Min @ nat @ ( vEBT_set_vebt @ T2 ) ) ) ) ) ) ) ).
% set_vebt_mint''
thf(fact_722_set__vebt__maxt_H_H,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( ( vEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( ( vEBT_vebt_maxt @ T2 )
= ( none @ nat ) ) )
& ( ( ( vEBT_set_vebt @ T2 )
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ ( lattic643756798349783984er_Max @ nat @ ( vEBT_set_vebt @ T2 ) ) ) ) ) ) ) ).
% set_vebt_maxt''
thf(fact_723_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_724_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_725_Max__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Max_singleton
thf(fact_726_Min__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Min_singleton
thf(fact_727_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X4 @ X ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_728_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less @ A @ X4 @ X ) ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_729_Min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X @ X4 ) ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_730_Min__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less @ A @ X @ X4 ) ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_731_Min_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ A2 ) ) ) ) ).
% Min.coboundedI
thf(fact_732_Min__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( member @ A @ X @ A5 )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= X ) ) ) ) ) ).
% Min_eqI
thf(fact_733_Min__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ X ) ) ) ) ).
% Min_le
thf(fact_734_Min__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ A5 ) ) ) ) ).
% Min_in
thf(fact_735_Max_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ord_less_eq @ A @ A2 @ ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ).
% Max.coboundedI
thf(fact_736_Max__eq__if,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ B5 )
& ( ord_less_eq @ A @ X3 @ Xa ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ A5 )
& ( ord_less_eq @ A @ X3 @ Xa ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= ( lattic643756798349783984er_Max @ A @ B5 ) ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_737_Max__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( member @ A @ X @ A5 )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= X ) ) ) ) ) ).
% Max_eqI
thf(fact_738_Max__ge,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ).
% Max_ge
thf(fact_739_Max__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ A5 ) ) ) ) ).
% Max_in
thf(fact_740_Min__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,M: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798350308766er_Min @ A @ A5 )
= M )
= ( ( member @ A @ M @ A5 )
& ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ M @ X4 ) ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_741_Min__le__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ X )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less_eq @ A @ X4 @ X ) ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_742_eq__Min__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,M: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( ( member @ A @ M @ A5 )
& ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ M @ X4 ) ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_743_Min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) )
=> ! [A9: A] :
( ( member @ A @ A9 @ A5 )
=> ( ord_less_eq @ A @ X @ A9 ) ) ) ) ) ) ).
% Min.boundedE
thf(fact_744_Min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( ord_less_eq @ A @ X @ A4 ) )
=> ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min.boundedI
thf(fact_745_Min__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ X )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less @ A @ X4 @ X ) ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_746_Min__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ A5 )
=> ( ord_less_eq @ A @ A2 @ B4 ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ A2 @ A5 ) )
= A2 ) ) ) ) ).
% Min_insert2
thf(fact_747_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,M: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798349783984er_Max @ A @ A5 )
= M )
= ( ( member @ A @ M @ A5 )
& ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X4 @ M ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_748_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less_eq @ A @ X @ X4 ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_749_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,M: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( ( member @ A @ M @ A5 )
& ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X4 @ M ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_750_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X )
=> ! [A9: A] :
( ( member @ A @ A9 @ A5 )
=> ( ord_less_eq @ A @ A9 @ X ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_751_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( ord_less_eq @ A @ A4 @ X ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X ) ) ) ) ) ).
% Max.boundedI
thf(fact_752_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less @ A @ X @ X4 ) ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_753_Max__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ A5 )
=> ( ord_less_eq @ A @ B4 @ A2 ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ A2 @ A5 ) )
= A2 ) ) ) ) ).
% Max_insert2
thf(fact_754_Min__antimono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M7: set @ A,N8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M7 @ N8 )
=> ( ( M7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N8 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N8 ) @ ( lattic643756798350308766er_Min @ A @ M7 ) ) ) ) ) ) ).
% Min_antimono
thf(fact_755_Min_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B5 ) @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min.subset_imp
thf(fact_756_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M7: set @ A,N8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M7 @ N8 )
=> ( ( M7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N8 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M7 ) @ ( lattic643756798349783984er_Max @ A @ N8 ) ) ) ) ) ) ).
% Max_mono
thf(fact_757_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ ( lattic643756798349783984er_Max @ A @ B5 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_758_greater__shift,axiom,
( ( ord_less @ nat )
= ( ^ [Y5: nat,X4: nat] : ( vEBT_VEBT_greater @ ( some @ nat @ X4 ) @ ( some @ nat @ Y5 ) ) ) ) ).
% greater_shift
thf(fact_759_less__shift,axiom,
( ( ord_less @ nat )
= ( ^ [X4: nat,Y5: nat] : ( vEBT_VEBT_less @ ( some @ nat @ X4 ) @ ( some @ nat @ Y5 ) ) ) ) ).
% less_shift
thf(fact_760_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,
! [V3: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
thf(fact_761_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_762_T_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_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_T_m_i_n_N_u_l_l @ X )
= Y )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Uv: $o] :
( X
= ( vEBT_Leaf @ $true @ Uv ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Uu: $o] :
( X
= ( vEBT_Leaf @ Uu @ $true ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ( ? [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> ( Y
!= ( one_one @ nat ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
thf(fact_763_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
! [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Vi2 )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
thf(fact_764_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_765_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
! [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Vi2 )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
thf(fact_766_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
! [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Vj2 )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
thf(fact_767_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
! [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Vj2 )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
thf(fact_768_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
( A6
= ( insert @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_769_is__singletonI_H,axiom,
! [A: $tType,A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( X3 = Y3 ) ) )
=> ( is_singleton @ A @ A5 ) ) ) ).
% is_singletonI'
thf(fact_770_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_771_T_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_Osimps_I1_J,axiom,
( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $false @ $false ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(1)
thf(fact_772_T_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_Osimps_I2_J,axiom,
! [Uv2: $o] :
( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $true @ Uv2 ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
thf(fact_773_T_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_Osimps_I3_J,axiom,
! [Uu2: $o] :
( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ Uu2 @ $true ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
thf(fact_774_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_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
thf(fact_775_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_776_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
! [Uv2: $o,Uw2: $o,N: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N ) )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
thf(fact_777_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
! [A2: $o,B2: $o,Va2: nat] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va2 ) ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
thf(fact_778_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_779_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
! [Uy2: nat,Uz: list @ vEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz @ Va2 ) @ Vb )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
thf(fact_780_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
! [Uv2: $o,Uw2: $o,N: nat] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N ) )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
thf(fact_781_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
! [Uy2: nat,Uz: list @ vEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz @ Va2 ) @ Vb )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
thf(fact_782_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz ) @ Va2 )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
thf(fact_783_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_784_is__singletonE,axiom,
! [A: $tType,A5: set @ A] :
( ( is_singleton @ A @ A5 )
=> ~ ! [X3: A] :
( A5
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_785_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
? [X4: A] :
( A6
= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_786_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz ) @ Va2 )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
thf(fact_787_T_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_Osimps_I5_J,axiom,
! [Uz: product_prod @ nat @ nat,Va2: nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va2 @ Vb @ Vc ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
thf(fact_788_T_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_Osimps_I4_J,axiom,
! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
thf(fact_789_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_I2_J,axiom,
! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
thf(fact_790_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Vf2 )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
thf(fact_791_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_792_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
! [V3: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) @ Ve2 )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
thf(fact_793_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Vf2 )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
thf(fact_794_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
! [V3: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) @ Ve2 )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
thf(fact_795_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,
! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
thf(fact_796_set__removeAll,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( set2 @ A @ ( removeAll @ A @ X @ Xs ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_removeAll
thf(fact_797_vebt__maxt_Opelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_maxt @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( ( A4
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
=> ( ( Y
= ( some @ nat @ Ma2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_maxt.pelims
thf(fact_798_vebt__mint_Opelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_mint @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ X )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( ( A4
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
=> ( ( Y
= ( some @ nat @ Mi2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_mint.pelims
thf(fact_799_height__node,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ord_less_eq @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% height_node
thf(fact_800_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_801_insersimp_H,axiom,
! [T2: vEBT_VEBT,N: nat,Y: 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 @ Y ) @ ( one_one @ nat ) ) ) ) ).
% insersimp'
thf(fact_802_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: A] :
( ( ( archimedean_frac @ A @ X )
= A2 )
= ( ( member @ A @ ( minus_minus @ A @ X @ A2 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ A2 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_803_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_I4_J,axiom,
! [V3: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
thf(fact_804_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X4: A,A6: set @ A] : ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_805_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,X: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
thf(fact_806_member__remove,axiom,
! [A: $tType,X: A,Y: A,A5: set @ A] :
( ( member @ A @ X @ ( remove @ A @ Y @ A5 ) )
= ( ( member @ A @ X @ A5 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_807_frac__eq__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( archimedean_frac @ A @ X )
= ( zero_zero @ A ) )
= ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% frac_eq_0_iff
thf(fact_808_frac__gt__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X ) )
= ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ) ).
% frac_gt_0_iff
thf(fact_809_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_810_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_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
thf(fact_811_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,X: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts2 @ S ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
thf(fact_812_T_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_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_T_m_i_n_N_u_l_l @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
=> ( ! [Uv: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $true @ Uv ) ) ) )
=> ( ! [Uu: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ $true ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ Uu @ $true ) ) ) )
=> ( ! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
thf(fact_813_delete__bound__height_H,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% delete_bound_height'
thf(fact_814_height__compose__child,axiom,
! [T2: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Info: option @ ( product_prod @ nat @ nat ),Deg: nat,Summary: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ T2 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ) ).
% height_compose_child
thf(fact_815_height__compose__list,axiom,
! [T2: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ T2 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_height @ T2 ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary @ ( set2 @ vEBT_VEBT @ TreeList ) ) ) ) ) ) ).
% height_compose_list
thf(fact_816_height__compose__summary,axiom,
! [Summary: vEBT_VEBT,Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ Summary ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ).
% height_compose_summary
thf(fact_817_VEBT__internal_OminNull_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ( Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
=> ( ! [Uv: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv ) ) ) )
=> ( ! [Uu: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ $true ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu @ $true ) ) ) )
=> ( ! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) )
=> ( Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(1)
thf(fact_818_VEBT__internal_OminNull_Opelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(2)
thf(fact_819_VEBT__internal_OminNull_Opelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ! [Uv: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv ) ) )
=> ( ! [Uu: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ $true ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu @ $true ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(3)
thf(fact_820_Ints__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_0
thf(fact_821_enumerate__Suc_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) )
= ( infini527867602293511546merate @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ ( infini527867602293511546merate @ A @ S3 @ ( zero_zero @ nat ) ) @ ( bot_bot @ ( set @ A ) ) ) ) @ N ) ) ) ).
% enumerate_Suc'
thf(fact_822_even__odd__cases,axiom,
! [X: nat] :
( ! [N2: nat] :
( X
!= ( plus_plus @ nat @ N2 @ N2 ) )
=> ~ ! [N2: nat] :
( X
!= ( plus_plus @ nat @ N2 @ ( suc @ N2 ) ) ) ) ).
% even_odd_cases
thf(fact_823_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C3 @ A2 ) )
= ( B2 = C3 ) ) ) ).
% add_right_cancel
thf(fact_824_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C3 ) )
= ( B2 = C3 ) ) ) ).
% add_left_cancel
thf(fact_825_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A,X: B,A5: set @ B] :
( ( B2
= ( F3 @ X ) )
=> ( ( member @ B @ X @ A5 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A5 ) ) ) ) ).
% image_eqI
thf(fact_826_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_827_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_828_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_829_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_830_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_831_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_832_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_833_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_834_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X @ Y ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_835_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add_0
thf(fact_836_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_837_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_838_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_839_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_840_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_841_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_842_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_843_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_844_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_845_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_846_image__empty,axiom,
! [B: $tType,A: $tType,F3: B > A] :
( ( image @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_847_empty__is__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image @ B @ A @ F3 @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_848_image__is__empty,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( ( image @ B @ A @ F3 @ A5 )
= ( bot_bot @ ( set @ A ) ) )
= ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_849_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_850_finite__imageI,axiom,
! [B: $tType,A: $tType,F5: set @ A,H2: A > B] :
( ( finite_finite2 @ A @ F5 )
=> ( finite_finite2 @ B @ ( image @ A @ B @ H2 @ F5 ) ) ) ).
% finite_imageI
thf(fact_851_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_852_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_853_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_854_insert__image,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,F3: A > B] :
( ( member @ A @ X @ A5 )
=> ( ( insert @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ A5 ) )
= ( image @ A @ B @ F3 @ A5 ) ) ) ).
% insert_image
thf(fact_855_image__insert,axiom,
! [A: $tType,B: $tType,F3: B > A,A2: B,B5: set @ B] :
( ( image @ B @ A @ F3 @ ( insert @ B @ A2 @ B5 ) )
= ( insert @ A @ ( F3 @ A2 ) @ ( image @ B @ A @ F3 @ B5 ) ) ) ).
% image_insert
thf(fact_856_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_857_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_858_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_859_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_860_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_861_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_862_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_863_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_864_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_865_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_866_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_867_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_868_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_869_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_870_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_871_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_872_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_873_image__add__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S3 )
= S3 ) ) ).
% image_add_0
thf(fact_874_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_875_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_876_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_877_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_878_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_879_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_880_enumerate__mono__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,M: nat,N: nat] :
( ~ ( finite_finite2 @ A @ S3 )
=> ( ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ M ) @ ( infini527867602293511546merate @ A @ S3 @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ) ).
% enumerate_mono_iff
thf(fact_881_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C3 @ A2 ) )
=> ( B2 = C3 ) ) ) ).
% add_right_imp_eq
thf(fact_882_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C3 ) )
=> ( B2 = C3 ) ) ) ).
% add_left_imp_eq
thf(fact_883_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C3 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_884_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A8: A,B8: A] : ( plus_plus @ A @ B8 @ A8 ) ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_885_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C3 @ A2 ) )
= ( B2 = C3 ) ) ) ).
% add.right_cancel
thf(fact_886_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C3 ) )
= ( B2 = C3 ) ) ) ).
% add.left_cancel
thf(fact_887_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add.assoc
thf(fact_888_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B5: A,K: A,B2: A,A2: A] :
( ( B5
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B5 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_889_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: A,K: A,A2: A,B2: A] :
( ( A5
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A5 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_890_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_891_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,B2: B,F3: A > B] :
( ( member @ A @ X @ A5 )
=> ( ( B2
= ( F3 @ X ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F3 @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_892_ball__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P2: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F3 @ A5 ) )
=> ( P2 @ X3 ) )
=> ! [X5: B] :
( ( member @ B @ X5 @ A5 )
=> ( P2 @ ( F3 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_893_image__cong,axiom,
! [B: $tType,A: $tType,M7: set @ A,N8: set @ A,F3: A > B,G3: A > B] :
( ( M7 = N8 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N8 )
=> ( ( F3 @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( image @ A @ B @ F3 @ M7 )
= ( image @ A @ B @ G3 @ N8 ) ) ) ) ).
% image_cong
thf(fact_894_bex__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P2: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F3 @ A5 ) )
& ( P2 @ X5 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A5 )
& ( P2 @ ( F3 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_895_image__iff,axiom,
! [A: $tType,B: $tType,Z: A,F3: B > A,A5: set @ B] :
( ( member @ A @ Z @ ( image @ B @ A @ F3 @ A5 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( Z
= ( F3 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_896_imageI,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,F3: A > B] :
( ( member @ A @ X @ A5 )
=> ( member @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ A5 ) ) ) ).
% imageI
thf(fact_897_image__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ A,F3: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ ( image @ A @ B @ F3 @ B5 ) ) ) ).
% image_mono
thf(fact_898_image__subsetI,axiom,
! [A: $tType,B: $tType,A5: set @ A,F3: A > B,B5: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ B @ ( F3 @ X3 ) @ B5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ B5 ) ) ).
% image_subsetI
thf(fact_899_subset__imageE,axiom,
! [A: $tType,B: $tType,B5: set @ A,F3: B > A,A5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F3 @ A5 ) )
=> ~ ! [C6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C6 @ A5 )
=> ( B5
!= ( image @ B @ A @ F3 @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_900_image__subset__iff,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ A5 ) @ B5 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( member @ A @ ( F3 @ X4 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_901_subset__image__iff,axiom,
! [A: $tType,B: $tType,B5: set @ A,F3: B > A,A5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F3 @ A5 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A5 )
& ( B5
= ( image @ B @ A @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_902_all__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P2: ( set @ A ) > $o] :
( ( ! [B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F3 @ A5 ) )
=> ( P2 @ B7 ) ) )
= ( ! [B7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B7 @ A5 )
=> ( P2 @ ( image @ B @ A @ F3 @ B7 ) ) ) ) ) ).
% all_subset_image
thf(fact_903_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% verit_sum_simplify
thf(fact_904_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_905_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_906_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_907_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_908_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_909_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B8: A] :
? [C7: A] :
( B8
= ( plus_plus @ A @ A8 @ C7 ) ) ) ) ) ).
% le_iff_add
thf(fact_910_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add_right_mono
thf(fact_911_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ! [C2: A] :
( B2
!= ( plus_plus @ A @ A2 @ C2 ) ) ) ) ).
% less_eqE
thf(fact_912_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_913_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_mono
thf(fact_914_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_915_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_916_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_917_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_918_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_919_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_920_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_strict_mono
thf(fact_921_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_922_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add_strict_right_mono
thf(fact_923_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_924_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_925_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C3 )
= ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% diff_diff_eq
thf(fact_926_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ( plus_plus @ A @ C3 @ B2 )
= A2 )
=> ( C3
= ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% add_implies_diff
thf(fact_927_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C3 ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_928_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C3 )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C3 ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_929_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C3 ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_930_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% add_diff_eq
thf(fact_931_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( A2
= ( minus_minus @ A @ C3 @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= C3 ) ) ) ).
% eq_diff_eq
thf(fact_932_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= C3 )
= ( A2
= ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_933_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A5: A,K: A,A2: A,B2: A] :
( ( A5
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( minus_minus @ A @ A5 @ B2 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_934_nat__arith_Osuc1,axiom,
! [A5: nat,K: nat,A2: nat] :
( ( A5
= ( plus_plus @ nat @ K @ A2 ) )
=> ( ( suc @ A5 )
= ( plus_plus @ nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_935_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_936_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_937_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_938_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_939_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_940_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_941_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_942_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_943_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_944_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_945_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_946_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_947_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M5: nat,N6: nat] :
? [K3: nat] :
( N6
= ( plus_plus @ nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_948_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_949_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_950_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_951_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_952_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_953_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_954_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_955_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_956_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_957_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_958_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_959_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_960_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_961_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_962_enumerate__in__set,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ S3 )
=> ( member @ A @ ( infini527867602293511546merate @ A @ S3 @ N ) @ S3 ) ) ) ).
% enumerate_in_set
thf(fact_963_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_964_enumerate__Ex,axiom,
! [S3: set @ nat,S: nat] :
( ~ ( finite_finite2 @ nat @ S3 )
=> ( ( member @ nat @ S @ S3 )
=> ? [N2: nat] :
( ( infini527867602293511546merate @ nat @ S3 @ N2 )
= S ) ) ) ).
% enumerate_Ex
thf(fact_965_all__finite__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P2: ( set @ A ) > $o] :
( ( ! [B7: set @ A] :
( ( ( finite_finite2 @ A @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F3 @ A5 ) ) )
=> ( P2 @ B7 ) ) )
= ( ! [B7: set @ B] :
( ( ( finite_finite2 @ B @ B7 )
& ( ord_less_eq @ ( set @ B ) @ B7 @ A5 ) )
=> ( P2 @ ( image @ B @ A @ F3 @ B7 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_966_ex__finite__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P2: ( set @ A ) > $o] :
( ( ? [B7: set @ A] :
( ( finite_finite2 @ A @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F3 @ A5 ) )
& ( P2 @ B7 ) ) )
= ( ? [B7: set @ B] :
( ( finite_finite2 @ B @ B7 )
& ( ord_less_eq @ ( set @ B ) @ B7 @ A5 )
& ( P2 @ ( image @ B @ A @ F3 @ B7 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_967_finite__subset__image,axiom,
! [A: $tType,B: $tType,B5: set @ A,F3: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F3 @ A5 ) )
=> ? [C6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C6 @ A5 )
& ( finite_finite2 @ B @ C6 )
& ( B5
= ( image @ B @ A @ F3 @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_968_finite__surj,axiom,
! [A: $tType,B: $tType,A5: set @ A,B5: set @ B,F3: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ ( image @ A @ B @ F3 @ A5 ) )
=> ( finite_finite2 @ B @ B5 ) ) ) ).
% finite_surj
thf(fact_969_image__diff__subset,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,B5: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image @ B @ A @ F3 @ A5 ) @ ( image @ B @ A @ F3 @ B5 ) ) @ ( image @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ B5 ) ) ) ).
% image_diff_subset
thf(fact_970_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_971_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_972_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_973_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_974_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C3 ) ) ) ) ) ).
% add_increasing2
thf(fact_975_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_976_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C3 ) ) ) ) ) ).
% add_increasing
thf(fact_977_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_978_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C3 ) ) ) ) ) ).
% pos_add_strict
thf(fact_979_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ! [C2: A] :
( ( B2
= ( plus_plus @ A @ A2 @ C2 ) )
=> ( C2
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_980_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_981_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_982_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X @ Y ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_983_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_984_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_985_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_986_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_987_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_988_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_989_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( minus_minus @ A @ B2 @ A2 )
= C3 )
= ( B2
= ( plus_plus @ A @ C3 @ A2 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_990_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_991_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ C3 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C3 @ A2 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_992_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C3 ) @ A2 )
= ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_993_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C3 )
= ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C3 ) @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_994_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C3 @ B2 ) @ A2 )
= ( plus_plus @ A @ C3 @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_995_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ C3 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C3 @ B2 ) @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_996_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_997_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ C3 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C3 ) @ A2 ) ) ) ) ).
% le_add_diff
thf(fact_998_ordered__cancel__comm__monoid__diff__class_Odiff__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 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_999_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( minus_minus @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% le_diff_eq
thf(fact_1000_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C3 )
= ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% diff_le_eq
thf(fact_1001_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_1002_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_1003_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C3 )
= ( ord_less @ A @ A2 @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% diff_less_eq
thf(fact_1004_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ A2 @ ( minus_minus @ A @ C3 @ B2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% less_diff_eq
thf(fact_1005_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_1006_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_1007_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_1008_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_1009_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ~ ! [Q5: nat] :
( N
!= ( suc @ ( plus_plus @ nat @ M @ Q5 ) ) ) ) ).
% less_natE
thf(fact_1010_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1011_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1012_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M5: nat,N6: nat] :
? [K3: nat] :
( N6
= ( suc @ ( plus_plus @ nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1013_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_1014_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_1015_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_1016_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1017_Suc__eq__plus1,axiom,
( suc
= ( ^ [N6: nat] : ( plus_plus @ nat @ N6 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_1018_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P2: A > $o,K: A,F3: A > nat,N: nat] :
( ( P2 @ K )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ? [Y4: A] :
( ( P2 @ Y4 )
& ~ ( ord_less_eq @ nat @ ( F3 @ Y4 ) @ ( F3 @ X3 ) ) ) )
=> ? [Y3: A] :
( ( P2 @ Y3 )
& ~ ( ord_less @ nat @ ( F3 @ Y3 ) @ ( plus_plus @ nat @ ( F3 @ K ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_1019_mono__nat__linear__lb,axiom,
! [F3: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( ord_less @ nat @ ( F3 @ M3 ) @ ( F3 @ N2 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F3 @ M ) @ K ) @ ( F3 @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1020_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_1021_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_1022_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_1023_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_1024_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_1025_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_1026_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_1027_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_1028_le__enumerate,axiom,
! [S3: set @ nat,N: nat] :
( ~ ( finite_finite2 @ nat @ S3 )
=> ( ord_less_eq @ nat @ N @ ( infini527867602293511546merate @ nat @ S3 @ N ) ) ) ).
% le_enumerate
thf(fact_1029_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B,X: A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( ( F3 @ Y3 )
= ( F3 @ X ) ) )
=> ( ( the_elem @ B @ ( image @ A @ B @ F3 @ A5 ) )
= ( F3 @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1030_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_1031_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C3 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_1032_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C3 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_1033_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_1034_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_1035_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_1036_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_1037_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_1038_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A8: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A8 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_1039_nat__diff__split__asm,axiom,
! [P2: nat > $o,A2: nat,B2: nat] :
( ( P2 @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( ~ ( ( ( ord_less @ nat @ A2 @ B2 )
& ~ ( P2 @ ( zero_zero @ nat ) ) )
| ? [D6: nat] :
( ( A2
= ( plus_plus @ nat @ B2 @ D6 ) )
& ~ ( P2 @ D6 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1040_nat__diff__split,axiom,
! [P2: nat > $o,A2: nat,B2: nat] :
( ( P2 @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( ( ( ord_less @ nat @ A2 @ B2 )
=> ( P2 @ ( zero_zero @ nat ) ) )
& ! [D6: nat] :
( ( A2
= ( plus_plus @ nat @ B2 @ D6 ) )
=> ( P2 @ D6 ) ) ) ) ).
% nat_diff_split
thf(fact_1041_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_1042_VEBT__internal_Oheight_Osimps_I2_J,axiom,
! [Uu2: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_VEBT_height @ ( vEBT_Node @ Uu2 @ Deg @ TreeList @ Summary ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary @ ( set2 @ vEBT_VEBT @ TreeList ) ) ) ) ) ) ).
% VEBT_internal.height.simps(2)
thf(fact_1043_enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ S3 )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ N ) @ ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) ) ) ) ) ).
% enumerate_step
thf(fact_1044_enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M: nat,N: nat,S3: set @ A] :
( ( ord_less @ nat @ M @ N )
=> ( ~ ( finite_finite2 @ A @ S3 )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ M ) @ ( infini527867602293511546merate @ A @ S3 @ N ) ) ) ) ) ).
% enumerate_mono
thf(fact_1045_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M5: nat,N6: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ N6
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) @ N6 ) ) ) ) ) ).
% add_eq_if
thf(fact_1046_frac__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% frac_add
thf(fact_1047_VEBT__internal_Oheight_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_height @ X )
= Y )
=> ( ( ? [A4: $o,B4: $o] :
( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
!= ( zero_zero @ nat ) ) )
=> ~ ! [Uu: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.height.elims
thf(fact_1048_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
! [A2: $o,B2: $o,Va2: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va2 ) ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B2 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
thf(fact_1049_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_1050_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_1051_pred__bound__height_H,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_p_r_e_d2 @ T2 @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% pred_bound_height'
thf(fact_1052_succ_H__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_s_u_c_c2 @ T2 @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% succ'_bound_height
thf(fact_1053_insert_H__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% insert'_bound_height
thf(fact_1054_member__bound__height_H,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_m_e_m_b_e_r2 @ T2 @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% member_bound_height'
thf(fact_1055_VEBT__internal_Oheight_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_height @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ X )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( zero_zero @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
=> ~ ! [Uu: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Node @ Uu @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% VEBT_internal.height.pelims
thf(fact_1056_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_1057_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ X @ ( plus_plus @ A @ Y @ E2 ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_epsilon
thf(fact_1058_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_T_m_i_n_t @ X )
= Y )
=> ( ! [A4: $o] :
( ? [B4: $o] :
( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ A4 @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
=> ( Y
!= ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
thf(fact_1059_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_T_m_a_x_t @ X )
= Y )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B4 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
=> ( Y
!= ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
thf(fact_1060_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_T_m_i_n_t @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ X )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ A4 @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( 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,Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
thf(fact_1061_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_T_m_a_x_t @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ X )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B4 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
thf(fact_1062_Euclid__induct,axiom,
! [P2: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( P2 @ A4 @ B4 )
= ( P2 @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P2 @ A4 @ ( zero_zero @ nat ) )
=> ( ! [A4: nat,B4: nat] :
( ( P2 @ A4 @ B4 )
=> ( P2 @ A4 @ ( plus_plus @ nat @ A4 @ B4 ) ) )
=> ( P2 @ A2 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_1063_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus @ nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_1064_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ( times_times @ A @ A2 @ C3 )
= ( times_times @ A @ B2 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_1065_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ( times_times @ A @ C3 @ A2 )
= ( times_times @ A @ C3 @ B2 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_1066_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_1067_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_1068_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_1069_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.right_neutral
thf(fact_1070_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% mult_1
thf(fact_1071_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_1072_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_1073_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_1074_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_1075_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_1076_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_1077_triangle__0,axiom,
( ( nat_triangle @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% triangle_0
thf(fact_1078_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A2: A,C3: A] :
( ( ( times_times @ A @ A2 @ C3 )
= C3 )
= ( ( C3
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_1079_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,B2: A] :
( ( C3
= ( times_times @ A @ B2 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_1080_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,A2: A] :
( ( ( times_times @ A @ C3 @ A2 )
= C3 )
= ( ( C3
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_1081_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,B2: A] :
( ( C3
= ( times_times @ A @ C3 @ B2 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_1082_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_1083_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_1084_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_1085_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_1086_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_1087_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_1088_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_1089_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C3 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% mult.assoc
thf(fact_1090_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A8: A,B8: A] : ( times_times @ A @ B8 @ A8 ) ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_1091_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C3 ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_1092_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C3 )
= ( times_times @ A @ B2 @ C3 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_1093_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C3 @ A2 )
= ( times_times @ A @ C3 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_1094_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_1095_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_1096_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_1097_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_1098_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_1099_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_1100_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_1101_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_1102_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_1103_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1104_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_1105_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_1106_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_1107_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_1108_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_1109_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_1110_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_1111_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_1112_zero__notin__Suc__image,axiom,
! [A5: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ A5 ) ) ).
% zero_notin_Suc_image
thf(fact_1113_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1114_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_1115_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_1116_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_1117_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_1118_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_1119_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_1120_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_1121_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% mult_right_mono
thf(fact_1122_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1123_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_left_mono
thf(fact_1124_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_1125_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1126_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_1127_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_1128_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1129_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_1130_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_1131_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_1132_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_1133_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_1134_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_1135_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_1136_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_1137_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_1138_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_1139_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_1140_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1141_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1142_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1143_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1144_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1145_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1146_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1147_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1148_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1149_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_1150_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_1151_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_1152_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_1153_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_1154_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_1155_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_1156_mlex__snd__decrI,axiom,
! [A2: nat,A3: nat,B2: nat,B3: nat,N8: nat] :
( ( A2 = A3 )
=> ( ( ord_less @ nat @ B2 @ B3 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N8 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N8 ) @ B3 ) ) ) ) ).
% mlex_snd_decrI
thf(fact_1157_mlex__fst__decrI,axiom,
! [A2: nat,A3: nat,B2: nat,N8: nat,B3: nat] :
( ( ord_less @ nat @ A2 @ A3 )
=> ( ( ord_less @ nat @ B2 @ N8 )
=> ( ( ord_less @ nat @ B3 @ N8 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N8 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N8 ) @ B3 ) ) ) ) ) ).
% mlex_fst_decrI
thf(fact_1158_mlex__bound,axiom,
! [A2: nat,A5: nat,B2: nat,N8: nat] :
( ( ord_less @ nat @ A2 @ A5 )
=> ( ( ord_less @ nat @ B2 @ N8 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N8 ) @ B2 ) @ ( times_times @ nat @ A5 @ N8 ) ) ) ) ).
% mlex_bound
thf(fact_1159_mlex__leI,axiom,
! [A2: nat,A3: nat,B2: nat,B3: nat,N8: nat] :
( ( ord_less_eq @ nat @ A2 @ A3 )
=> ( ( ord_less_eq @ nat @ B2 @ B3 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N8 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N8 ) @ B3 ) ) ) ) ).
% mlex_leI
thf(fact_1160_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less @ A @ Z3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X ) @ Y ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_mult_one_interval
thf(fact_1161_in__image__insert__iff,axiom,
! [A: $tType,B5: set @ ( set @ A ),X: A,A5: set @ A] :
( ! [C6: set @ A] :
( ( member @ ( set @ A ) @ C6 @ B5 )
=> ~ ( member @ A @ X @ C6 ) )
=> ( ( member @ ( set @ A ) @ A5 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X ) @ B5 ) )
= ( ( member @ A @ X @ A5 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1162_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1163_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1164_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_1165_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1166_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1167_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_1168_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1169_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1170_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1171_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1172_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_1173_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_1174_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1175_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1176_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_1177_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C3: A,A2: A] :
( ( ord_less_eq @ A @ C3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ A2 ) ) ) ) ).
% mult_left_le
thf(fact_1178_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_1179_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X @ Y ) @ X ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1180_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y @ X ) @ X ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1181_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_squares_lt_zero
thf(fact_1182_ordered__ring__class_Ole__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E3: A,C3: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E3 ) @ C3 ) @ D2 ) ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1183_ordered__ring__class_Ole__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E3: A,C3: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) )
= ( ord_less_eq @ A @ C3 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1184_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E3: A,C3: A,B2: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) )
= ( ord_less @ A @ C3 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ).
% less_add_iff2
thf(fact_1185_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E3: A,C3: A,B2: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E3 ) @ C3 ) @ D2 ) ) ) ).
% less_add_iff1
thf(fact_1186_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_1187_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_1188_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_1189_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X5: A] :
? [X_1: A] : ( ord_less @ A @ X5 @ X_1 ) ) ).
% linordered_field_no_ub
thf(fact_1190_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X5: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X5 ) ) ).
% linordered_field_no_lb
thf(fact_1191_mult__le__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1192_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A2 ) @ C3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1193_mult__le__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1194_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ C3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1195_mult__less__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1196_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A2: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A2 ) @ C3 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1197_mult__less__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1198_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C3: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ C3 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1199_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X: A,A2: A,Y: A,U: A,V3: A] :
( ( ord_less_eq @ A @ X @ A2 )
=> ( ( ord_less_eq @ A @ Y @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V3 )
=> ( ( ( plus_plus @ A @ U @ V3 )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V3 @ Y ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1200_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M5: nat,N6: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N6 @ ( times_times @ nat @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) @ N6 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1201_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_T_m_a_x_t @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B2 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
thf(fact_1202_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
thf(fact_1203_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
thf(fact_1204_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X: A,A2: A,Y: A,U: A,V3: A] :
( ( ord_less @ A @ X @ A2 )
=> ( ( ord_less @ A @ Y @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V3 )
=> ( ( ( plus_plus @ A @ U @ V3 )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V3 @ Y ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1205_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
thf(fact_1206_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_1207_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
thf(fact_1208_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_1209_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_1210_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1211_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_1212_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_1213_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_1214_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_1215_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_1216_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_1217_star__false__left,axiom,
! [P2: assn] :
( ( times_times @ assn @ ( bot_bot @ assn ) @ P2 )
= ( bot_bot @ assn ) ) ).
% star_false_left
thf(fact_1218_star__false__right,axiom,
! [P2: assn] :
( ( times_times @ assn @ P2 @ ( bot_bot @ assn ) )
= ( bot_bot @ assn ) ) ).
% star_false_right
thf(fact_1219_snga__same__false,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P3: array @ A,X: list @ A,Y: list @ A] :
( ( times_times @ assn @ ( snga_assn @ A @ P3 @ X ) @ ( snga_assn @ A @ P3 @ Y ) )
= ( bot_bot @ assn ) ) ) ).
% snga_same_false
thf(fact_1220_sngr__same__false,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P3: ref @ A,X: A,Y: A] :
( ( times_times @ assn @ ( sngr_assn @ A @ P3 @ X ) @ ( sngr_assn @ A @ P3 @ Y ) )
= ( bot_bot @ assn ) ) ) ).
% sngr_same_false
thf(fact_1221_mod__h__bot__iff_I5_J,axiom,
! [P2: assn,Q2: assn,H2: heap_ext @ product_unit] :
( ( rep_assn @ ( times_times @ assn @ P2 @ Q2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
& ( rep_assn @ Q2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(5)
thf(fact_1222_mod__starE,axiom,
! [A2: assn,B2: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A2 @ B2 ) @ H2 )
=> ~ ( ? [X_1: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : ( rep_assn @ A2 @ X_1 )
=> ! [H_2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ B2 @ H_2 ) ) ) ).
% mod_starE
thf(fact_1223_mod__starD,axiom,
! [A5: assn,B5: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A5 @ B5 ) @ H2 )
=> ? [H1: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),H22: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ A5 @ H1 )
& ( rep_assn @ B5 @ H22 ) ) ) ).
% mod_starD
thf(fact_1224_assn__one__left,axiom,
! [P2: assn] :
( ( times_times @ assn @ ( one_one @ assn ) @ P2 )
= P2 ) ).
% assn_one_left
thf(fact_1225_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_1226_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_1227_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_1228_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1229_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1230_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_1231_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_1232_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_1233_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ X ) @ ( times_times @ A @ Z @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1234_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ Y @ Z ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1235_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R2: A,A2: A,B2: A,C3: A,D2: A] :
( ( R2
!= ( zero_zero @ A ) )
=> ( ( ( A2 = B2 )
& ( C3 != D2 ) )
=> ( ( plus_plus @ A @ A2 @ ( times_times @ A @ R2 @ C3 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R2 @ D2 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1236_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ Y @ Z ) )
= ( ord_less @ A @ X @ Y ) ) ) ) ).
% mult_less_iff1
thf(fact_1237_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_1238_mul__shift,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( times_times @ nat @ X @ Y )
= Z )
= ( ( vEBT_VEBT_mul @ ( some @ nat @ X ) @ ( some @ nat @ Y ) )
= ( some @ nat @ Z ) ) ) ).
% mul_shift
thf(fact_1239_minus__Max__eq__Min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798349783984er_Max @ A @ S3 ) )
= ( lattic643756798350308766er_Min @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ S3 ) ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_1240_minus__Min__eq__Max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798350308766er_Min @ A @ S3 ) )
= ( lattic643756798349783984er_Max @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ S3 ) ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_1241_add__shift,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( plus_plus @ nat @ X @ Y )
= Z )
= ( ( vEBT_VEBT_add @ ( some @ nat @ X ) @ ( some @ nat @ Y ) )
= ( some @ nat @ Z ) ) ) ).
% add_shift
thf(fact_1242_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_1243_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_1244_Compl__subset__Compl__iff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1245_Compl__anti__mono,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% Compl_anti_mono
thf(fact_1246_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_1247_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_1248_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_1249_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_1250_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_1251_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_1252_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% compl_le_compl_iff
thf(fact_1253_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_1254_compl__less__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% compl_less_compl_iff
thf(fact_1255_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_1256_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_1257_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_1258_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_1259_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_1260_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_1261_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_1262_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_1263_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_1264_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_1265_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_1266_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_1267_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_1268_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_1269_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_1270_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_1271_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_1272_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_1273_subset__Compl__singleton,axiom,
! [A: $tType,A5: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B2 @ A5 ) ) ) ).
% subset_Compl_singleton
thf(fact_1274_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_1275_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_1276_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_1277_add__def,axiom,
( vEBT_VEBT_add
= ( vEBT_V2048590022279873568_shift @ nat @ ( plus_plus @ nat ) ) ) ).
% add_def
thf(fact_1278_mul__def,axiom,
( vEBT_VEBT_mul
= ( vEBT_V2048590022279873568_shift @ nat @ ( times_times @ nat ) ) ) ).
% mul_def
thf(fact_1279_assn__times__comm,axiom,
( ( times_times @ assn )
= ( ^ [P5: assn,Q6: assn] : ( times_times @ assn @ Q6 @ P5 ) ) ) ).
% assn_times_comm
thf(fact_1280_assn__times__assoc,axiom,
! [P2: assn,Q2: assn,R3: assn] :
( ( times_times @ assn @ ( times_times @ assn @ P2 @ Q2 ) @ R3 )
= ( times_times @ assn @ P2 @ ( times_times @ assn @ Q2 @ R3 ) ) ) ).
% assn_times_assoc
thf(fact_1281_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_1282_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_1283_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_1284_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_1285_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_1286_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_1287_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_1288_compl__mono,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% compl_mono
thf(fact_1289_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_1290_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_1291_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_1292_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_less_swap2
thf(fact_1293_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_less_swap1
thf(fact_1294_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A5: A,K: A,A2: A] :
( ( A5
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( uminus_uminus @ A @ A5 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_1295_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_1296_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_1297_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_1298_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_1299_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_1300_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_1301_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_1302_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_1303_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_1304_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_1305_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_1306_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_1307_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B5: A,K: A,B2: A,A2: A] :
( ( B5
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( minus_minus @ A @ A2 @ B5 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_1308_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A8: A,B8: A] : ( plus_plus @ A @ A8 @ ( uminus_uminus @ A @ B8 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1309_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A8: A,B8: A] : ( plus_plus @ A @ A8 @ ( uminus_uminus @ A @ B8 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1310_subset__Compl__self__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_1311_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_1312_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_1313_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_1314_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_1315_Compl__insert,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ A5 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_1316_frac__neg,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
= ( zero_zero @ A ) ) )
& ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X ) ) ) ) ) ) ).
% frac_neg
thf(fact_1317_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_1318_foldr__one,axiom,
! [D2: nat,Ys: list @ nat] : ( ord_less_eq @ nat @ D2 @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Ys @ D2 ) ) ).
% foldr_one
thf(fact_1319_Gcd__remove0__nat,axiom,
! [M7: set @ nat] :
( ( finite_finite2 @ nat @ M7 )
=> ( ( gcd_Gcd @ nat @ M7 )
= ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M7 @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_1320_Gcd__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( ( gcd_Gcd @ A @ A5 )
= ( zero_zero @ A ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Gcd_0_iff
thf(fact_1321_power_Opower__eq__if,axiom,
! [A: $tType] :
( ( power2 @ A )
= ( ^ [One: A,Times: A > A > A,P7: A,M5: nat] :
( if @ A
@ ( M5
= ( zero_zero @ nat ) )
@ One
@ ( Times @ P7 @ ( power2 @ A @ One @ Times @ P7 @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% power.power_eq_if
thf(fact_1322_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( X
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
thf(fact_1323_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,X: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
thf(fact_1324_the__dflt__None__empty,axiom,
! [A: $tType] :
( ( dflt_None_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( none @ ( set @ A ) ) ) ).
% the_dflt_None_empty
thf(fact_1325_ComplI,axiom,
! [A: $tType,C3: A,A5: set @ A] :
( ~ ( member @ A @ C3 @ A5 )
=> ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% ComplI
thf(fact_1326_Compl__iff,axiom,
! [A: $tType,C3: A,A5: set @ A] :
( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( ~ ( member @ A @ C3 @ A5 ) ) ) ).
% Compl_iff
thf(fact_1327_Compl__eq__Compl__iff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A5 )
= ( uminus_uminus @ ( set @ A ) @ B5 ) )
= ( A5 = B5 ) ) ).
% Compl_eq_Compl_iff
thf(fact_1328_the__dflt__None__nonempty,axiom,
! [A: $tType,S3: set @ A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( dflt_None_set @ A @ S3 )
= ( some @ ( set @ A ) @ S3 ) ) ) ).
% the_dflt_None_nonempty
thf(fact_1329_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_1330_ComplD,axiom,
! [A: $tType,C3: A,A5: set @ A] :
( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
=> ~ ( member @ A @ C3 @ A5 ) ) ).
% ComplD
thf(fact_1331_double__complement,axiom,
! [A: $tType,A5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= A5 ) ).
% double_complement
thf(fact_1332_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_1333_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_1334_dflt__None__set__def,axiom,
! [A: $tType] :
( ( dflt_None_set @ A )
= ( ^ [S6: set @ A] :
( if @ ( option @ ( set @ A ) )
@ ( S6
= ( bot_bot @ ( set @ A ) ) )
@ ( none @ ( set @ A ) )
@ ( some @ ( set @ A ) @ S6 ) ) ) ) ).
% dflt_None_set_def
thf(fact_1335_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,X: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts2 @ S ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
thf(fact_1336_the__dflt__None__set,axiom,
! [A: $tType,X: set @ A] :
( ( the_default @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( dflt_None_set @ A @ X ) )
= X ) ).
% the_dflt_None_set
thf(fact_1337_foldr__same__int,axiom,
! [Xs: list @ nat,Y: nat] :
( ! [X3: nat,Y3: nat] :
( ( member @ nat @ X3 @ ( set2 @ nat @ Xs ) )
=> ( ( member @ nat @ Y3 @ ( set2 @ nat @ Xs ) )
=> ( X3 = Y3 ) ) )
=> ( ! [X3: nat] :
( ( member @ nat @ X3 @ ( set2 @ nat @ Xs ) )
=> ( X3 = Y ) )
=> ( ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Xs @ ( zero_zero @ nat ) )
= ( times_times @ nat @ ( size_size @ ( list @ nat ) @ Xs ) @ Y ) ) ) ) ).
% foldr_same_int
thf(fact_1338_Gcd__fin__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ( ( semiring_gcd_Gcd_fin @ A @ A5 )
= ( zero_zero @ A ) )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) )
& ( finite_finite2 @ A @ A5 ) ) ) ) ).
% Gcd_fin_0_iff
thf(fact_1339_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_1340_card__insert__le__m1,axiom,
! [A: $tType,N: nat,Y: set @ A,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert @ A @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_1341_count__notin,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( count_list @ A @ Xs @ X )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_1342_image__Fpow__mono,axiom,
! [B: $tType,A: $tType,F3: B > A,A5: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ A5 ) @ B5 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F3 ) @ ( finite_Fpow @ B @ A5 ) ) @ ( finite_Fpow @ A @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_1343_set__to__map__empty,axiom,
! [B: $tType,A: $tType] :
( ( set_to_map @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ^ [X4: A] : ( none @ B ) ) ) ).
% set_to_map_empty
thf(fact_1344_power__shift,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( power_power @ nat @ X @ Y )
= Z )
= ( ( vEBT_VEBT_power @ ( some @ nat @ X ) @ ( some @ nat @ Y ) )
= ( some @ nat @ Z ) ) ) ).
% power_shift
thf(fact_1345_local_Opower__def,axiom,
( vEBT_VEBT_power
= ( vEBT_V2048590022279873568_shift @ nat @ ( power_power @ nat ) ) ) ).
% local.power_def
thf(fact_1346_power__Suc__0,axiom,
! [N: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_1347_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power @ nat @ X @ M )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( X
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1348_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1349_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_1350_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_1351_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_1352_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_1353_card_Oinfinite,axiom,
! [A: $tType,A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ A @ A5 )
= ( zero_zero @ nat ) ) ) ).
% card.infinite
thf(fact_1354_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_1355_Gcd__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A5 )
= ( one_one @ A ) ) ) ) ).
% Gcd_fin.infinite
thf(fact_1356_Gcd__fin__eq__Gcd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A5 )
= ( gcd_Gcd @ A @ A5 ) ) ) ) ).
% Gcd_fin_eq_Gcd
thf(fact_1357_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less @ nat @ X @ Y ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_1358_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_1359_card__0__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( finite_card @ A @ A5 )
= ( zero_zero @ nat ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_1360_card__insert__disjoint,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A5 ) )
= ( suc @ ( finite_card @ A @ A5 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1361_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_1362_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_1363_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less_eq @ nat @ X @ Y ) ) ) ) ).
% power_increasing_iff
thf(fact_1364_card__Diff__insert,axiom,
! [A: $tType,A2: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( ~ ( member @ A @ A2 @ B5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ B5 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_1365_finite__enumerate__mono__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,M: nat,N: nat] :
( ( finite_finite2 @ A @ S3 )
=> ( ( ord_less @ nat @ M @ ( finite_card @ A @ S3 ) )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S3 ) )
=> ( ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ M ) @ ( infini527867602293511546merate @ A @ S3 @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_1366_Fpow__not__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_Fpow @ A @ A5 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Fpow_not_empty
thf(fact_1367_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X: A,Y: A] :
( ( ( size_size @ A @ X )
!= ( size_size @ A @ Y ) )
=> ( X != Y ) ) ) ).
% size_neq_size_imp_neq
thf(fact_1368_count__le__length,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% count_le_length
thf(fact_1369_card__length,axiom,
! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( set2 @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% card_length
thf(fact_1370_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A,N: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ A2 @ N )
!= ( zero_zero @ A ) ) ) ) ).
% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_1371_length__induct,axiom,
! [A: $tType,P2: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs3: list @ A] :
( ! [Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs3 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_1372_finite__maxlen,axiom,
! [A: $tType,M7: set @ ( list @ A )] :
( ( finite_finite2 @ ( list @ A ) @ M7 )
=> ? [N2: nat] :
! [X5: list @ A] :
( ( member @ ( list @ A ) @ X5 @ M7 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X5 ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_1373_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_1374_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_1375_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_1376_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_1377_card__subset__eq,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ( finite_card @ A @ A5 )
= ( finite_card @ A @ B5 ) )
=> ( A5 = B5 ) ) ) ) ).
% card_subset_eq
thf(fact_1378_infinite__arbitrarily__large,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ A5 )
=> ? [B9: set @ A] :
( ( finite_finite2 @ A @ B9 )
& ( ( finite_card @ A @ B9 )
= N )
& ( ord_less_eq @ ( set @ A ) @ B9 @ A5 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1379_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B5: set @ A,A5: set @ B,R2: B > A > $o] :
( ( finite_finite2 @ A @ B5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ A5 )
=> ? [B10: A] :
( ( member @ A @ B10 @ B5 )
& ( R2 @ A4 @ B10 ) ) )
=> ( ! [A1: B,A22: B,B4: A] :
( ( member @ B @ A1 @ A5 )
=> ( ( member @ B @ A22 @ A5 )
=> ( ( member @ A @ B4 @ B5 )
=> ( ( R2 @ A1 @ B4 )
=> ( ( R2 @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1380_empty__in__Fpow,axiom,
! [A: $tType,A5: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( finite_Fpow @ A @ A5 ) ) ).
% empty_in_Fpow
thf(fact_1381_card__insert__le,axiom,
! [A: $tType,A5: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ ( insert @ A @ X @ A5 ) ) ) ).
% card_insert_le
thf(fact_1382_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_1383_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_1384_Fpow__mono,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A5 ) @ ( finite_Fpow @ A @ B5 ) ) ) ).
% Fpow_mono
thf(fact_1385_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_1386_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_1387_length__removeAll__less__eq,axiom,
! [A: $tType,X: A,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_1388_is__singleton__altdef,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
( ( finite_card @ A @ A6 )
= ( one_one @ nat ) ) ) ) ).
% is_singleton_altdef
thf(fact_1389_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_1390_card__ge__0__finite,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A5 ) )
=> ( finite_finite2 @ A @ A5 ) ) ).
% card_ge_0_finite
thf(fact_1391_card__eq__0__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( ( finite_card @ A @ A5 )
= ( zero_zero @ nat ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A5 ) ) ) ).
% card_eq_0_iff
thf(fact_1392_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_1393_card__insert__if,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A5 ) )
= ( finite_card @ A @ A5 ) ) )
& ( ~ ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A5 ) )
= ( suc @ ( finite_card @ A @ A5 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_1394_card__Suc__eq__finite,axiom,
! [A: $tType,A5: set @ A,K: nat] :
( ( ( finite_card @ A @ A5 )
= ( suc @ K ) )
= ( ? [B8: A,B7: set @ A] :
( ( A5
= ( insert @ A @ B8 @ B7 ) )
& ~ ( member @ A @ B8 @ B7 )
& ( ( finite_card @ A @ B7 )
= K )
& ( finite_finite2 @ A @ B7 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_1395_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_1396_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_1397_card__image__le,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( image @ A @ B @ F3 @ A5 ) ) @ ( finite_card @ A @ A5 ) ) ) ).
% card_image_le
thf(fact_1398_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_1399_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_1400_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F5: set @ A,C4: nat] :
( ! [G5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G5 @ F5 )
=> ( ( finite_finite2 @ A @ G5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G5 ) @ C4 ) ) )
=> ( ( finite_finite2 @ A @ F5 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F5 ) @ C4 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1401_card__seteq,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B5 ) @ ( finite_card @ A @ A5 ) )
=> ( A5 = B5 ) ) ) ) ).
% card_seteq
thf(fact_1402_card__mono,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ) ).
% card_mono
thf(fact_1403_obtain__subset__with__card__n,axiom,
! [A: $tType,N: nat,S3: set @ A] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S3 ) )
=> ~ ! [T5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T5 @ S3 )
=> ( ( ( finite_card @ A @ T5 )
= N )
=> ~ ( finite_finite2 @ A @ T5 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1404_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_1405_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_1406_card__1__singletonE,axiom,
! [A: $tType,A5: set @ A] :
( ( ( finite_card @ A @ A5 )
= ( one_one @ nat ) )
=> ~ ! [X3: A] :
( A5
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_1407_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N8: nat,A2: A] :
( ( ord_less @ nat @ N @ N8 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N8 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_1408_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_1409_card__less__sym__Diff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B5 @ A5 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1410_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_1411_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N8: nat,A2: A] :
( ( ord_less_eq @ nat @ N @ N8 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N8 ) ) ) ) ) ).
% power_increasing
thf(fact_1412_card__le__sym__Diff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B5 @ A5 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1413_psubset__card__mono,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ) ).
% psubset_card_mono
thf(fact_1414_length__pos__if__in__set,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1415_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_1416_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_1417_finite__enum__ext,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X6: set @ A,Y7: set @ A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( finite_card @ A @ X6 ) )
=> ( ( infini527867602293511546merate @ A @ X6 @ I2 )
= ( infini527867602293511546merate @ A @ Y7 @ I2 ) ) )
=> ( ( finite_finite2 @ A @ X6 )
=> ( ( finite_finite2 @ A @ Y7 )
=> ( ( ( finite_card @ A @ X6 )
= ( finite_card @ A @ Y7 ) )
=> ( X6 = Y7 ) ) ) ) ) ) ).
% finite_enum_ext
thf(fact_1418_finite__enumerate__Ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,S: A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( member @ A @ S @ S3 )
=> ? [N2: nat] :
( ( ord_less @ nat @ N2 @ ( finite_card @ A @ S3 ) )
& ( ( infini527867602293511546merate @ A @ S3 @ N2 )
= S ) ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_1419_finite__enumerate__in__set,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ( finite_finite2 @ A @ S3 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S3 ) )
=> ( member @ A @ ( infini527867602293511546merate @ A @ S3 @ N ) @ S3 ) ) ) ) ).
% finite_enumerate_in_set
thf(fact_1420_length__removeAll__less,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_1421_card__gt__0__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A5 ) )
= ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite2 @ A @ A5 ) ) ) ).
% card_gt_0_iff
thf(fact_1422_card__le__Suc0__iff__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ A5 )
=> ( X4 = Y5 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1423_card__Suc__eq,axiom,
! [A: $tType,A5: set @ A,K: nat] :
( ( ( finite_card @ A @ A5 )
= ( suc @ K ) )
= ( ? [B8: A,B7: set @ A] :
( ( A5
= ( insert @ A @ B8 @ B7 ) )
& ~ ( member @ A @ B8 @ B7 )
& ( ( finite_card @ A @ B7 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B7
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1424_card__eq__SucD,axiom,
! [A: $tType,A5: set @ A,K: nat] :
( ( ( finite_card @ A @ A5 )
= ( suc @ K ) )
=> ? [B4: A,B9: set @ A] :
( ( A5
= ( insert @ A @ B4 @ B9 ) )
& ~ ( member @ A @ B4 @ B9 )
& ( ( finite_card @ A @ B9 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B9
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_1425_card__1__singleton__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( ( finite_card @ A @ A5 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X4: A] :
( A5
= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1426_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_1427_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_1428_card__le__Suc__iff,axiom,
! [A: $tType,N: nat,A5: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( finite_card @ A @ A5 ) )
= ( ? [A8: A,B7: set @ A] :
( ( A5
= ( insert @ A @ A8 @ B7 ) )
& ~ ( member @ A @ A8 @ B7 )
& ( ord_less_eq @ nat @ N @ ( finite_card @ A @ B7 ) )
& ( finite_finite2 @ A @ B7 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_1429_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_1430_set__to__map__empty__iff_I1_J,axiom,
! [B: $tType,A: $tType,S3: set @ ( product_prod @ A @ B )] :
( ( ( set_to_map @ A @ B @ S3 )
= ( ^ [X4: A] : ( none @ B ) ) )
= ( S3
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% set_to_map_empty_iff(1)
thf(fact_1431_surj__card__le,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B,F3: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ ( image @ A @ B @ F3 @ A5 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ B5 ) @ ( finite_card @ A @ A5 ) ) ) ) ).
% surj_card_le
thf(fact_1432_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N8: nat,A2: A] :
( ( ord_less @ nat @ N @ N8 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N8 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1433_card__1__singletonI,axiom,
! [A: $tType,S3: set @ A,X: A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( ( finite_card @ A @ S3 )
= ( one_one @ nat ) )
=> ( ( member @ A @ X @ S3 )
=> ( S3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_1_singletonI
thf(fact_1434_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N8: nat,A2: A] :
( ( ord_less_eq @ nat @ N @ N8 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N8 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_decreasing
thf(fact_1435_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_1436_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_1437_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_1438_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_1439_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_1440_card__Diff1__le,axiom,
! [A: $tType,A5: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) ) ).
% card_Diff1_le
thf(fact_1441_card__Diff__subset,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1442_diff__card__le__card__Diff,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1443_card__psubset,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) )
=> ( ord_less @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% card_psubset
thf(fact_1444_finite__enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M: nat,N: nat,S3: set @ A] :
( ( ord_less @ nat @ M @ N )
=> ( ( finite_finite2 @ A @ S3 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S3 ) )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ M ) @ ( infini527867602293511546merate @ A @ S3 @ N ) ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_1445_nat__mult__power__less__eq,axiom,
! [B2: nat,A2: nat,N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ B2 )
=> ( ( ord_less @ nat @ ( times_times @ nat @ A2 @ ( power_power @ nat @ B2 @ N ) ) @ ( power_power @ nat @ B2 @ M ) )
= ( ord_less @ nat @ A2 @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ).
% nat_mult_power_less_eq
thf(fact_1446_card__le__Suc__Max,axiom,
! [S3: set @ nat] :
( ( finite_finite2 @ nat @ S3 )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ S3 ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S3 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_1447_finite__le__enumerate,axiom,
! [S3: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ S3 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ nat @ S3 ) )
=> ( ord_less_eq @ nat @ N @ ( infini527867602293511546merate @ nat @ S3 @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_1448_card__insert__disjoint_H,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( minus_minus @ nat @ ( finite_card @ A @ ( insert @ A @ X @ A5 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_insert_disjoint'
thf(fact_1449_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_1450_card_Oremove,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ A5 )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% card.remove
thf(fact_1451_card_Oinsert__remove,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A5 ) )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_1452_card__Suc__Diff1,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_1453_card__Diff1__less,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) ) ) ) ).
% card_Diff1_less
thf(fact_1454_card__Diff2__less,axiom,
! [A: $tType,A5: set @ A,X: A,Y: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y @ A5 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1455_card__Diff1__less__iff,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) )
= ( ( finite_finite2 @ A @ A5 )
& ( member @ A @ X @ A5 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1456_card__Diff__singleton__if,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1457_card__Diff__singleton,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_1458_the__default_Osimps_I1_J,axiom,
! [A: $tType,Uu2: A,X: A] :
( ( the_default @ A @ Uu2 @ ( some @ A @ X ) )
= X ) ).
% the_default.simps(1)
thf(fact_1459_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P7: A,M5: nat] :
( if @ A
@ ( M5
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P7 @ ( power_power @ A @ P7 @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1460_the__default_Osimps_I2_J,axiom,
! [A: $tType,X: A] :
( ( the_default @ A @ X @ ( none @ A ) )
= X ) ).
% the_default.simps(2)
thf(fact_1461_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_1462_finite__enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ( finite_finite2 @ A @ S3 )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( finite_card @ A @ S3 ) )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ N ) @ ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_1463_finite__enum__subset,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X6: set @ A,Y7: set @ A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( finite_card @ A @ X6 ) )
=> ( ( infini527867602293511546merate @ A @ X6 @ I2 )
= ( infini527867602293511546merate @ A @ Y7 @ I2 ) ) )
=> ( ( finite_finite2 @ A @ X6 )
=> ( ( finite_finite2 @ A @ Y7 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ X6 ) @ ( finite_card @ A @ Y7 ) )
=> ( ord_less_eq @ ( set @ A ) @ X6 @ Y7 ) ) ) ) ) ) ).
% finite_enum_subset
thf(fact_1464_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_1465_inthall,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o,N: nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P2 @ X3 ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P2 @ ( nth @ A @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_1466_slice__len,axiom,
! [A: $tType,From: nat,To: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ From @ To )
=> ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( slice @ A @ From @ To @ Xs ) )
= ( minus_minus @ nat @ To @ From ) ) ) ) ).
% slice_len
thf(fact_1467_card__Min__le__sum,axiom,
! [A: $tType,A5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( finite_card @ A @ A5 ) @ ( lattic643756798350308766er_Min @ nat @ ( image @ A @ nat @ F3 @ A5 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) ) ) ).
% card_Min_le_sum
thf(fact_1468_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_1469_realpow__pos__nth__unique,axiom,
! [N: nat,A2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
& ( ( power_power @ real @ X3 @ N )
= A2 )
& ! [Y4: real] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y4 )
& ( ( power_power @ real @ Y4 @ N )
= A2 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1470_length__mul__elem,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N: nat] :
( ! [X3: list @ A] :
( ( member @ ( list @ A ) @ X3 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ X3 )
= N ) )
=> ( ( size_size @ ( list @ A ) @ ( concat @ A @ Xs ) )
= ( times_times @ nat @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) @ N ) ) ) ).
% length_mul_elem
thf(fact_1471_rotate1__length01,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( ( rotate1 @ A @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_1472_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less @ real @ ( zero_zero @ real ) @ ( times_times @ real @ X @ X ) ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% not_real_square_gt_zero
thf(fact_1473_slice__complete,axiom,
! [A: $tType,Xs: list @ A] :
( ( slice @ A @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) @ Xs )
= Xs ) ).
% slice_complete
thf(fact_1474_real__add__le__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ X @ Y ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ Y @ ( uminus_uminus @ real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_1475_real__0__le__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ Y ) )
= ( ord_less_eq @ real @ ( uminus_uminus @ real @ X ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_1476_less__eq__real__def,axiom,
( ( ord_less_eq @ real )
= ( ^ [X4: real,Y5: real] :
( ( ord_less @ real @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_1477_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ? [N2: nat] : ( ord_less @ real @ Y @ ( power_power @ real @ X @ N2 ) ) ) ).
% real_arch_pow
thf(fact_1478_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ? [N2: nat] : ( ord_less @ real @ ( power_power @ real @ X @ N2 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1479_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( plus_plus @ real @ X @ Y ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ Y @ ( uminus_uminus @ real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_1480_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ Y ) )
= ( ord_less @ real @ ( uminus_uminus @ real @ X ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_1481_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_1482_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P2: nat > A > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ? [X8: A] : ( P2 @ I3 @ X8 ) ) )
= ( ? [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ( P2 @ I3 @ ( nth @ A @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1483_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y6: list @ A,Z4: list @ A] : Y6 = Z4 )
= ( ^ [Xs2: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I3 )
= ( nth @ A @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1484_obtain__list__from__elements,axiom,
! [A: $tType,N: nat,P2: A > nat > $o] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ? [Li: A] : ( P2 @ Li @ I2 ) )
=> ~ ! [L2: list @ A] :
( ( ( size_size @ ( list @ A ) @ L2 )
= N )
=> ~ ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( P2 @ ( nth @ A @ L2 @ I4 ) @ I4 ) ) ) ) ).
% obtain_list_from_elements
thf(fact_1485_slice__nth,axiom,
! [A: $tType,From: nat,To: nat,Xs: list @ A,I: nat] :
( ( ord_less @ nat @ From @ To )
=> ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ To @ From ) )
=> ( ( nth @ A @ ( slice @ A @ From @ To @ Xs ) @ I )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ From @ I ) ) ) ) ) ) ).
% slice_nth
thf(fact_1486_all__set__conv__nth,axiom,
! [A: $tType,L: list @ A,P2: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ L ) )
=> ( P2 @ X4 ) ) )
= ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) )
=> ( P2 @ ( nth @ A @ L @ I3 ) ) ) ) ) ).
% all_set_conv_nth
thf(fact_1487_nth__mem,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ ( nth @ A @ Xs @ N ) @ ( set2 @ A @ Xs ) ) ) ).
% nth_mem
thf(fact_1488_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P2: A > $o] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P2 @ X3 ) )
=> ( P2 @ ( nth @ A @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1489_in__set__conv__nth,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1490_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o,X: A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P2 @ ( nth @ A @ Xs @ I2 ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1491_all__set__conv__all__nth,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P2 @ X4 ) ) )
= ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P2 @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1492_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_1493_VEBT_Osize_I4_J,axiom,
! [X21: $o,X22: $o] :
( ( size_size @ vEBT_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
= ( zero_zero @ nat ) ) ).
% VEBT.size(4)
thf(fact_1494_set__image__eq__pointwiseI,axiom,
! [B: $tType,A: $tType,L: list @ A,L3: list @ A,F3: A > B] :
( ( ( size_size @ ( list @ A ) @ L )
= ( size_size @ ( list @ A ) @ L3 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F3 @ ( nth @ A @ L @ I2 ) )
= ( F3 @ ( nth @ A @ L3 @ I2 ) ) ) )
=> ( ( image @ A @ B @ F3 @ ( set2 @ A @ L ) )
= ( image @ A @ B @ F3 @ ( set2 @ A @ L3 ) ) ) ) ) ).
% set_image_eq_pointwiseI
thf(fact_1495_in__set__image__conv__nth,axiom,
! [B: $tType,A: $tType,F3: B > A,X: B,L: list @ B] :
( ( member @ A @ ( F3 @ X ) @ ( image @ B @ A @ F3 @ ( set2 @ B @ L ) ) )
= ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ B ) @ L ) )
& ( ( F3 @ ( nth @ B @ L @ I3 ) )
= ( F3 @ X ) ) ) ) ) ).
% in_set_image_conv_nth
thf(fact_1496_sum__count__set,axiom,
! [A: $tType,Xs: list @ A,X6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X6 )
=> ( ( finite_finite2 @ A @ X6 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs ) @ X6 )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_1497_sum__le__card__Max,axiom,
! [A: $tType,A5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ord_less_eq @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) @ ( times_times @ nat @ ( finite_card @ A @ A5 ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ A @ nat @ F3 @ A5 ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_1498_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ~ ( member @ B @ X @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( plus_plus @ A @ ( G3 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) ) ) ) ) ) ).
% sum.insert
thf(fact_1499_sum_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A] :
( ~ ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( zero_zero @ A ) ) ) ) ).
% sum.infinite
thf(fact_1500_sum__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [F5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ F5 )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ F5 )
= ( zero_zero @ A ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ F5 )
=> ( ( F3 @ X4 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_1501_sum_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > A] :
( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty
thf(fact_1502_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I: C,A5: set @ C,F3: C > B] :
( ( member @ C @ I @ A5 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ ( minus_minus @ ( set @ C ) @ A5 @ ( insert @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ X3 ) ) )
=> ( ( finite_finite2 @ C @ A5 )
=> ( ord_less_eq @ B @ ( F3 @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F3 @ A5 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_1503_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B5: set @ A,A5: set @ A,B2: A,F3: A > B] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B5 @ A5 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F3 @ B2 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ X3 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ B5 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_1504_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A5: set @ B,A2: B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( member @ B @ A2 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( F3 @ A2 ) ) ) )
& ( ~ ( member @ B @ A2 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) ) ) ) ) ) ).
% sum_diff1
thf(fact_1505_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A,X: B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( plus_plus @ A @ ( G3 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_1506_complete__real,axiom,
! [S3: set @ real] :
( ? [X5: real] : ( member @ real @ X5 @ S3 )
=> ( ? [Z5: real] :
! [X3: real] :
( ( member @ real @ X3 @ S3 )
=> ( ord_less_eq @ real @ X3 @ Z5 ) )
=> ? [Y3: real] :
( ! [X5: real] :
( ( member @ real @ X5 @ S3 )
=> ( ord_less_eq @ real @ X5 @ Y3 ) )
& ! [Z5: real] :
( ! [X3: real] :
( ( member @ real @ X3 @ S3 )
=> ( ord_less_eq @ real @ X3 @ Z5 ) )
=> ( ord_less_eq @ real @ Y3 @ Z5 ) ) ) ) ) ).
% complete_real
thf(fact_1507_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( times_times @ real @ U @ U ) ) @ ( times_times @ real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1508_sum__cong__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ nat,F3: nat > A,G3: nat > A] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ A5 )
=> ( ! [X3: nat] :
( ( member @ nat @ ( suc @ X3 ) @ A5 )
=> ( ( F3 @ ( suc @ X3 ) )
= ( G3 @ ( suc @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ A5 )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ A5 ) ) ) ) ) ).
% sum_cong_Suc
thf(fact_1509_sum_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( zero_zero @ A ) ) ) ) ).
% sum.neutral
thf(fact_1510_sum_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > A,A5: set @ B] :
( ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
!= ( zero_zero @ A ) )
=> ~ ! [A4: B] :
( ( member @ B @ A4 @ A5 )
=> ( ( G3 @ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_1511_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A5: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_1512_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A5: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) ) ) ) ).
% sum_nonneg
thf(fact_1513_sum__mono__inv,axiom,
! [A: $tType,I5: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [F3: I5 > A,I6: set @ I5,G3: I5 > A,I: I5] :
( ( ( groups7311177749621191930dd_sum @ I5 @ A @ F3 @ I6 )
= ( groups7311177749621191930dd_sum @ I5 @ A @ G3 @ I6 ) )
=> ( ! [I2: I5] :
( ( member @ I5 @ I2 @ I6 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( G3 @ I2 ) ) )
=> ( ( member @ I5 @ I @ I6 )
=> ( ( finite_finite2 @ I5 @ I6 )
=> ( ( F3 @ I )
= ( G3 @ I ) ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1514_sum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 )
= ( zero_zero @ A ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( F3 @ X4 )
= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1515_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,T2: set @ C,G3: C > A,I: C > B,F3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ T2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G3 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ? [Xa: C] :
( ( member @ C @ Xa @ T2 )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ Xa ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S ) @ ( groups7311177749621191930dd_sum @ C @ A @ G3 @ T2 ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1516_sum__strict__mono__ex1,axiom,
! [A: $tType,I5: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A5: set @ I5,F3: I5 > A,G3: I5 > A] :
( ( finite_finite2 @ I5 @ A5 )
=> ( ! [X3: I5] :
( ( member @ I5 @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ? [X5: I5] :
( ( member @ I5 @ X5 @ A5 )
& ( ord_less @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I5 @ A @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ I5 @ A @ G3 @ A5 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_1517_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R3: A > A > $o,S3: set @ B,H2: B > A,G3: B > A] :
( ( R3 @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X15: A,Y15: A,X23: A,Y23: A] :
( ( ( R3 @ X15 @ X23 )
& ( R3 @ Y15 @ Y23 ) )
=> ( R3 @ ( plus_plus @ A @ X15 @ Y15 ) @ ( plus_plus @ A @ X23 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( R3 @ ( H2 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( R3 @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S3 ) ) ) ) ) ) ) ).
% sum.related
thf(fact_1518_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A5: set @ B,F3: B > A,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_1519_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( member @ B @ X @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) ) )
& ( ~ ( member @ B @ X @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( plus_plus @ A @ ( G3 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_1520_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S7: set @ B,T6: set @ C,S3: set @ B,I: C > B,J: B > C,T4: set @ C,G3: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S7 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) )
=> ( member @ C @ ( J @ A4 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( member @ B @ ( I @ B4 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S7 )
=> ( ( G3 @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T6 )
=> ( ( H2 @ B4 )
= ( zero_zero @ A ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S3 )
=> ( ( H2 @ ( J @ A4 ) )
= ( G3 @ A4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S3 )
= ( groups7311177749621191930dd_sum @ C @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_1521_sum__SucD,axiom,
! [A: $tType,F3: A > nat,A5: set @ A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 )
= ( suc @ N ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F3 @ X3 ) ) ) ) ).
% sum_SucD
thf(fact_1522_sum__eq__Suc0__iff,axiom,
! [A: $tType,A5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ( F3 @ X4 )
= ( suc @ ( zero_zero @ nat ) ) )
& ! [Y5: A] :
( ( member @ A @ Y5 @ A5 )
=> ( ( X4 != Y5 )
=> ( ( F3 @ Y5 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_1523_sum__eq__1__iff,axiom,
! [A: $tType,A5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 )
= ( one_one @ nat ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ( F3 @ X4 )
= ( one_one @ nat ) )
& ! [Y5: A] :
( ( member @ A @ Y5 @ A5 )
=> ( ( X4 != Y5 )
=> ( ( F3 @ Y5 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_1524_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,I: B,F3: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( member @ B @ I @ I6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ I6 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1525_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I6 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ I6 ) ) ) ) ) ) ).
% sum_pos
thf(fact_1526_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C4: set @ B,A5: set @ B,B5: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C4 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C4 @ A5 ) )
=> ( ( G3 @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C4 @ B5 ) )
=> ( ( H2 @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B5 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ C4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C4 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_1527_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C4: set @ B,A5: set @ B,B5: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C4 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C4 @ A5 ) )
=> ( ( G3 @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C4 @ B5 ) )
=> ( ( H2 @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ C4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C4 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B5 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_1528_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S3: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ G3 @ T4 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_1529_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S3: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_1530_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S3: set @ B,H2: B > A,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( H2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G3 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_1531_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S3: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G3 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_1532_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B5: set @ B,A5: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B5 @ A5 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ B5 ) ) ) ) ) ) ).
% sum.subset_diff
thf(fact_1533_sum__diff,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A5: set @ B,B5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ B5 ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B5 ) ) ) ) ) ) ).
% sum_diff
thf(fact_1534_sum__diff__nat,axiom,
! [A: $tType,B5: set @ A,A5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ B5 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_1535_sum__diff1__nat,axiom,
! [A: $tType,A2: A,A5: set @ A,F3: A > nat] :
( ( ( member @ A @ A2 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) @ ( F3 @ A2 ) ) ) )
& ( ~ ( member @ A @ A2 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) ) ) ) ).
% sum_diff1_nat
thf(fact_1536_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B5: set @ B,A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B5 )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ B5 @ A5 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ B4 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B5 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_1537_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( member @ B @ X @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( plus_plus @ A @ ( G3 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_1538_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_1539_cnt__non__neg,axiom,
! [T2: vEBT_VEBT] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( vEBT_VEBT_cnt @ T2 ) ) ).
% cnt_non_neg
thf(fact_1540_word__size__gt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( word @ A ) @ W2 ) ) ) ).
% word_size_gt_0
thf(fact_1541_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_1542_Bolzano,axiom,
! [A2: real,B2: real,P2: real > real > $o] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [A4: real,B4: real,C2: real] :
( ( P2 @ A4 @ B4 )
=> ( ( P2 @ B4 @ C2 )
=> ( ( ord_less_eq @ real @ A4 @ B4 )
=> ( ( ord_less_eq @ real @ B4 @ C2 )
=> ( P2 @ A4 @ C2 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [A4: real,B4: real] :
( ( ( ord_less_eq @ real @ A4 @ X3 )
& ( ord_less_eq @ real @ X3 @ B4 )
& ( ord_less @ real @ ( minus_minus @ real @ B4 @ A4 ) @ D4 ) )
=> ( P2 @ A4 @ B4 ) ) ) ) )
=> ( P2 @ A2 @ B2 ) ) ) ) ).
% Bolzano
thf(fact_1543_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_1544_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_1545_size__empty,axiom,
! [A: $tType] :
( ( size_size @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) )
= ( zero_zero @ nat ) ) ).
% size_empty
thf(fact_1546_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% max_bot
thf(fact_1547_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% max_bot2
thf(fact_1548_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_1549_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_1550_max__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A2 )
= A2 ) ).
% max_nat.left_neutral
thf(fact_1551_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_1552_max__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ A2 @ ( zero_zero @ nat ) )
= A2 ) ).
% max_nat.right_neutral
thf(fact_1553_max__0L,axiom,
! [N: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% max_0L
thf(fact_1554_max__0R,axiom,
! [N: nat] :
( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
= N ) ).
% max_0R
thf(fact_1555_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_1556_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_1557_size__eq__0__iff__empty,axiom,
! [A: $tType,M7: multiset @ A] :
( ( ( size_size @ ( multiset @ A ) @ M7 )
= ( zero_zero @ nat ) )
= ( M7
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% size_eq_0_iff_empty
thf(fact_1558_Max__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A5 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ) ).
% Max_insert
thf(fact_1559_union__diff__assoc,axiom,
! [A: $tType,C4: multiset @ A,B5: multiset @ A,A5: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ C4 @ B5 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) @ C4 )
= ( plus_plus @ ( multiset @ A ) @ A5 @ ( minus_minus @ ( multiset @ A ) @ B5 @ C4 ) ) ) ) ).
% union_diff_assoc
thf(fact_1560_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A8: A,B8: A] : ( if @ A @ ( ord_less_eq @ A @ A8 @ B8 ) @ B8 @ A8 ) ) ) ) ).
% max_def
thf(fact_1561_max__absorb1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_max @ A @ X @ Y )
= X ) ) ) ).
% max_absorb1
thf(fact_1562_max__absorb2,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_max @ A @ X @ Y )
= Y ) ) ) ).
% max_absorb2
thf(fact_1563_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ X @ ( ord_max @ A @ Y @ Z ) )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z ) ) ) ) ).
% max_add_distrib_right
thf(fact_1564_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X @ Y ) @ Z )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Z ) @ ( plus_plus @ A @ Y @ Z ) ) ) ) ).
% max_add_distrib_left
thf(fact_1565_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A,Z: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X @ Y ) @ Z )
= ( ord_max @ A @ ( minus_minus @ A @ X @ Z ) @ ( minus_minus @ A @ Y @ Z ) ) ) ) ).
% max_diff_distrib_left
thf(fact_1566_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_1567_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_1568_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_1569_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_1570_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_1571_Max_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ).
% Max.in_idem
thf(fact_1572_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_VEBT_cnt @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( one_one @ real ) ) ).
% VEBT_internal.cnt.simps(1)
thf(fact_1573_le__minus,axiom,
! [Aa: $tType,A: $tType] :
( ( ( type_len @ A )
& ( order @ Aa ) )
=> ! [Y: Aa,X: Aa,A2: word @ A,C3: word @ A,B2: word @ A] :
( ( ord_less_eq @ Aa @ Y @ X )
=> ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ C3 ) @ B2 )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ C3 ) )
=> ( ord_less_eq @ ( word @ A ) @ C3 @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) ) ) ) ) ) ).
% le_minus
thf(fact_1574_size__0__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,V3: word @ A] :
( ( ( size_size @ ( word @ A ) @ W2 )
= ( zero_zero @ nat ) )
=> ( V3 = W2 ) ) ) ).
% size_0_eq
thf(fact_1575_lens__not__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A] :
( ( size_size @ ( word @ A ) @ W2 )
!= ( zero_zero @ nat ) ) ) ).
% lens_not_0
thf(fact_1576_size__0__same_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,V3: word @ A] :
( ( ( size_size @ ( word @ A ) @ W2 )
= ( zero_zero @ nat ) )
=> ( W2 = V3 ) ) ) ).
% size_0_same'
thf(fact_1577_option_Osize__neq,axiom,
! [A: $tType,X: option @ A] :
( ( size_size @ ( option @ A ) @ X )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_1578_sum__eq__empty__iff,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > ( multiset @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ ( multiset @ B ) @ F3 @ A5 )
= ( zero_zero @ ( multiset @ B ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ( F3 @ X4 )
= ( zero_zero @ ( multiset @ B ) ) ) ) ) ) ) ).
% sum_eq_empty_iff
thf(fact_1579_hom__Max__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H2: A > A,N8: set @ A] :
( ! [X3: A,Y3: A] :
( ( H2 @ ( ord_max @ A @ X3 @ Y3 ) )
= ( ord_max @ A @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N8 )
=> ( ( N8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic643756798349783984er_Max @ A @ N8 ) )
= ( lattic643756798349783984er_Max @ A @ ( image @ A @ A @ H2 @ N8 ) ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_1580_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B5 ) @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ) ).
% Max.subset
thf(fact_1581_Max_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] : ( member @ A @ ( ord_max @ A @ X3 @ Y3 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Max.closed
thf(fact_1582_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A5 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_1583_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A5 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_1584_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_1585_nonempty__has__size,axiom,
! [A: $tType,S3: multiset @ A] :
( ( S3
!= ( zero_zero @ ( multiset @ A ) ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( multiset @ A ) @ S3 ) ) ) ).
% nonempty_has_size
thf(fact_1586_diff__size__le__size__Diff,axiom,
! [A: $tType,M7: multiset @ A,M9: multiset @ A] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( size_size @ ( multiset @ A ) @ M7 ) @ ( size_size @ ( multiset @ A ) @ M9 ) ) @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M7 @ M9 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_1587_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_1588_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_1589_max__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ ( ord_max @ A @ X @ Y ) @ Z )
= ( ( ord_less @ A @ X @ Z )
& ( ord_less @ A @ Y @ Z ) ) ) ) ).
% max_less_iff_conj
thf(fact_1590_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C3 ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% max.bounded_iff
thf(fact_1591_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_1592_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_1593_option_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X2: A] :
( ( size_option @ A @ X @ ( some @ A @ X2 ) )
= ( plus_plus @ nat @ ( X @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% option.size_gen(2)
thf(fact_1594_option_Osize__gen_I1_J,axiom,
! [A: $tType,X: A > nat] :
( ( size_option @ A @ X @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size_gen(1)
thf(fact_1595_word__coorder_Oextremum__unique,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A2 @ ( zero_zero @ ( word @ A ) ) )
= ( A2
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_coorder.extremum_unique
thf(fact_1596_word__le__0__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( zero_zero @ ( word @ A ) ) )
= ( X
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_le_0_iff
thf(fact_1597_word__order_Oextremum,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] : ( ord_less_eq @ ( word @ A ) @ A2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_order.extremum
thf(fact_1598_word__order_Oextremum__unique,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ A2 )
= ( A2
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_order.extremum_unique
thf(fact_1599_word__n1__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A] : ( ord_less_eq @ ( word @ A ) @ Y @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_n1_ge
thf(fact_1600_max__word__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A] : ( ord_less_eq @ ( word @ A ) @ N @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% max_word_max
thf(fact_1601_word__coorder_Oextremum,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ A2 ) ) ).
% word_coorder.extremum
thf(fact_1602_word__coorder_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A2 @ ( zero_zero @ ( word @ A ) ) )
=> ( A2
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_coorder.extremum_uniqueI
thf(fact_1603_word__zero__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y ) ) ).
% word_zero_le
thf(fact_1604_word__le__less__eq,axiom,
! [Z6: $tType] :
( ( type_len @ Z6 )
=> ( ( ord_less_eq @ ( word @ Z6 ) )
= ( ^ [X4: word @ Z6,Y5: word @ Z6] :
( ( X4 = Y5 )
| ( ord_less @ ( word @ Z6 ) @ X4 @ Y5 ) ) ) ) ) ).
% word_le_less_eq
thf(fact_1605_word__le__sub1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X )
= ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_le_sub1
thf(fact_1606_word__less__minus__mono__left,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% word_less_minus_mono_left
thf(fact_1607_word__less__minus__cancel,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less @ ( word @ A ) @ Y @ Z ) ) ) ) ).
% word_less_minus_cancel
thf(fact_1608_sub__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ X @ Z ) )
= ( ( Z
= ( zero_zero @ ( word @ A ) ) )
| ( ord_less @ ( word @ A ) @ X @ Z ) ) ) ) ).
% sub_wrap
thf(fact_1609_plus__le__left__cancel__nowrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y8: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y8 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
=> ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y8 ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ ( word @ A ) @ Y8 @ Y ) ) ) ) ) ).
% plus_le_left_cancel_nowrap
thf(fact_1610_inc__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,M: word @ A] :
( ( ord_less @ ( word @ A ) @ I @ M )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ ( one_one @ ( word @ A ) ) ) @ M ) ) ) ).
% inc_le
thf(fact_1611_inc__i,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,M: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ I )
=> ( ( ord_less @ ( word @ A ) @ I @ M )
=> ( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( plus_plus @ ( word @ A ) @ I @ ( one_one @ ( word @ A ) ) ) )
& ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ ( one_one @ ( word @ A ) ) ) @ M ) ) ) ) ) ).
% inc_i
thf(fact_1612_plus__minus__not__NULL__ab,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Ab: word @ A,C3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Ab @ C3 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C3 @ Ab )
=> ( ( C3
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X @ C3 )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% plus_minus_not_NULL_ab
thf(fact_1613_plus__minus__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Ab: word @ A,C3: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Ab @ C3 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C3 @ Ab )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ C3 ) ) ) ) ) ).
% plus_minus_no_overflow
thf(fact_1614_word__less__sub__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ Y @ Z ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y @ X )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ Z ) ) ) ) ).
% word_less_sub_right
thf(fact_1615_word__less__add__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Y @ Z ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Z @ Y )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Z ) @ Y ) ) ) ) ).
% word_less_add_right
thf(fact_1616_word__plus__mcs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V3: word @ A,Xb: word @ A,X: word @ A,W2: word @ A,Xa2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V3 @ Xb ) @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V3 @ Xb ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) ) ) ) ) ).
% word_plus_mcs(1)
thf(fact_1617_word__plus__mcs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V3: word @ A,Xb: word @ A,W2: word @ A,Xa2: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V3 @ Xb ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ V3 @ Xb ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V3 @ Xb ) @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) @ X ) ) ) ) ) ).
% word_plus_mcs(2)
thf(fact_1618_Word_Oword__l__diffs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,Xa2: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) @ Z ) ) ) ) ).
% Word.word_l_diffs(1)
thf(fact_1619_Word_Oword__l__diffs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,Xa2: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% Word.word_l_diffs(2)
thf(fact_1620_word__diff__ls_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,W2: word @ A,Xa2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) )
=> ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) ) ) ) ) ).
% word_diff_ls(1)
thf(fact_1621_word__diff__ls_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,W2: word @ A,Xa2: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) @ X ) ) ) ) ) ).
% word_diff_ls(2)
thf(fact_1622_word__less__nowrapI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Z: word @ A,K: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Z @ K ) )
=> ( ( ord_less_eq @ ( word @ A ) @ K @ Z )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K )
=> ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ K ) ) ) ) ) ) ).
% word_less_nowrapI
thf(fact_1623_plus__minus__not__NULL,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Ab: word @ A,C3: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Ab @ C3 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C3 @ Ab )
=> ( ( C3
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X @ C3 )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% plus_minus_not_NULL
thf(fact_1624_word__induct__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: ( word @ A ) > $o,M: word @ A] :
( ( P2 @ ( zero_zero @ ( word @ A ) ) )
=> ( ! [N2: word @ A] :
( ( ord_less @ ( word @ A ) @ N2 @ M )
=> ( ( P2 @ N2 )
=> ( P2 @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N2 ) ) ) )
=> ( P2 @ M ) ) ) ) ).
% word_induct_less
thf(fact_1625_plus__le__left__cancel__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y8: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y8 ) @ X )
=> ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) @ X )
=> ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y8 ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ ( word @ A ) @ Y8 @ Y ) ) ) ) ) ).
% plus_le_left_cancel_wrap
thf(fact_1626_union__less__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A5: multiset @ A,C4: multiset @ A,B5: multiset @ A,D5: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ A5 @ C4 )
=> ( ( ord_less @ ( multiset @ A ) @ B5 @ D5 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) @ ( plus_plus @ ( multiset @ A ) @ C4 @ D5 ) ) ) ) ) ).
% union_less_mono
thf(fact_1627_union__le__mono2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B5: multiset @ A,D5: multiset @ A,C4: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ B5 @ D5 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C4 @ B5 ) @ ( plus_plus @ ( multiset @ A ) @ C4 @ D5 ) ) ) ) ).
% union_le_mono2
thf(fact_1628_union__le__mono1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B5: multiset @ A,D5: multiset @ A,C4: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ B5 @ D5 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B5 @ C4 ) @ ( plus_plus @ ( multiset @ A ) @ D5 @ C4 ) ) ) ) ).
% union_le_mono1
thf(fact_1629_word__less__sub1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X )
= ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_less_sub1
thf(fact_1630_plus__minus__no__overflow__ab,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Ab: word @ A,C3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Ab @ C3 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C3 @ Ab )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ C3 ) ) ) ) ) ).
% plus_minus_no_overflow_ab
thf(fact_1631_le__minus_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,C3: word @ A,B2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ C3 ) @ B2 )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ C3 ) )
=> ( ord_less_eq @ ( word @ A ) @ C3 @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) ) ) ) ) ).
% le_minus'
thf(fact_1632_le__plus_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,C3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( word @ A ) @ C3 @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ C3 ) @ B2 ) ) ) ) ).
% le_plus'
thf(fact_1633_le__plus,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [C3: word @ A,B2: word @ A,A2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ C3 @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ C3 ) @ B2 ) ) ) ) ).
% le_plus
thf(fact_1634_word__plus__mcs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V3: word @ A,Xb: word @ A,X: word @ A,W2: word @ A,Xa2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V3 @ Xb ) @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V3 @ Xb ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) ) ) ) ) ).
% word_plus_mcs(3)
thf(fact_1635_word__plus__mcs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V3: word @ A,Xb: word @ A,W2: word @ A,Xa2: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V3 @ Xb ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ V3 @ Xb ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V3 @ Xb ) @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) @ X ) ) ) ) ) ).
% word_plus_mcs(4)
thf(fact_1636_Word_Oword__l__diffs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,Xa2: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) @ Z ) ) ) ) ).
% Word.word_l_diffs(3)
thf(fact_1637_Word_Oword__l__diffs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,Xa2: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% Word.word_l_diffs(4)
thf(fact_1638_word__diff__ls_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,W2: word @ A,Xa2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) )
=> ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) ) ) ) ) ).
% word_diff_ls(3)
thf(fact_1639_word__diff__ls_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,W2: word @ A,Xa2: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ Xa2 ) @ X ) ) ) ) ) ).
% word_diff_ls(4)
thf(fact_1640_word__plus__mono__right2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,C3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C3 @ B2 )
=> ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ C3 ) ) ) ) ) ).
% word_plus_mono_right2
thf(fact_1641_word__plus__mono__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) @ ( plus_plus @ ( word @ A ) @ X @ Z ) ) ) ) ) ).
% word_plus_mono_right
thf(fact_1642_word__le__plus__either,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
| ( ord_less_eq @ ( word @ A ) @ X @ Z ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ Y @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ Y @ Z ) ) ) ) ) ).
% word_le_plus_either
thf(fact_1643_le__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,B2: word @ A,A2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ B2 )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) ) ) ) ) ).
% le_no_overflow
thf(fact_1644_olen__add__eqv,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ Y @ X ) )
= ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ Y @ X ) ) ) ) ).
% olen_add_eqv
thf(fact_1645_word__le__minus__mono__left,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% word_le_minus_mono_left
thf(fact_1646_word__le__minus__cancel,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less_eq @ ( word @ A ) @ Y @ Z ) ) ) ) ).
% word_le_minus_cancel
thf(fact_1647_word__le__minus__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,C3: word @ A,D2: word @ A,B2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A2 @ C3 )
=> ( ( ord_less_eq @ ( word @ A ) @ D2 @ B2 )
=> ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ A2 )
=> ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ C3 @ D2 ) @ C3 )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( word @ A ) @ C3 @ D2 ) ) ) ) ) ) ) ).
% word_le_minus_mono
thf(fact_1648_word__sub__le__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ X )
= ( ord_less_eq @ ( word @ A ) @ Y @ X ) ) ) ).
% word_sub_le_iff
thf(fact_1649_word__sub__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ X )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ X ) ) ) ).
% word_sub_le
thf(fact_1650_word__order_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ A2 )
=> ( A2
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_order.extremum_uniqueI
thf(fact_1651_sub__wrap__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ X @ Z ) )
= ( ord_less @ ( word @ A ) @ X @ Z ) ) ) ).
% sub_wrap_lt
thf(fact_1652_word__sub__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ ( word @ A ) @ X @ Y ) ) ) ).
% word_sub_less_iff
thf(fact_1653_word__less__minus__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,C3: word @ A,D2: word @ A,B2: word @ A] :
( ( ord_less @ ( word @ A ) @ A2 @ C3 )
=> ( ( ord_less @ ( word @ A ) @ D2 @ B2 )
=> ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ A2 )
=> ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ C3 @ D2 ) @ C3 )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( word @ A ) @ C3 @ D2 ) ) ) ) ) ) ) ).
% word_less_minus_mono
thf(fact_1654_mset__distrib,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A,M7: multiset @ A,N8: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ A5 @ B5 )
= ( plus_plus @ ( multiset @ A ) @ M7 @ N8 ) )
=> ~ ! [Am: multiset @ A,An: multiset @ A] :
( ( A5
= ( plus_plus @ ( multiset @ A ) @ Am @ An ) )
=> ! [Bm: multiset @ A,Bn: multiset @ A] :
( ( B5
= ( plus_plus @ ( multiset @ A ) @ Bm @ Bn ) )
=> ( ( M7
= ( plus_plus @ ( multiset @ A ) @ Am @ Bm ) )
=> ( N8
!= ( plus_plus @ ( multiset @ A ) @ An @ Bn ) ) ) ) ) ) ).
% mset_distrib
thf(fact_1655_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A2: A,D2: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ A2 )
=> ( ( ord_less_eq @ A @ D2 @ B2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C3 @ D2 ) @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ) ).
% max.mono
thf(fact_1656_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_1657_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_1658_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C3 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% max.boundedE
thf(fact_1659_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C3 ) @ A2 ) ) ) ) ).
% max.boundedI
thf(fact_1660_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B8: A,A8: A] :
( A8
= ( ord_max @ A @ A8 @ B8 ) ) ) ) ) ).
% max.order_iff
thf(fact_1661_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_1662_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_1663_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less_eq @ A @ Z @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less_eq @ A @ Z @ X )
| ( ord_less_eq @ A @ Z @ Y ) ) ) ) ).
% le_max_iff_disj
thf(fact_1664_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B8: A,A8: A] :
( ( ord_max @ A @ A8 @ B8 )
= A8 ) ) ) ) ).
% max.absorb_iff1
thf(fact_1665_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B8: A] :
( ( ord_max @ A @ A8 @ B8 )
= B8 ) ) ) ) ).
% max.absorb_iff2
thf(fact_1666_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ A2 )
=> ( ord_less_eq @ A @ C3 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.coboundedI1
thf(fact_1667_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.coboundedI2
thf(fact_1668_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.strict_coboundedI2
thf(fact_1669_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ C3 @ A2 )
=> ( ord_less @ A @ C3 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.strict_coboundedI1
thf(fact_1670_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B8: A,A8: A] :
( ( A8
= ( ord_max @ A @ A8 @ B8 ) )
& ( A8 != B8 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_1671_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less @ A @ ( ord_max @ A @ B2 @ C3 ) @ A2 )
=> ~ ( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% max.strict_boundedE
thf(fact_1672_less__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z @ X )
| ( ord_less @ A @ Z @ Y ) ) ) ) ).
% less_max_iff_disj
thf(fact_1673_word__sub__plus__one__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: word @ A,N: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ N3 @ N )
=> ( ( N3
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N @ N3 ) @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_sub_plus_one_nonzero
thf(fact_1674_word__diff__ls_H_H_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,W2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ X ) @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W2 @ X ) )
=> ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W2 @ X ) ) ) ) ) ).
% word_diff_ls''(1)
thf(fact_1675_word__diff__ls_H_H_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,W2: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W2 @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ X ) @ X ) ) ) ) ) ).
% word_diff_ls''(2)
thf(fact_1676_word__l__diffs_H_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ X ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ X ) @ Z ) ) ) ) ).
% word_l_diffs'(1)
thf(fact_1677_word__l__diffs_H_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ X ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W2 @ X ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ X ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% word_l_diffs'(2)
thf(fact_1678_word__diff__ls_H_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,W2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ W2 )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W2 @ X ) )
=> ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W2 @ X ) ) ) ) ) ).
% word_diff_ls'(1)
thf(fact_1679_word__diff__ls_H_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,W2: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W2 @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ W2 ) ) ) ) ).
% word_diff_ls'(2)
thf(fact_1680_More__Word_Oword__l__diffs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ W2 @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ X ) @ Z ) ) ) ) ).
% More_Word.word_l_diffs(1)
thf(fact_1681_word__less__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
= ( X
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_less_1
thf(fact_1682_word__minus__one__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X )
= ( X
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_minus_one_le
thf(fact_1683_max__word__less__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ W2 )
= ( W2
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% max_word_less_eq_iff
thf(fact_1684_less__eq__multiset__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less_eq @ ( multiset @ A ) )
= ( ^ [M10: multiset @ A,N7: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ M10 @ N7 )
| ( M10 = N7 ) ) ) ) ) ).
% less_eq_multiset_def
thf(fact_1685_max__word__not__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X ) ) ).
% max_word_not_less
thf(fact_1686_mset__le__not__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M7: multiset @ A] :
~ ( ord_less @ ( multiset @ A ) @ M7 @ M7 ) ) ).
% mset_le_not_refl
thf(fact_1687_mset__le__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M7: multiset @ A,N8: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ M7 @ N8 )
=> ~ ( ord_less @ ( multiset @ A ) @ N8 @ M7 ) ) ) ).
% mset_le_not_sym
thf(fact_1688_mset__le__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M7: multiset @ A] :
~ ( ord_less @ ( multiset @ A ) @ M7 @ M7 ) ) ).
% mset_le_irrefl
thf(fact_1689_mset__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K4: multiset @ A,M7: multiset @ A,N8: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ K4 @ M7 )
=> ( ( ord_less @ ( multiset @ A ) @ M7 @ N8 )
=> ( ord_less @ ( multiset @ A ) @ K4 @ N8 ) ) ) ) ).
% mset_le_trans
thf(fact_1690_mset__le__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M7: multiset @ A,N8: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ M7 @ N8 )
=> ~ ( ord_less @ ( multiset @ A ) @ N8 @ M7 ) ) ) ).
% mset_le_asym
thf(fact_1691_word__not__simps_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ( ord_less @ ( word @ A ) @ X @ ( zero_zero @ ( word @ A ) ) ) ) ).
% word_not_simps(1)
thf(fact_1692_word__coorder_Oextremum__strict,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
~ ( ord_less @ ( word @ A ) @ A2 @ ( zero_zero @ ( word @ A ) ) ) ) ).
% word_coorder.extremum_strict
thf(fact_1693_word__gt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y )
= ( ( zero_zero @ ( word @ A ) )
!= Y ) ) ) ).
% word_gt_0
thf(fact_1694_word__neq__0__conv,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A] :
( ( W2
!= ( zero_zero @ ( word @ A ) ) )
= ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ W2 ) ) ) ).
% word_neq_0_conv
thf(fact_1695_word__gt__a__gt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,N: word @ A] :
( ( ord_less @ ( word @ A ) @ A2 @ N )
=> ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N ) ) ) ).
% word_gt_a_gt_0
thf(fact_1696_word__greater__zero__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ A2 )
= ( A2
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_greater_zero_iff
thf(fact_1697_word__not__simps_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A] :
~ ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ Y ) ) ).
% word_not_simps(3)
thf(fact_1698_word__order_Oextremum__strict,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
~ ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ A2 ) ) ).
% word_order.extremum_strict
thf(fact_1699_word__order_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( A2
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ ( word @ A ) @ A2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_order.not_eq_extremum
thf(fact_1700_word__le__not__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ ( word @ A ) )
= ( ^ [B8: word @ A,A8: word @ A] :
~ ( ord_less @ ( word @ A ) @ A8 @ B8 ) ) ) ) ).
% word_le_not_less
thf(fact_1701_word__not__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ~ ( ord_less_eq @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ ( word @ A ) @ Y @ X ) ) ) ).
% word_not_le
thf(fact_1702_lt1__neq0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X )
= ( X
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% lt1_neq0
thf(fact_1703_word__add__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_add_no_overflow
thf(fact_1704_less__x__plus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( X
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) )
= ( ( ord_less @ ( word @ A ) @ Y @ X )
| ( Y = X ) ) ) ) ) ).
% less_x_plus_1
thf(fact_1705_less__is__non__zero__p1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,K: word @ A] :
( ( ord_less @ ( word @ A ) @ A2 @ K )
=> ( ( plus_plus @ ( word @ A ) @ A2 @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% less_is_non_zero_p1
thf(fact_1706_word__gr0__conv__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ M )
=> ? [N2: word @ A] :
( M
= ( plus_plus @ ( word @ A ) @ N2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_gr0_conv_Suc
thf(fact_1707_word__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) )
| ( ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_overflow
thf(fact_1708_gt0__iff__gem1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
= ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ X ) ) ) ).
% gt0_iff_gem1
thf(fact_1709_word__less__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ( X
= ( minus_minus @ ( word @ A ) @ Y @ ( one_one @ ( word @ A ) ) ) )
| ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Y @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_less_cases
thf(fact_1710_no__plus__overflow__neg,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( uminus_uminus @ ( word @ A ) @ Y ) )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) ) ) ) ).
% no_plus_overflow_neg
thf(fact_1711_word__le__make__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( Y
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
= ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ Y @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_le_make_less
thf(fact_1712_word__Suc__leq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: word @ A,X: word @ A] :
( ( K
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ K @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less_eq @ ( word @ A ) @ X @ K ) ) ) ) ).
% word_Suc_leq
thf(fact_1713_word__Suc__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,K: word @ A] :
( ( X
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ K )
= ( ord_less @ ( word @ A ) @ X @ K ) ) ) ) ).
% word_Suc_le
thf(fact_1714_word__plus__strict__mono__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Z ) )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) @ ( plus_plus @ ( word @ A ) @ X @ Z ) ) ) ) ) ).
% word_plus_strict_mono_right
thf(fact_1715_plus__one__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ X @ N ) ) ) ).
% plus_one_helper
thf(fact_1716_word__le__plus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,N: word @ A,A2: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ Y @ N ) )
=> ( ( ord_less @ ( word @ A ) @ A2 @ N )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y @ A2 ) @ ( plus_plus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y @ A2 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_le_plus_1
thf(fact_1717_word__le__plus,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,C3: word @ A] :
( ( ord_less @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
=> ( ( ord_less @ ( word @ A ) @ C3 @ B2 )
=> ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ C3 ) ) ) ) ) ).
% word_le_plus
thf(fact_1718_word__plus__mono__left,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y @ X ) @ ( plus_plus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% word_plus_mono_left
thf(fact_1719_word__add__increasing,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P3: word @ A,W2: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P3 @ W2 ) @ X )
=> ( ( ord_less_eq @ ( word @ A ) @ P3 @ ( plus_plus @ ( word @ A ) @ P3 @ W2 ) )
=> ( ord_less_eq @ ( word @ A ) @ P3 @ X ) ) ) ) ).
% word_add_increasing
thf(fact_1720_word__plus__mcs__4_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,V3: word @ A,W2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ V3 ) @ ( plus_plus @ ( word @ A ) @ X @ W2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ V3 ) )
=> ( ord_less_eq @ ( word @ A ) @ V3 @ W2 ) ) ) ) ).
% word_plus_mcs_4'
thf(fact_1721_word__add__le__iff2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,J: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ ( plus_plus @ ( word @ A ) @ I @ K ) )
=> ( ( ord_less_eq @ ( word @ A ) @ J @ ( plus_plus @ ( word @ A ) @ J @ K ) )
=> ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ K ) @ ( plus_plus @ ( word @ A ) @ J @ K ) )
= ( ord_less_eq @ ( word @ A ) @ I @ J ) ) ) ) ) ).
% word_add_le_iff2
thf(fact_1722_word__plus__mcs__4,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V3: word @ A,X: word @ A,W2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V3 @ X ) @ ( plus_plus @ ( word @ A ) @ W2 @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ V3 @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ V3 @ W2 ) ) ) ) ).
% word_plus_mcs_4
thf(fact_1723_word__plus__mcs__3,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V3: word @ A,W2: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ V3 @ W2 )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W2 @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V3 @ X ) @ ( plus_plus @ ( word @ A ) @ W2 @ X ) ) ) ) ) ).
% word_plus_mcs_3
thf(fact_1724_word__sub__mono2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,C3: word @ A,D2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) @ ( plus_plus @ ( word @ A ) @ C3 @ D2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C3 @ A2 )
=> ( ( ord_less_eq @ ( word @ A ) @ B2 @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ D2 @ ( plus_plus @ ( word @ A ) @ C3 @ D2 ) )
=> ( ord_less_eq @ ( word @ A ) @ B2 @ D2 ) ) ) ) ) ) ).
% word_sub_mono2
thf(fact_1725_word__random,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P3: word @ A,X9: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ P3 @ ( plus_plus @ ( word @ A ) @ P3 @ X9 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ X9 )
=> ( ord_less_eq @ ( word @ A ) @ P3 @ ( plus_plus @ ( word @ A ) @ P3 @ X ) ) ) ) ) ).
% word_random
thf(fact_1726_plus__1__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ X )
= ( X
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% plus_1_less
thf(fact_1727_overflow__plus__one__self,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ P3 ) @ P3 )
= ( P3
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% overflow_plus_one_self
thf(fact_1728_plus__one__helper2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ N )
=> ( ( ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% plus_one_helper2
thf(fact_1729_word__plus__one__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
=> ( ( Y
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_plus_one_nonzero
thf(fact_1730_neq__0__no__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
=> ( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X @ Y )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% neq_0_no_wrap
thf(fact_1731_word__le__imp__diff__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: word @ A,N: word @ A,M: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ K @ N )
=> ( ( ord_less_eq @ ( word @ A ) @ N @ M )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N @ K ) @ M ) ) ) ) ).
% word_le_imp_diff_le
thf(fact_1732_word__le__minus__mono__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Z: word @ A,Y: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Z @ Y )
=> ( ( ord_less_eq @ ( word @ A ) @ Y @ X )
=> ( ( ord_less_eq @ ( word @ A ) @ Z @ X )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ ( word @ A ) @ X @ Z ) ) ) ) ) ) ).
% word_le_minus_mono_right
thf(fact_1733_le__step__down__word__2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Y @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% le_step_down_word_2
thf(fact_1734_le__step__down__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,N: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ N )
=> ( ( I != N )
=> ( ord_less_eq @ ( word @ A ) @ I @ ( minus_minus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% le_step_down_word
thf(fact_1735_word__leq__minus__one__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( Y
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Y @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ X @ Y ) ) ) ) ).
% word_leq_minus_one_le
thf(fact_1736_word__leq__le__minus__one,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ Y ) ) ) ) ).
% word_leq_le_minus_one
thf(fact_1737_word__diff__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,M: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ M )
=> ( ( ord_less_eq @ ( word @ A ) @ N @ M )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ M @ N ) @ M ) ) ) ) ) ).
% word_diff_less
thf(fact_1738_le__m1__iff__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
= ( ( ord_less_eq @ ( word @ A ) @ Y @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ ( word @ A ) @ Y @ X ) ) ) ) ).
% le_m1_iff_lt
thf(fact_1739_less__1__simp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,M: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) @ M )
= ( ( ord_less_eq @ ( word @ A ) @ N @ M )
& ( N
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% less_1_simp
thf(fact_1740_word__le__minus__one__leq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Y @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_le_minus_one_leq
thf(fact_1741_word__minus__one__le__leq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ Y )
=> ( ord_less_eq @ ( word @ A ) @ X @ Y ) ) ) ).
% word_minus_one_le_leq
thf(fact_1742_word__less__imp__diff__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: word @ A,N: word @ A,M: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ K @ N )
=> ( ( ord_less @ ( word @ A ) @ N @ M )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N @ K ) @ M ) ) ) ) ).
% word_less_imp_diff_less
thf(fact_1743_word__sub__1__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ X ) ) ) ).
% word_sub_1_le
thf(fact_1744_word__must__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ N @ X )
=> ( N
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_must_wrap
thf(fact_1745_More__Word_Oword__l__diffs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ X ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W2 @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ W2 @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% More_Word.word_l_diffs(4)
thf(fact_1746_More__Word_Oword__l__diffs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ W2 @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ X ) @ Z ) ) ) ) ).
% More_Word.word_l_diffs(3)
thf(fact_1747_word__diff__ls_H_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,W2: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W2 @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ W2 ) ) ) ) ).
% word_diff_ls'(4)
thf(fact_1748_word__diff__ls_H_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,W2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ W2 )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W2 @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W2 @ X ) ) ) ) ) ).
% word_diff_ls'(3)
thf(fact_1749_word__l__diffs_H_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ X ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W2 @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ X ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% word_l_diffs'(4)
thf(fact_1750_word__l__diffs_H_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ X ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ X ) @ Z ) ) ) ) ).
% word_l_diffs'(3)
thf(fact_1751_word__diff__ls_H_H_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,W2: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W2 @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ X ) @ X ) ) ) ) ) ).
% word_diff_ls''(4)
thf(fact_1752_word__diff__ls_H_H_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,W2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ X ) @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W2 @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W2 @ X ) ) ) ) ) ).
% word_diff_ls''(3)
thf(fact_1753_word__less__nowrapI_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Z: word @ A,K: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Z @ K ) )
=> ( ( ord_less_eq @ ( word @ A ) @ K @ Z )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K )
=> ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ K ) ) ) ) ) ) ).
% word_less_nowrapI'
thf(fact_1754_More__Word_Oword__l__diffs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W2 @ X ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W2 @ X ) )
=> ( ord_less @ ( word @ A ) @ W2 @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% More_Word.word_l_diffs(2)
thf(fact_1755_sorted__in__between,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: nat,J: nat,L: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ I )
=> ( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ I ) @ X )
=> ( ( ord_less @ A @ X @ ( nth @ A @ L @ J ) )
=> ~ ! [K2: nat] :
( ( ord_less_eq @ nat @ I @ K2 )
=> ( ( ord_less @ nat @ K2 @ J )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ K2 ) @ X )
=> ~ ( ord_less @ A @ X @ ( nth @ A @ L @ ( plus_plus @ nat @ K2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% sorted_in_between
thf(fact_1756_udvd__incr2__K,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P3: word @ A,A2: word @ A,S: word @ A,K4: word @ A] :
( ( ord_less @ ( word @ A ) @ P3 @ ( plus_plus @ ( word @ A ) @ A2 @ S ) )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ S ) )
=> ( ( udvd @ A @ K4 @ S )
=> ( ( udvd @ A @ K4 @ ( minus_minus @ ( word @ A ) @ P3 @ A2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ P3 )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K4 )
=> ( ( ord_less_eq @ ( word @ A ) @ P3 @ ( plus_plus @ ( word @ A ) @ P3 @ K4 ) )
& ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P3 @ K4 ) @ ( plus_plus @ ( word @ A ) @ A2 @ S ) ) ) ) ) ) ) ) ) ) ).
% udvd_incr2_K
thf(fact_1757_nth__enumerate__eq,axiom,
! [A: $tType,M: nat,Xs: list @ A,N: nat] :
( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ M )
= ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N @ M ) @ ( nth @ A @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1758_word__order_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ordering_top @ ( word @ A ) @ ( ord_less_eq @ ( word @ A ) ) @ ( ord_less @ ( word @ A ) ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_order.ordering_top_axioms
thf(fact_1759_listI__assn__extract,axiom,
! [A: $tType,B: $tType,I: nat,I6: set @ nat,Xs: list @ A,A5: A > B > assn,Xsi: list @ B] :
( ( member @ nat @ I @ I6 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ I6 @ A5 @ Xs @ Xsi )
= ( times_times @ assn @ ( A5 @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ ( minus_minus @ ( set @ nat ) @ I6 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A5 @ Xs @ Xsi ) ) ) ) ) ).
% listI_assn_extract
thf(fact_1760_nth__step__trancl,axiom,
! [A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ A ),N: nat,M: nat] :
( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ ( suc @ N2 ) ) @ ( nth @ A @ Xs @ N2 ) ) @ R3 ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ M @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N ) @ ( nth @ A @ Xs @ M ) ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ) ).
% nth_step_trancl
thf(fact_1761_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_1762_length__nth__simps_I4_J,axiom,
! [B: $tType,X: B,Xs: list @ B,N: nat] :
( ( nth @ B @ ( cons @ B @ X @ Xs ) @ ( suc @ N ) )
= ( nth @ B @ Xs @ N ) ) ).
% length_nth_simps(4)
thf(fact_1763_nth__Cons__Suc,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( suc @ N ) )
= ( nth @ A @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1764_length__nth__simps_I3_J,axiom,
! [B: $tType,X: B,Xs: list @ B] :
( ( nth @ B @ ( cons @ B @ X @ Xs ) @ ( zero_zero @ nat ) )
= X ) ).
% length_nth_simps(3)
thf(fact_1765_nth__Cons__0,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) )
= X ) ).
% nth_Cons_0
thf(fact_1766_trancl__single,axiom,
! [A: $tType,A2: A,B2: A] :
( ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) )
= ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trancl_single
thf(fact_1767_listI__assn__finite,axiom,
! [B: $tType,A: $tType,I6: set @ nat,A5: A > B > assn,Xs: list @ A,Xsi: list @ B] :
( ~ ( finite_finite2 @ nat @ I6 )
=> ( ( vEBT_List_listI_assn @ A @ B @ I6 @ A5 @ Xs @ Xsi )
= ( bot_bot @ assn ) ) ) ).
% listI_assn_finite
thf(fact_1768_enumerate__simps_I2_J,axiom,
! [B: $tType,N: nat,X: B,Xs: list @ B] :
( ( enumerate @ B @ N @ ( cons @ B @ X @ Xs ) )
= ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N @ X ) @ ( enumerate @ B @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_1769_list__tail__coinc,axiom,
! [A: $tType,N1: A,R1: list @ A,N22: A,R22: list @ A] :
( ( ( cons @ A @ N1 @ R1 )
= ( cons @ A @ N22 @ R22 ) )
=> ( ( N1 = N22 )
& ( R1 = R22 ) ) ) ).
% list_tail_coinc
thf(fact_1770_sorted2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Zs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% sorted2
thf(fact_1771_sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ Ys ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X @ X4 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys ) ) ) ) ).
% sorted_simps(2)
thf(fact_1772_strict__sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ ( cons @ A @ X @ Ys ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( ord_less @ A @ X @ X4 ) )
& ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys ) ) ) ) ).
% strict_sorted_simps(2)
thf(fact_1773_trancl__mono__mp,axiom,
! [A: $tType,U2: set @ ( product_prod @ A @ A ),V2: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U2 @ V2 )
=> ( ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_trancl @ A @ U2 ) )
=> ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_trancl @ A @ V2 ) ) ) ) ).
% trancl_mono_mp
thf(fact_1774_trancl__sub,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( transitive_trancl @ A @ R3 ) ) ).
% trancl_sub
thf(fact_1775_trancl__sub__insert__trancl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 ) @ ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ X @ R3 ) ) ) ).
% trancl_sub_insert_trancl
thf(fact_1776_strict__sorted__imp__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% strict_sorted_imp_sorted
thf(fact_1777_strict__sorted__equal,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys )
=> ( ( ( set2 @ A @ Ys )
= ( set2 @ A @ Xs ) )
=> ( Ys = Xs ) ) ) ) ) ).
% strict_sorted_equal
thf(fact_1778_length__nth__simps_I2_J,axiom,
! [B: $tType,X: B,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( cons @ B @ X @ Xs ) )
= ( suc @ ( size_size @ ( list @ B ) @ Xs ) ) ) ).
% length_nth_simps(2)
thf(fact_1779_length__Suc__conv,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y5: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y5 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1780_Suc__length__conv,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( suc @ N )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [Y5: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y5 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1781_impossible__Cons,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs
!= ( cons @ A @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1782_set__subset__Cons,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1783_listI__assn__cong,axiom,
! [A: $tType,B: $tType,I6: set @ nat,I7: set @ nat,Xs: list @ A,Xs4: list @ A,Xsi: list @ B,Xsi2: list @ B,A5: A > B > assn,A10: A > B > assn] :
( ( I6 = I7 )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Xs4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Xsi )
= ( size_size @ ( list @ B ) @ Xsi2 ) )
=> ( ! [I2: nat] :
( ( member @ nat @ I2 @ I6 )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Xsi ) )
=> ( ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Xs4 @ I2 ) )
& ( ( nth @ B @ Xsi @ I2 )
= ( nth @ B @ Xsi2 @ I2 ) )
& ( ( A5 @ ( nth @ A @ Xs @ I2 ) @ ( nth @ B @ Xsi @ I2 ) )
= ( A10 @ ( nth @ A @ Xs4 @ I2 ) @ ( nth @ B @ Xsi2 @ I2 ) ) ) ) ) ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ I6 @ A5 @ Xs @ Xsi )
= ( vEBT_List_listI_assn @ A @ B @ I7 @ A10 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).
% listI_assn_cong
thf(fact_1784_listI__assn__weak__cong,axiom,
! [A: $tType,B: $tType,I6: set @ nat,I7: set @ nat,A5: A > B > assn,A10: A > B > assn,Xs: list @ A,Xs4: list @ A,Xsi: list @ B,Xsi2: list @ B] :
( ( I6 = I7 )
=> ( ( A5 = A10 )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Xs4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Xsi )
= ( size_size @ ( list @ B ) @ Xsi2 ) )
=> ( ! [I2: nat] :
( ( member @ nat @ I2 @ I6 )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Xsi ) )
=> ( ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Xs4 @ I2 ) )
& ( ( nth @ B @ Xsi @ I2 )
= ( nth @ B @ Xsi2 @ I2 ) ) ) ) ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ I6 @ A5 @ Xs @ Xsi )
= ( vEBT_List_listI_assn @ A @ B @ I7 @ A10 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).
% listI_assn_weak_cong
thf(fact_1785_sorted__wrt01,axiom,
! [A: $tType,Xs: list @ A,P2: A > A > $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ P2 @ Xs ) ) ).
% sorted_wrt01
thf(fact_1786_sorted__wrt__iff__nth__less,axiom,
! [A: $tType] :
( ( sorted_wrt @ A )
= ( ^ [P5: A > A > $o,Xs2: list @ A] :
! [I3: nat,J3: nat] :
( ( ord_less @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P5 @ ( nth @ A @ Xs2 @ I3 ) @ ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_1787_sorted__wrt__nth__less,axiom,
! [A: $tType,P2: A > A > $o,Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ P2 @ Xs )
=> ( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P2 @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_1788_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_1789_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [X4: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ X4 @ Ys3 ) )
& ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1790_sorted01,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted01
thf(fact_1791_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I3: nat,J3: nat] :
( ( ord_less @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_1792_list_Osize_I4_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X21 @ X22 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size(4)
thf(fact_1793_nth__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= X ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_1794_listI__assn__insert,axiom,
! [A: $tType,B: $tType,I: nat,I6: set @ nat,Xs: list @ A,A5: A > B > assn,Xsi: list @ B] :
( ~ ( member @ nat @ I @ I6 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ ( insert @ nat @ I @ I6 ) @ A5 @ Xs @ Xsi )
= ( times_times @ assn @ ( A5 @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ I6 @ A5 @ Xs @ Xsi ) ) ) ) ) ).
% listI_assn_insert
thf(fact_1795_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I3: nat] :
( ( ord_less @ nat @ ( suc @ I3 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ ( suc @ I3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_1796_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_1797_sorted__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_1798_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ~ ! [L2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ L2 )
=> ( ( ( set2 @ A @ L2 )
= A5 )
=> ( ( size_size @ ( list @ A ) @ L2 )
!= ( finite_card @ A @ A5 ) ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_1799_udvd__minus__le_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xy: word @ A,K: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ Xy @ K )
=> ( ( udvd @ A @ Z @ Xy )
=> ( ( udvd @ A @ Z @ K )
=> ( ord_less_eq @ ( word @ A ) @ Xy @ ( minus_minus @ ( word @ A ) @ K @ Z ) ) ) ) ) ) ).
% udvd_minus_le'
thf(fact_1800_nth__non__equal__first__eq,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A,N: nat] :
( ( X != Y )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= Y )
= ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1801_nth__equal__first__eq,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= X )
= ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1802_slice__Cons,axiom,
! [A: $tType,Begin: nat,End: nat,X: A,Xs: list @ A] :
( ( ( ( Begin
= ( zero_zero @ nat ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
=> ( ( slice @ A @ Begin @ End @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( slice @ A @ Begin @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) )
& ( ~ ( ( Begin
= ( zero_zero @ nat ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
=> ( ( slice @ A @ Begin @ End @ ( cons @ A @ X @ Xs ) )
= ( slice @ A @ ( minus_minus @ nat @ Begin @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% slice_Cons
thf(fact_1803_length__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X @ Xs ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_Cons
thf(fact_1804_listI__assn__reinsert,axiom,
! [B: $tType,A: $tType,P2: assn,A5: A > B > assn,Xs: list @ A,I: nat,Xsi: list @ B,I6: set @ nat,F5: assn,Q2: assn] :
( ( entails @ P2 @ ( times_times @ assn @ ( times_times @ assn @ ( A5 @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ ( minus_minus @ ( set @ nat ) @ I6 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A5 @ Xs @ Xsi ) ) @ F5 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( member @ nat @ I @ I6 )
=> ( ( entails @ ( times_times @ assn @ ( vEBT_List_listI_assn @ A @ B @ I6 @ A5 @ Xs @ Xsi ) @ F5 ) @ Q2 )
=> ( entails @ P2 @ Q2 ) ) ) ) ) ).
% listI_assn_reinsert
thf(fact_1805_sorted__list__of__set__nonempty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( linord4507533701916653071of_set @ A @ A5 )
= ( cons @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_1806_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,L: list @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
& ( ( set2 @ A @ L )
= A5 )
& ( ( size_size @ ( list @ A ) @ L )
= ( finite_card @ A @ A5 ) ) )
= ( ( linord4507533701916653071of_set @ A @ A5 )
= L ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_1807_trancl__mono,axiom,
! [A: $tType,P3: product_prod @ A @ A,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P3 @ ( transitive_trancl @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S )
=> ( member @ ( product_prod @ A @ A ) @ P3 @ ( transitive_trancl @ A @ S ) ) ) ) ).
% trancl_mono
thf(fact_1808_converse__trancl__induct,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),P2: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ B2 ) @ R2 )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( P2 @ Z3 )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% converse_trancl_induct
thf(fact_1809_trancl__trans__induct,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),P2: A > A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 )
=> ( P2 @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( P2 @ X3 @ Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( P2 @ Y3 @ Z3 )
=> ( P2 @ X3 @ Z3 ) ) ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_1810_groupy,axiom,
! [A5: assn,B5: assn,C4: assn,D5: assn,X6: assn] :
( ( entails @ ( times_times @ assn @ ( times_times @ assn @ A5 @ B5 ) @ ( times_times @ assn @ C4 @ D5 ) ) @ X6 )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ A5 @ B5 ) @ C4 ) @ D5 ) @ X6 ) ) ).
% groupy
thf(fact_1811_midextr,axiom,
! [P2: assn,Q2: assn,Q4: assn,R3: assn,X6: assn] :
( ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ P2 @ Q2 ) @ Q4 ) @ R3 ) @ X6 )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ P2 @ R3 ) @ Q2 ) @ Q4 ) @ X6 ) ) ).
% midextr
thf(fact_1812_swappa,axiom,
! [B5: assn,A5: assn,C4: assn,X6: assn] :
( ( entails @ ( times_times @ assn @ ( times_times @ assn @ B5 @ A5 ) @ C4 ) @ X6 )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ A5 @ B5 ) @ C4 ) @ X6 ) ) ).
% swappa
thf(fact_1813_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( set2 @ A @ ( linord4507533701916653071of_set @ A @ A5 ) )
= A5 ) ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_1814_ent__false__iff,axiom,
! [P2: assn] :
( ( entails @ P2 @ ( bot_bot @ assn ) )
= ( ! [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ P2 @ H4 ) ) ) ).
% ent_false_iff
thf(fact_1815_ent__trans,axiom,
! [P2: assn,Q2: assn,R3: assn] :
( ( entails @ P2 @ Q2 )
=> ( ( entails @ Q2 @ R3 )
=> ( entails @ P2 @ R3 ) ) ) ).
% ent_trans
thf(fact_1816_ent__refl,axiom,
! [P2: assn] : ( entails @ P2 @ P2 ) ).
% ent_refl
thf(fact_1817_ent__iffI,axiom,
! [A5: assn,B5: assn] :
( ( entails @ A5 @ B5 )
=> ( ( entails @ B5 @ A5 )
=> ( A5 = B5 ) ) ) ).
% ent_iffI
thf(fact_1818_ent__star__mono,axiom,
! [P2: assn,P6: assn,Q2: assn,Q4: assn] :
( ( entails @ P2 @ P6 )
=> ( ( entails @ Q2 @ Q4 )
=> ( entails @ ( times_times @ assn @ P2 @ Q2 ) @ ( times_times @ assn @ P6 @ Q4 ) ) ) ) ).
% ent_star_mono
thf(fact_1819_entails__def,axiom,
( entails
= ( ^ [P5: assn,Q6: assn] :
! [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P5 @ H4 )
=> ( rep_assn @ Q6 @ H4 ) ) ) ) ).
% entails_def
thf(fact_1820_entailsI,axiom,
! [P2: assn,Q2: assn] :
( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P2 @ H )
=> ( rep_assn @ Q2 @ H ) )
=> ( entails @ P2 @ Q2 ) ) ).
% entailsI
thf(fact_1821_entailsD,axiom,
! [P2: assn,Q2: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( entails @ P2 @ Q2 )
=> ( ( rep_assn @ P2 @ H2 )
=> ( rep_assn @ Q2 @ H2 ) ) ) ).
% entailsD
thf(fact_1822_ent__fwd,axiom,
! [P2: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Q2: assn] :
( ( rep_assn @ P2 @ H2 )
=> ( ( entails @ P2 @ Q2 )
=> ( rep_assn @ Q2 @ H2 ) ) ) ).
% ent_fwd
thf(fact_1823_ent__false,axiom,
! [P2: assn] : ( entails @ ( bot_bot @ assn ) @ P2 ) ).
% ent_false
thf(fact_1824_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( ( linord4507533701916653071of_set @ A @ A5 )
= ( linord4507533701916653071of_set @ A @ B5 ) )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( A5 = B5 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_1825_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_1826_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] : ( sorted_wrt @ A @ ( ord_less @ A ) @ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_1827_trancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P2: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ! [A4: A,B4: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A4 @ B4 ) ) @ R2 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: A,B4: B,Aa2: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A4 @ B4 ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R2 )
=> ( ( P2 @ A4 @ B4 )
=> ( P2 @ Aa2 @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_1828_trancl_Ocases,axiom,
! [A: $tType,A12: A,A23: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ A23 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ A23 ) @ R2 )
=> ~ ! [B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B4 ) @ ( transitive_trancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A23 ) @ R2 ) ) ) ) ).
% trancl.cases
thf(fact_1829_trancl_Osimps,axiom,
! [A: $tType,A12: A,A23: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ A23 ) @ ( transitive_trancl @ A @ R2 ) )
= ( ? [A8: A,B8: A] :
( ( A12 = A8 )
& ( A23 = B8 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B8 ) @ R2 ) )
| ? [A8: A,B8: A,C7: A] :
( ( A12 = A8 )
& ( A23 = C7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B8 ) @ ( transitive_trancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ C7 ) @ R2 ) ) ) ) ).
% trancl.simps
thf(fact_1830_trancl_Or__into__trancl,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ).
% trancl.r_into_trancl
thf(fact_1831_tranclE,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ~ ! [C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ C2 @ B2 ) @ R2 ) ) ) ) ).
% tranclE
thf(fact_1832_trancl__trans,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_trans
thf(fact_1833_trancl__induct,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),P2: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ Y3 ) @ R2 )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ Y3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z3 ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% trancl_induct
thf(fact_1834_r__r__into__trancl,axiom,
! [A: $tType,A2: A,B2: A,R3: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% r_r_into_trancl
thf(fact_1835_converse__tranclE,axiom,
! [A: $tType,X: A,Z: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R2 )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ) ).
% converse_tranclE
thf(fact_1836_irrefl__trancl__rD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_1837_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C3 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_1838_trancl__into__trancl2,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C3 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_1839_local_Oext,axiom,
! [Y: nat,TreeList: list @ vEBT_VEBT,X13: array @ vEBT_VEBTi,Tree_is: list @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
( ( ord_less @ nat @ Y @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( entails @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is ) @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( nth @ vEBT_VEBT @ TreeList @ Y ) @ ( nth @ vEBT_VEBTi @ Tree_is @ Y ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ Y @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ Y @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ ( nth @ vEBT_VEBT @ TreeList @ Y ) @ ( nth @ vEBT_VEBTi @ Tree_is @ Y ) ) ) ) ) ).
% local.ext
thf(fact_1840_mod__frame__fwd,axiom,
! [Ps: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),P2: assn,R3: assn,F5: assn] :
( ( rep_assn @ Ps @ H2 )
=> ( ( entails @ P2 @ R3 )
=> ( ( entails @ Ps @ ( times_times @ assn @ P2 @ F5 ) )
=> ( rep_assn @ ( times_times @ assn @ R3 @ F5 ) @ H2 ) ) ) ) ).
% mod_frame_fwd
thf(fact_1841_assnle,axiom,
! [TreeList: list @ vEBT_VEBT,Tree_is: list @ vEBT_VEBTi,X13: array @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] : ( entails @ ( times_times @ assn @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) ) @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ).
% assnle
thf(fact_1842_listI__assn__reinsert_H,axiom,
! [A: $tType,B: $tType,C: $tType,P2: assn,A5: A > B > assn,Xs: list @ A,I: nat,Xsi: list @ B,I6: set @ nat,F5: assn,C3: heap_Time_Heap @ C,Q2: C > assn] :
( ( entails @ P2 @ ( times_times @ assn @ ( times_times @ assn @ ( A5 @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ ( minus_minus @ ( set @ nat ) @ I6 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A5 @ Xs @ Xsi ) ) @ F5 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( member @ nat @ I @ I6 )
=> ( ( hoare_hoare_triple @ C @ ( times_times @ assn @ ( vEBT_List_listI_assn @ A @ B @ I6 @ A5 @ Xs @ Xsi ) @ F5 ) @ C3 @ Q2 )
=> ( hoare_hoare_triple @ C @ P2 @ C3 @ Q2 ) ) ) ) ) ).
% listI_assn_reinsert'
thf(fact_1843_listI__assn__reinsert__upd,axiom,
! [D: $tType,C: $tType,P2: assn,A5: C > D > assn,X: C,Xi: D,I6: set @ nat,I: nat,Xs: list @ C,Xsi: list @ D,F5: assn,Q2: assn] :
( ( entails @ P2 @ ( times_times @ assn @ ( times_times @ assn @ ( A5 @ X @ Xi ) @ ( vEBT_List_listI_assn @ C @ D @ ( minus_minus @ ( set @ nat ) @ I6 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A5 @ Xs @ Xsi ) ) @ F5 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ C ) @ Xs ) )
=> ( ( member @ nat @ I @ I6 )
=> ( ( entails @ ( times_times @ assn @ ( vEBT_List_listI_assn @ C @ D @ I6 @ A5 @ ( list_update @ C @ Xs @ I @ X ) @ ( list_update @ D @ Xsi @ I @ Xi ) ) @ F5 ) @ Q2 )
=> ( entails @ P2 @ Q2 ) ) ) ) ) ).
% listI_assn_reinsert_upd
thf(fact_1844_recomp,axiom,
! [I: nat,TreeList: list @ vEBT_VEBT,Tree_is: list @ vEBT_VEBTi,X13: array @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( nth @ vEBT_VEBT @ TreeList @ I ) @ ( nth @ vEBT_VEBTi @ Tree_is @ I ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) ).
% recomp
thf(fact_1845_repack,axiom,
! [I: nat,TreeList: list @ vEBT_VEBT,Tree_is: list @ vEBT_VEBTi,Rest: assn,X13: array @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( nth @ vEBT_VEBT @ TreeList @ I ) @ ( nth @ vEBT_VEBTi @ Tree_is @ I ) ) @ Rest ) @ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ Rest @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) ).
% repack
thf(fact_1846_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_1847_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_1848_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_1849_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_1850_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_1851_infinite__Ico__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ico_iff
thf(fact_1852_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_1853_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_1854_list__update__beyond,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( list_update @ A @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_1855_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] :
( ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) )
= ( set_or7035219750837199246ssThan @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_1856_tcd,axiom,
! [A: $tType,I: nat,TreeList: list @ vEBT_VEBT,TreeList3: list @ A,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array @ vEBT_VEBTi,Tree_is: list @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( size_size @ ( list @ A ) @ TreeList3 ) )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ Y @ X ) @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is @ I @ X ) ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ ( list_update @ vEBT_VEBT @ TreeList @ I @ Y ) @ ( list_update @ vEBT_VEBTi @ Tree_is @ I @ X ) ) ) @ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is @ I @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ ( list_update @ vEBT_VEBT @ TreeList @ I @ Y ) @ ( list_update @ vEBT_VEBTi @ Tree_is @ I @ X ) ) ) ) ) ) ).
% tcd
thf(fact_1857_nth__update__invalid,axiom,
! [A: $tType,I: nat,L: list @ A,J: nat,X: A] :
( ~ ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ J @ X ) @ I )
= ( nth @ A @ L @ I ) ) ) ).
% nth_update_invalid
thf(fact_1858_nth__list__update__eq,axiom,
! [A: $tType,I: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_1859_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) )
= ( insert @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_1860_txe,axiom,
! [Y: nat,TreeList: list @ vEBT_VEBT,Tree_is: list @ vEBT_VEBTi,X13: array @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
( ( ord_less @ nat @ Y @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( nth @ vEBT_VEBT @ TreeList @ Y ) @ ( nth @ vEBT_VEBTi @ Tree_is @ Y ) ) @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ Y @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) ).
% txe
thf(fact_1861_set__swap,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) )
= ( set2 @ A @ Xs ) ) ) ) ).
% set_swap
thf(fact_1862_sum_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G3: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G3 @ N ) ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_1863_nth__image__indices,axiom,
! [A: $tType,L: list @ A] :
( ( image @ nat @ A @ ( nth @ A @ L ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
= ( set2 @ A @ L ) ) ).
% nth_image_indices
thf(fact_1864_deleti_H__rf__abstr,axiom,
! [T2: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( hoare_hoare_triple @ vEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_V1365221501068881998eletei @ T2 @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ T2 @ X ) ) ) ) ).
% deleti'_rf_abstr
thf(fact_1865_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
= ( ( A2 = C3 )
& ( B2 = D2 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_1866_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( A2 = C3 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_1867_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ D2 )
=> ( B2 = D2 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_1868_listI__assn__conv,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,A5: A > B > assn,Xsi: list @ B] :
( ( N
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) @ A5 @ Xs @ Xsi )
= ( vEBT_List_list_assn @ A @ B @ A5 @ Xs @ Xsi ) ) ) ).
% listI_assn_conv
thf(fact_1869_list__assn__conv__idx,axiom,
! [B: $tType,A: $tType] :
( ( vEBT_List_list_assn @ A @ B )
= ( ^ [A6: A > B > assn,Xs2: list @ A] : ( vEBT_List_listI_assn @ A @ B @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ A6 @ Xs2 ) ) ) ).
% list_assn_conv_idx
thf(fact_1870_list__update__code_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,I: nat,Y: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons @ A @ X @ ( list_update @ A @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_1871_list__update__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) @ Y )
= ( cons @ A @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1872_set__update__subsetI,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A,X: A,I: nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X ) ) @ A5 ) ) ) ).
% set_update_subsetI
thf(fact_1873_infinite__Ico,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ico
thf(fact_1874_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_1875_ex__nat__less__eq,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less @ nat @ M5 @ N )
& ( P2 @ M5 ) ) )
= ( ? [X4: nat] :
( ( member @ nat @ X4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P2 @ X4 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_1876_all__nat__less__eq,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less @ nat @ M5 @ N )
=> ( P2 @ M5 ) ) )
= ( ! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P2 @ X4 ) ) ) ) ).
% all_nat_less_eq
thf(fact_1877_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_1878_listI__assn__conv_H,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,A5: A > B > assn,Xsi: list @ B,F5: assn] :
( ( N
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( times_times @ assn @ ( vEBT_List_listI_assn @ A @ B @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) @ A5 @ Xs @ Xsi ) @ F5 )
= ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ A5 @ Xs @ Xsi ) @ F5 ) ) ) ).
% listI_assn_conv'
thf(fact_1879_subset__eq__atLeast0__lessThan__finite,axiom,
! [N8: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N8 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N8 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_1880_subset__card__intvl__is__intvl,axiom,
! [A5: set @ nat,K: nat] :
( ( ord_less_eq @ ( set @ nat ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A5 ) ) ) )
=> ( A5
= ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A5 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_1881_extract__pre__list__assn__lengthD,axiom,
! [B: $tType,A: $tType,A5: A > B > assn,Xs: list @ A,Xsi: list @ B,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( vEBT_List_list_assn @ A @ B @ A5 @ Xs @ Xsi ) @ H2 )
=> ( ( size_size @ ( list @ B ) @ Xsi )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% extract_pre_list_assn_lengthD
thf(fact_1882_set__update__memI,axiom,
! [A: $tType,N: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1883_in__set__upd__eq,axiom,
! [A: $tType,I: nat,L: list @ A,X: A,Y: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
= ( ( X = Y )
| ( ( member @ A @ X @ ( set2 @ A @ L ) )
& ! [Y5: A] : ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y5 ) ) ) ) ) ) ) ).
% in_set_upd_eq
thf(fact_1884_in__set__upd__cases,axiom,
! [A: $tType,X: A,L: list @ A,I: nat,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
=> ( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( X != Y ) )
=> ( member @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% in_set_upd_cases
thf(fact_1885_in__set__upd__eq__aux,axiom,
! [A: $tType,I: nat,L: list @ A,X: A,Y: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
= ( ( X = Y )
| ! [Y5: A] : ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y5 ) ) ) ) ) ) ).
% in_set_upd_eq_aux
thf(fact_1886_nth__list__update_H,axiom,
! [A: $tType,I: nat,J: nat,L: list @ A,X: A] :
( ( ( ( I = J )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ I @ X ) @ J )
= X ) )
& ( ~ ( ( I = J )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ I @ X ) @ J )
= ( nth @ A @ L @ J ) ) ) ) ).
% nth_list_update'
thf(fact_1887_list__update__same__conv,axiom,
! [A: $tType,I: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( list_update @ A @ Xs @ I @ X )
= Xs )
= ( ( nth @ A @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_1888_nth__list__update,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_1889_set__update__subset__insert,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X ) ) @ ( insert @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_1890_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A2: B,C3: B,B2: B,D2: B,G3: B > A,H2: B > A] :
( ( A2 = C3 )
=> ( ( B2 = D2 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C3 @ X3 )
=> ( ( ord_less @ B @ X3 @ D2 )
=> ( ( G3 @ X3 )
= ( H2 @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C3 @ D2 ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_1891_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,P3: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P3 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ P3 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ P3 ) ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_1892_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,P3: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P3 )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M @ P3 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ N @ P3 ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_1893_subst__not__in,axiom,
! [A: $tType,B: $tType,I: nat,I6: set @ nat,Xs: list @ A,A5: A > B > assn,X1: A,Xsi: list @ B,X2: B] :
( ~ ( member @ nat @ I @ I6 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ I6 @ A5 @ ( list_update @ A @ Xs @ I @ X1 ) @ ( list_update @ B @ Xsi @ I @ X2 ) )
= ( vEBT_List_listI_assn @ A @ B @ I6 @ A5 @ Xs @ Xsi ) ) ) ) ).
% subst_not_in
thf(fact_1894_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_1895_subset__eq__atLeast0__lessThan__card,axiom,
! [N8: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N8 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ N8 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_1896_listI__assn__reinsert__upd_H,axiom,
! [C: $tType,D: $tType,E: $tType,P2: assn,A5: C > D > assn,X: C,Xi: D,I6: set @ nat,I: nat,Xs: list @ C,Xsi: list @ D,F5: assn,C3: heap_Time_Heap @ E,Q2: E > assn] :
( ( entails @ P2 @ ( times_times @ assn @ ( times_times @ assn @ ( A5 @ X @ Xi ) @ ( vEBT_List_listI_assn @ C @ D @ ( minus_minus @ ( set @ nat ) @ I6 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A5 @ Xs @ Xsi ) ) @ F5 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ C ) @ Xs ) )
=> ( ( member @ nat @ I @ I6 )
=> ( ( hoare_hoare_triple @ E @ ( times_times @ assn @ ( vEBT_List_listI_assn @ C @ D @ I6 @ A5 @ ( list_update @ C @ Xs @ I @ X ) @ ( list_update @ D @ Xsi @ I @ Xi ) ) @ F5 ) @ C3 @ Q2 )
=> ( hoare_hoare_triple @ E @ P2 @ C3 @ Q2 ) ) ) ) ) ).
% listI_assn_reinsert_upd'
thf(fact_1897_sum__shift__lb__Suc0__0__upt,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,K: nat] :
( ( ( F3 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_1898_sum_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G3 @ N ) ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_1899_sum_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G3: nat > A] :
( ( ord_less @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G3 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_1900_sum_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat,B2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G3 @ B2 ) ) ) ) ) ).
% sum.atLeastLessThan_Suc
thf(fact_1901_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_1902_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_1903_insert__swap__set__eq,axiom,
! [A: $tType,I: nat,L: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( insert @ A @ ( nth @ A @ L @ I ) @ ( set2 @ A @ ( list_update @ A @ L @ I @ X ) ) )
= ( insert @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% insert_swap_set_eq
thf(fact_1904_listI__assn__subst,axiom,
! [A: $tType,B: $tType,I: nat,I6: set @ nat,Xs: list @ A,A5: A > B > assn,X1: A,Xsi: list @ B,X2: B] :
( ~ ( member @ nat @ I @ I6 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ ( insert @ nat @ I @ I6 ) @ A5 @ ( list_update @ A @ Xs @ I @ X1 ) @ ( list_update @ B @ Xsi @ I @ X2 ) )
= ( times_times @ assn @ ( A5 @ X1 @ X2 ) @ ( vEBT_List_listI_assn @ A @ B @ I6 @ A5 @ Xs @ Xsi ) ) ) ) ) ).
% listI_assn_subst
thf(fact_1905_big__assn__simp_H,axiom,
! [H2: nat,TreeList: list @ vEBT_VEBT,Xaa: vEBT_VEBT,L: nat,X: vEBT_VEBTi,Xb: option @ nat,X13: array @ vEBT_VEBTi,Tree_is: list @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
( ( ord_less @ nat @ H2 @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( Xaa
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( entails
@ ( times_times @ assn
@ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ Xaa @ X )
@ ( pure_assn
@ ( Xb
= ( vEBT_vebt_mint @ Xaa ) ) ) )
@ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) )
@ ( times_times @ assn
@ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) )
@ ( pure_assn
@ ( Xb
= ( vEBT_vebt_mint @ Xaa ) ) ) )
@ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Xaa ) @ ( list_update @ vEBT_VEBTi @ Tree_is @ H2 @ X ) ) ) ) ) ) ).
% big_assn_simp'
thf(fact_1906_big__assn__simp,axiom,
! [H2: nat,TreeList: list @ vEBT_VEBT,L: nat,X: vEBT_VEBTi,Xaa: option @ nat,X13: array @ vEBT_VEBTi,Tree_is: list @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
( ( ord_less @ nat @ H2 @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( entails
@ ( times_times @ assn
@ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) @ X )
@ ( pure_assn
@ ( Xaa
= ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) ) ) )
@ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) )
@ ( times_times @ assn
@ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) )
@ ( pure_assn
@ ( Xaa
= ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) ) ) )
@ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( list_update @ vEBT_VEBTi @ Tree_is @ H2 @ X ) ) ) ) ) ).
% big_assn_simp
thf(fact_1907_listI__assn__wrap__insert,axiom,
! [E: $tType,P2: assn,Uu2: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I6: set @ nat,I: nat,Xs: list @ vEBT_VEBT,Xsi: list @ vEBT_VEBTi,F5: assn,C3: heap_Time_Heap @ E,Q2: E > assn] :
( ( entails @ P2 @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ Uu2 @ Uua ) @ Xi ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ I6 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ Xs @ Xsi ) ) @ F5 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ Xs ) )
=> ( ( member @ nat @ I @ I6 )
=> ( ( hoare_hoare_triple @ E @ ( times_times @ assn @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ I6 @ vEBT_vebt_assn_raw @ ( list_update @ vEBT_VEBT @ Xs @ I @ ( vEBT_vebt_insert @ Uu2 @ Uua ) ) @ ( list_update @ vEBT_VEBTi @ Xsi @ I @ Xi ) ) @ F5 ) @ C3 @ Q2 )
=> ( hoare_hoare_triple @ E @ P2 @ C3 @ Q2 ) ) ) ) ) ).
% listI_assn_wrap_insert
thf(fact_1908_set__update__distinct,axiom,
! [A: $tType,Xs: list @ A,N: nat,X: A] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ Xs @ N @ X ) )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ N ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_1909_hoare__triple__preI,axiom,
! [A: $tType,P2: assn,C3: heap_Time_Heap @ A,Q2: A > assn] :
( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P2 @ H )
=> ( hoare_hoare_triple @ A @ P2 @ C3 @ Q2 ) )
=> ( hoare_hoare_triple @ A @ P2 @ C3 @ Q2 ) ) ).
% hoare_triple_preI
thf(fact_1910_subset__mset_Osum__list__update,axiom,
! [A: $tType,K: nat,Xs: list @ ( multiset @ A ),X: multiset @ A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ ( multiset @ A ) ) @ Xs ) )
=> ( ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ ( list_update @ ( multiset @ A ) @ Xs @ K @ X ) )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) @ X ) @ ( nth @ ( multiset @ A ) @ Xs @ K ) ) ) ) ).
% subset_mset.sum_list_update
thf(fact_1911_nth__image,axiom,
! [A: $tType,L: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( image @ nat @ A @ ( nth @ A @ Xs ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ L ) )
= ( set2 @ A @ ( take @ A @ L @ Xs ) ) ) ) ).
% nth_image
thf(fact_1912_pure__assn__eq__conv,axiom,
! [P2: $o,Q2: $o] :
( ( ( pure_assn @ P2 )
= ( pure_assn @ Q2 ) )
= ( P2 = Q2 ) ) ).
% pure_assn_eq_conv
thf(fact_1913_merge__pure__star,axiom,
! [A2: $o,B2: $o] :
( ( times_times @ assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B2 ) )
= ( pure_assn
@ ( A2
& B2 ) ) ) ).
% merge_pure_star
thf(fact_1914_take__update,axiom,
! [A: $tType,N: nat,L: list @ A,I: nat,X: A] :
( ( take @ A @ N @ ( list_update @ A @ L @ I @ X ) )
= ( list_update @ A @ ( take @ A @ N @ L ) @ I @ X ) ) ).
% take_update
thf(fact_1915_pure__assn__eq__emp__iff,axiom,
! [P2: $o] :
( ( ( pure_assn @ P2 )
= ( one_one @ assn ) )
= P2 ) ).
% pure_assn_eq_emp_iff
thf(fact_1916_pure__true,axiom,
( ( pure_assn @ $true )
= ( one_one @ assn ) ) ).
% pure_true
thf(fact_1917_pure__false,axiom,
( ( pure_assn @ $false )
= ( bot_bot @ assn ) ) ).
% pure_false
thf(fact_1918_pure__assn__eq__false__iff,axiom,
! [P2: $o] :
( ( ( pure_assn @ P2 )
= ( bot_bot @ assn ) )
= ~ P2 ) ).
% pure_assn_eq_false_iff
thf(fact_1919_take__Suc__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( take @ A @ ( suc @ N ) @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( take @ A @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_1920_take__all,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
=> ( ( take @ A @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_1921_take__all__iff,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( take @ A @ N @ Xs )
= Xs )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_1922_nth__take,axiom,
! [A: $tType,I: nat,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( take @ A @ N @ Xs ) @ I )
= ( nth @ A @ Xs @ I ) ) ) ).
% nth_take
thf(fact_1923_take__update__cancel,axiom,
! [A: $tType,N: nat,M: nat,Xs: list @ A,Y: A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( take @ A @ N @ ( list_update @ A @ Xs @ M @ Y ) )
= ( take @ A @ N @ Xs ) ) ) ).
% take_update_cancel
thf(fact_1924_ent__pure__pre__iff,axiom,
! [P2: assn,B2: $o,Q2: assn] :
( ( entails @ ( times_times @ assn @ P2 @ ( pure_assn @ B2 ) ) @ Q2 )
= ( B2
=> ( entails @ P2 @ Q2 ) ) ) ).
% ent_pure_pre_iff
thf(fact_1925_mod__pure__star__dist,axiom,
! [P2: assn,B2: $o,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P2 @ ( pure_assn @ B2 ) ) @ H2 )
= ( ( rep_assn @ P2 @ H2 )
& B2 ) ) ).
% mod_pure_star_dist
thf(fact_1926_ent__pure__pre__iff__sng,axiom,
! [B2: $o,Q2: assn] :
( ( entails @ ( pure_assn @ B2 ) @ Q2 )
= ( B2
=> ( entails @ ( one_one @ assn ) @ Q2 ) ) ) ).
% ent_pure_pre_iff_sng
thf(fact_1927_ent__pure__post__iff,axiom,
! [P2: assn,Q2: assn,B2: $o] :
( ( entails @ P2 @ ( times_times @ assn @ Q2 @ ( pure_assn @ B2 ) ) )
= ( ! [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P2 @ H4 )
=> B2 )
& ( entails @ P2 @ Q2 ) ) ) ).
% ent_pure_post_iff
thf(fact_1928_mod__h__bot__iff_I1_J,axiom,
! [B2: $o,H2: heap_ext @ product_unit] :
( ( rep_assn @ ( pure_assn @ B2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= B2 ) ).
% mod_h_bot_iff(1)
thf(fact_1929_ent__pure__post__iff__sng,axiom,
! [P2: assn,B2: $o] :
( ( entails @ P2 @ ( pure_assn @ B2 ) )
= ( ! [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P2 @ H4 )
=> B2 )
& ( entails @ P2 @ ( one_one @ assn ) ) ) ) ).
% ent_pure_post_iff_sng
thf(fact_1930_distinct__swap,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) )
= ( distinct @ A @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_1931_VEBT__internal_OperInsTrans_Ointros_I2_J,axiom,
! [T2: vEBT_VEBT,S: vEBT_VEBT,X: nat,U: vEBT_VEBT] :
( ( T2
= ( vEBT_vebt_insert @ S @ X ) )
=> ( ( vEBT_V6289311342943941716sTrans @ T2 @ U )
=> ( vEBT_V6289311342943941716sTrans @ S @ U ) ) ) ).
% VEBT_internal.perInsTrans.intros(2)
thf(fact_1932_VEBT__internal_OperInsTrans_Ocases,axiom,
! [A12: vEBT_VEBT,A23: vEBT_VEBT] :
( ( vEBT_V6289311342943941716sTrans @ A12 @ A23 )
=> ( ( A23 != A12 )
=> ~ ! [T7: vEBT_VEBT] :
( ? [X3: nat] :
( T7
= ( vEBT_vebt_insert @ A12 @ X3 ) )
=> ~ ( vEBT_V6289311342943941716sTrans @ T7 @ A23 ) ) ) ) ).
% VEBT_internal.perInsTrans.cases
thf(fact_1933_VEBT__internal_OperInsTrans_Osimps,axiom,
( vEBT_V6289311342943941716sTrans
= ( ^ [A13: vEBT_VEBT,A24: vEBT_VEBT] :
( ? [T3: vEBT_VEBT] :
( ( A13 = T3 )
& ( A24 = T3 ) )
| ? [T3: vEBT_VEBT,S8: vEBT_VEBT,X4: nat,U3: vEBT_VEBT] :
( ( A13 = S8 )
& ( A24 = U3 )
& ( T3
= ( vEBT_vebt_insert @ S8 @ X4 ) )
& ( vEBT_V6289311342943941716sTrans @ T3 @ U3 ) ) ) ) ) ).
% VEBT_internal.perInsTrans.simps
thf(fact_1934_set__take__subset,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ N @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_take_subset
thf(fact_1935_sorted__take,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( take @ A @ N @ Xs ) ) ) ) ).
% sorted_take
thf(fact_1936_finite__distinct__list,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [Xs3: list @ A] :
( ( ( set2 @ A @ Xs3 )
= A5 )
& ( distinct @ A @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_1937_finite__set__image,axiom,
! [A: $tType,A5: set @ ( list @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A5 ) )
=> ( ! [Xs3: list @ A] :
( ( member @ ( list @ A ) @ Xs3 @ A5 )
=> ( distinct @ A @ Xs3 ) )
=> ( finite_finite2 @ ( list @ A ) @ A5 ) ) ) ).
% finite_set_image
thf(fact_1938_set__take__subset__set__take,axiom,
! [A: $tType,M: nat,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ M @ Xs ) ) @ ( set2 @ A @ ( take @ A @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1939_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_1940_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs2: list @ A] :
! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( I3 != J3 )
=> ( ( nth @ A @ Xs2 @ I3 )
!= ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_1941_nth__eq__iff__index__eq,axiom,
! [A: $tType,Xs: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I )
= ( nth @ A @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_1942_distinct__length__le,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( distinct @ A @ Ys )
=> ( ( ( set2 @ A @ Ys )
= ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% distinct_length_le
thf(fact_1943_sorted__distinct__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( distinct @ A @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
=> ( ( distinct @ A @ Ys )
=> ( ( ( set2 @ A @ Xs )
= ( set2 @ A @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_1944_VEBT__internal_Oinsert_H_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_VEBT_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X ) ) ).
% VEBT_internal.insert'.simps(1)
thf(fact_1945_vebt__insert__code_I1_J,axiom,
! [X: nat,A2: $o,B2: $o] :
( ( ( X
= ( zero_zero @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_Leaf @ $true @ B2 ) ) )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_Leaf @ A2 @ $true ) ) )
& ( ( X
!= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_Leaf @ A2 @ B2 ) ) ) ) ) ) ).
% vebt_insert_code(1)
thf(fact_1946_nth__take__lemma,axiom,
! [A: $tType,K: nat,Xs: list @ A,Ys: list @ A] :
( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Ys @ I2 ) ) )
=> ( ( take @ A @ K @ Xs )
= ( take @ A @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_1947_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [X3: list @ A] :
( ( ( set2 @ A @ X3 )
= A5 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ X3 )
& ( distinct @ A @ X3 )
& ! [Y4: list @ A] :
( ( ( ( set2 @ A @ Y4 )
= A5 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Y4 )
& ( distinct @ A @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_1948_distinct__Ex1,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [X3: nat] :
( ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ X3 )
= X )
& ! [Y4: nat] :
( ( ( ord_less @ nat @ Y4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ Y4 )
= X ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_1949_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( distinct @ A @ Xs )
=> ( ( linord4507533701916653071of_set @ A @ ( set2 @ A @ Xs ) )
= Xs ) ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_1950_distinct__sorted__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( distinct @ A @ L )
=> ( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ord_less @ A @ ( nth @ A @ L @ I ) @ ( nth @ A @ L @ J ) ) ) ) ) ) ) ).
% distinct_sorted_mono
thf(fact_1951_distinct__sorted__strict__mono__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,I: nat,J: nat] :
( ( distinct @ A @ L )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less @ A @ ( nth @ A @ L @ I ) @ ( nth @ A @ L @ J ) )
= ( ord_less @ nat @ I @ J ) ) ) ) ) ) ) ).
% distinct_sorted_strict_mono_iff
thf(fact_1952_distinct__sorted__mono__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,I: nat,J: nat] :
( ( distinct @ A @ L )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ I ) @ ( nth @ A @ L @ J ) )
= ( ord_less_eq @ nat @ I @ J ) ) ) ) ) ) ) ).
% distinct_sorted_mono_iff
thf(fact_1953_distinct__list__update,axiom,
! [A: $tType,Xs: list @ A,A2: A,I: nat] :
( ( distinct @ A @ Xs )
=> ( ~ ( member @ A @ A2 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ I ) @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( distinct @ A @ ( list_update @ A @ Xs @ I @ A2 ) ) ) ) ).
% distinct_list_update
thf(fact_1954_vebt__insert_Osimps_I4_J,axiom,
! [V3: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ X ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList @ Summary ) ) ).
% vebt_insert.simps(4)
thf(fact_1955_heaphelp,axiom,
! [A: $tType,Xa2: array @ vEBT_VEBTi,Tree_is: list @ vEBT_VEBTi,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Xb: vEBT_VEBTi,N: nat,Xc: vEBT_VEBTi,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn
@ ( times_times @ assn
@ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ Xa2 @ Tree_is ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ Xb ) )
@ ( pure_assn
@ ( ( ( none @ A )
= ( none @ A ) )
& ( N = N ) ) ) )
@ ( pure_assn
@ ( Xc
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ N @ Xa2 @ Xb ) ) ) )
@ H2 )
=> ( rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ N @ TreeList @ Summary ) @ Xc ) @ H2 ) ) ).
% heaphelp
thf(fact_1956_vebt__insert_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) @ X )
= ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) ) ).
% vebt_insert.simps(3)
thf(fact_1957_vebt__insert_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts2 @ S ) @ X )
= ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts2 @ S ) ) ).
% vebt_insert.simps(2)
thf(fact_1958_mergesort__remdups__correct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( distinct @ A @ ( mergesort_remdups @ A @ L ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( mergesort_remdups @ A @ L ) )
& ( ( set2 @ A @ ( mergesort_remdups @ A @ L ) )
= ( set2 @ A @ L ) ) ) ) ).
% mergesort_remdups_correct
thf(fact_1959_subset__mset_Oelem__le__sum__list,axiom,
! [A: $tType,K: nat,Ns: list @ ( multiset @ A )] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ ( multiset @ A ) ) @ Ns ) )
=> ( subseteq_mset @ A @ ( nth @ ( multiset @ A ) @ Ns @ K ) @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Ns ) ) ) ).
% subset_mset.elem_le_sum_list
thf(fact_1960_last__take__nth__conv,axiom,
! [A: $tType,N: nat,L: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( take @ A @ N @ L ) )
= ( nth @ A @ L @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% last_take_nth_conv
thf(fact_1961_VEBTi_Oinject_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: array @ vEBT_VEBTi,X14: vEBT_VEBTi,Y11: option @ ( product_prod @ nat @ nat ),Y12: nat,Y13: array @ vEBT_VEBTi,Y14: vEBT_VEBTi] :
( ( ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 )
= ( vEBT_Nodei @ Y11 @ Y12 @ Y13 @ Y14 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 )
& ( X13 = Y13 )
& ( X14 = Y14 ) ) ) ).
% VEBTi.inject(1)
thf(fact_1962_mset__le__incr__right2,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C3: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ( subseteq_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ C3 @ B2 ) ) ) ).
% mset_le_incr_right2
thf(fact_1963_mset__le__incr__right1,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C3: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ( subseteq_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ B2 @ C3 ) ) ) ).
% mset_le_incr_right1
thf(fact_1964_mset__le__decr__left2,axiom,
! [A: $tType,C3: multiset @ A,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C3 @ A2 ) @ B2 )
=> ( subseteq_mset @ A @ A2 @ B2 ) ) ).
% mset_le_decr_left2
thf(fact_1965_mset__le__decr__left1,axiom,
! [A: $tType,A2: multiset @ A,C3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C3 ) @ B2 )
=> ( subseteq_mset @ A @ A2 @ B2 ) ) ).
% mset_le_decr_left1
thf(fact_1966_mset__union__subset,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A,C4: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) @ C4 )
=> ( ( subseteq_mset @ A @ A5 @ C4 )
& ( subseteq_mset @ A @ B5 @ C4 ) ) ) ).
% mset_union_subset
thf(fact_1967_mset__le__distrib,axiom,
! [A: $tType,X6: multiset @ A,A5: multiset @ A,B5: multiset @ A] :
( ( subseteq_mset @ A @ X6 @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) )
=> ~ ! [Xa3: multiset @ A,Xb2: multiset @ A] :
( ( X6
= ( plus_plus @ ( multiset @ A ) @ Xa3 @ Xb2 ) )
=> ( ( subseteq_mset @ A @ Xa3 @ A5 )
=> ~ ( subseteq_mset @ A @ Xb2 @ B5 ) ) ) ) ).
% mset_le_distrib
thf(fact_1968_mset__le__addE,axiom,
! [A: $tType,Xs: multiset @ A,Ys: multiset @ A] :
( ( subseteq_mset @ A @ Xs @ Ys )
=> ~ ! [Zs2: multiset @ A] :
( Ys
!= ( plus_plus @ ( multiset @ A ) @ Xs @ Zs2 ) ) ) ).
% mset_le_addE
thf(fact_1969_mset__le__subtract,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A,C4: multiset @ A] :
( ( subseteq_mset @ A @ A5 @ B5 )
=> ( subseteq_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A5 @ C4 ) @ ( minus_minus @ ( multiset @ A ) @ B5 @ C4 ) ) ) ).
% mset_le_subtract
thf(fact_1970_subset__mset_Olift__Suc__mono__le,axiom,
! [A: $tType,F3: nat > ( multiset @ A ),N: nat,N3: nat] :
( ! [N2: nat] : ( subseteq_mset @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N3 )
=> ( subseteq_mset @ A @ ( F3 @ N ) @ ( F3 @ N3 ) ) ) ) ).
% subset_mset.lift_Suc_mono_le
thf(fact_1971_subset__mset_Olift__Suc__antimono__le,axiom,
! [A: $tType,F3: nat > ( multiset @ A ),N: nat,N3: nat] :
( ! [N2: nat] : ( subseteq_mset @ A @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
=> ( ( ord_less_eq @ nat @ N @ N3 )
=> ( subseteq_mset @ A @ ( F3 @ N3 ) @ ( F3 @ N ) ) ) ) ).
% subset_mset.lift_Suc_antimono_le
thf(fact_1972_mset__le__subtract__right,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A,X6: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) @ X6 )
=> ( ( subseteq_mset @ A @ A5 @ ( minus_minus @ ( multiset @ A ) @ X6 @ B5 ) )
& ( subseteq_mset @ A @ B5 @ X6 ) ) ) ).
% mset_le_subtract_right
thf(fact_1973_mset__le__subtract__left,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A,X6: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) @ X6 )
=> ( ( subseteq_mset @ A @ B5 @ ( minus_minus @ ( multiset @ A ) @ X6 @ A5 ) )
& ( subseteq_mset @ A @ A5 @ X6 ) ) ) ).
% mset_le_subtract_left
thf(fact_1974_size__mset__mono,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A] :
( ( subseteq_mset @ A @ A5 @ B5 )
=> ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ A5 ) @ ( size_size @ ( multiset @ A ) @ B5 ) ) ) ).
% size_mset_mono
thf(fact_1975_vebt__assn__raw_Osimps_I4_J,axiom,
! [Vd2: $o,Ve2: $o,V3: option @ ( product_prod @ nat @ nat ),Va2: nat,Vb: array @ vEBT_VEBTi,Vc: vEBT_VEBTi] :
( ( vEBT_vebt_assn_raw @ ( vEBT_Leaf @ Vd2 @ Ve2 ) @ ( vEBT_Nodei @ V3 @ Va2 @ Vb @ Vc ) )
= ( bot_bot @ assn ) ) ).
% vebt_assn_raw.simps(4)
thf(fact_1976_subset__mset_Osum__mono2,axiom,
! [A: $tType,B: $tType,B5: set @ B,A5: set @ B,F3: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B5 )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ B5 @ A5 ) )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F3 @ B4 ) ) )
=> ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F3 @ A5 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F3 @ B5 ) ) ) ) ) ).
% subset_mset.sum_mono2
thf(fact_1977_subset__subseqs,axiom,
! [A: $tType,X6: set @ A,Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ X6 @ ( set2 @ A @ Xs ) )
=> ( member @ ( set @ A ) @ X6 @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_1978_set__remove1__eq,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( set2 @ A @ ( remove1 @ A @ X @ Xs ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_remove1_eq
thf(fact_1979_distinct__foldl__invar,axiom,
! [B: $tType,A: $tType,S3: list @ A,I6: ( set @ A ) > B > $o,Sigma_0: B,F3: B > A > B] :
( ( distinct @ A @ S3 )
=> ( ( I6 @ ( set2 @ A @ S3 ) @ Sigma_0 )
=> ( ! [X3: A,It: set @ A,Sigma: B] :
( ( member @ A @ X3 @ It )
=> ( ( ord_less_eq @ ( set @ A ) @ It @ ( set2 @ A @ S3 ) )
=> ( ( I6 @ It @ Sigma )
=> ( I6 @ ( minus_minus @ ( set @ A ) @ It @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( F3 @ Sigma @ X3 ) ) ) ) )
=> ( I6 @ ( bot_bot @ ( set @ A ) ) @ ( foldl @ B @ A @ F3 @ Sigma_0 @ S3 ) ) ) ) ) ).
% distinct_foldl_invar
thf(fact_1980_lex__take__index,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R2 ) )
=> ~ ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( take @ A @ I2 @ Xs )
= ( take @ A @ I2 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Ys @ I2 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_1981_take__butlast__conv,axiom,
! [A: $tType,L: list @ A] :
( ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L )
= ( butlast @ A @ L ) ) ).
% take_butlast_conv
thf(fact_1982_take__minus__one__conv__butlast,axiom,
! [A: $tType,N: nat,L: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( take @ A @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ L )
= ( butlast @ A @ ( take @ A @ N @ L ) ) ) ) ).
% take_minus_one_conv_butlast
thf(fact_1983_subset__mset_Osum__eq__0__iff,axiom,
! [A: $tType,B: $tType,F5: set @ B,F3: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ F5 )
=> ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F3 @ F5 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ F5 )
=> ( ( F3 @ X4 )
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% subset_mset.sum_eq_0_iff
thf(fact_1984_Cons__in__lex,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( lex @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
& ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) )
| ( ( X = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_1985_foldl__A1__eq,axiom,
! [A: $tType,F3: A > A > A,N: A,I: A,Ww: list @ A] :
( ! [E2: A] :
( ( F3 @ N @ E2 )
= E2 )
=> ( ! [E2: A] :
( ( F3 @ E2 @ N )
= E2 )
=> ( ! [A4: A,B4: A,C2: A] :
( ( F3 @ A4 @ ( F3 @ B4 @ C2 ) )
= ( F3 @ ( F3 @ A4 @ B4 ) @ C2 ) )
=> ( ( foldl @ A @ A @ F3 @ I @ Ww )
= ( F3 @ I @ ( foldl @ A @ A @ F3 @ N @ Ww ) ) ) ) ) ) ).
% foldl_A1_eq
thf(fact_1986_foldl__foldl__conv__concat,axiom,
! [A: $tType,B: $tType,F3: A > B > A,A2: A,Xs: list @ ( list @ B )] :
( ( foldl @ A @ ( list @ B ) @ ( foldl @ A @ B @ F3 ) @ A2 @ Xs )
= ( foldl @ A @ B @ F3 @ A2 @ ( concat @ B @ Xs ) ) ) ).
% foldl_foldl_conv_concat
thf(fact_1987_butlast__update_H,axiom,
! [A: $tType,L: list @ A,I: nat,X: A] :
( ( list_update @ A @ ( butlast @ A @ L ) @ I @ X )
= ( butlast @ A @ ( list_update @ A @ L @ I @ X ) ) ) ).
% butlast_update'
thf(fact_1988_foldl__absorb1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Zs: list @ A] :
( ( times_times @ A @ X @ ( foldl @ A @ A @ ( times_times @ A ) @ ( one_one @ A ) @ Zs ) )
= ( foldl @ A @ A @ ( times_times @ A ) @ X @ Zs ) ) ) ).
% foldl_absorb1
thf(fact_1989_set__remove1__subset,axiom,
! [A: $tType,X: A,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( remove1 @ A @ X @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_remove1_subset
thf(fact_1990_sorted__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remove1 @ A @ A2 @ Xs ) ) ) ) ).
% sorted_remove1
thf(fact_1991_nth__butlast,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) ) )
=> ( ( nth @ A @ ( butlast @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_1992_take__butlast,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( take @ A @ N @ ( butlast @ A @ Xs ) )
= ( take @ A @ N @ Xs ) ) ) ).
% take_butlast
thf(fact_1993_distinct__butlast__swap,axiom,
! [A: $tType,Pq: list @ A,I: nat] :
( ( distinct @ A @ Pq )
=> ( distinct @ A @ ( butlast @ A @ ( list_update @ A @ Pq @ I @ ( last @ A @ Pq ) ) ) ) ) ).
% distinct_butlast_swap
thf(fact_1994_butlast__take,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( butlast @ A @ ( take @ A @ N @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% butlast_take
thf(fact_1995_subset__mset_Osum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType,S: set @ B,T2: set @ C,G3: C > ( multiset @ A ),I: C > B,F3: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ T2 )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( G3 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ? [Xa: C] :
( ( member @ C @ Xa @ T2 )
& ( ( I @ Xa )
= X3 )
& ( subseteq_mset @ A @ ( F3 @ X3 ) @ ( G3 @ Xa ) ) ) )
=> ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F3 @ S ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ C @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G3 @ T2 ) ) ) ) ) ) ).
% subset_mset.sum_le_included
thf(fact_1996_subset__mset_Osum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType,A5: set @ B,F3: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F3 @ X3 ) ) )
=> ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F3 @ A5 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( F3 @ X4 )
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ).
% subset_mset.sum_nonneg_eq_0_iff
thf(fact_1997_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( remove1 @ A @ X @ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_1998_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_1999_listrel1__iff__update,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
= ( ? [Y5: A,N6: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N6 ) @ Y5 ) @ R2 )
& ( ord_less @ nat @ N6 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( Ys
= ( list_update @ A @ Xs @ N6 @ Y5 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_2000_total__on__singleton,axiom,
! [A: $tType,X: A] : ( total_on @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% total_on_singleton
thf(fact_2001_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_2002_rule__at__index,axiom,
! [A: $tType,B: $tType,C: $tType,P2: assn,A5: A > B > assn,Xs: list @ A,Xsi: list @ B,F5: assn,I: nat,C3: heap_Time_Heap @ C,Q4: C > assn,F7: C > assn] :
( ( entails @ P2 @ ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ A5 @ Xs @ Xsi ) @ F5 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( hoare_hoare_triple @ C @ ( times_times @ assn @ ( times_times @ assn @ ( A5 @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A5 @ Xs @ Xsi ) ) @ F5 ) @ C3 @ Q4 )
=> ( ! [R: C] : ( entails @ ( Q4 @ R ) @ ( times_times @ assn @ ( times_times @ assn @ ( A5 @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A5 @ Xs @ Xsi ) ) @ ( F7 @ R ) ) )
=> ( hoare_hoare_triple @ C @ P2 @ C3
@ ^ [R5: C] : ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ A5 @ Xs @ Xsi ) @ ( F7 @ R5 ) ) ) ) ) ) ) ).
% rule_at_index
thf(fact_2003_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_2004_Bernoulli__inequality,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_2005_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% set_vebt'_def
thf(fact_2006_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_2007_image__ident,axiom,
! [A: $tType,Y7: set @ A] :
( ( image @ A @ A
@ ^ [X4: A] : X4
@ Y7 )
= Y7 ) ).
% image_ident
thf(fact_2008_finite__Collect__disjI,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_2009_finite__Collect__conjI,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
| ( finite_finite2 @ A @ ( collect @ A @ Q2 ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_2010_succ__empty,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X )
= ( none @ nat ) )
= ( ( collect @ nat
@ ^ [Y5: nat] :
( ( vEBT_vebt_member @ T2 @ Y5 )
& ( ord_less @ nat @ X @ Y5 ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% succ_empty
thf(fact_2011_pred__empty,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X )
= ( none @ nat ) )
= ( ( collect @ nat
@ ^ [Y5: nat] :
( ( vEBT_vebt_member @ T2 @ Y5 )
& ( ord_less @ nat @ Y5 @ X ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% pred_empty
thf(fact_2012_foldr__mono,axiom,
! [Xs: list @ nat,Ys: list @ nat,C3: nat,D2: nat] :
( ( ( size_size @ ( list @ nat ) @ Xs )
= ( size_size @ ( list @ nat ) @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ nat ) @ Xs ) )
=> ( ord_less @ nat @ ( nth @ nat @ Xs @ I2 ) @ ( nth @ nat @ Ys @ I2 ) ) )
=> ( ( ord_less_eq @ nat @ C3 @ D2 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Xs @ C3 ) @ ( size_size @ ( list @ nat ) @ Ys ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Ys @ D2 ) ) ) ) ) ).
% foldr_mono
thf(fact_2013_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_2014_zip__eq__zip__same__len,axiom,
! [A: $tType,B: $tType,A2: list @ A,B2: list @ B,A3: list @ A,B3: list @ B] :
( ( ( size_size @ ( list @ A ) @ A2 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B3 ) )
=> ( ( ( zip @ A @ B @ A2 @ B2 )
= ( zip @ A @ B @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ) ) ).
% zip_eq_zip_same_len
thf(fact_2015_finite__lessThan,axiom,
! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_lessThan @ nat @ K ) ) ).
% finite_lessThan
thf(fact_2016_foldr__zero,axiom,
! [Xs: list @ nat,D2: nat] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ nat ) @ Xs ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nth @ nat @ Xs @ I2 ) ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ nat ) @ Xs ) @ ( minus_minus @ nat @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Xs @ D2 ) @ D2 ) ) ) ).
% foldr_zero
thf(fact_2017_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [Uu3: B] : ( zero_zero @ A )
@ A5 )
= ( zero_zero @ A ) ) ) ).
% sum.neutral_const
thf(fact_2018_singleton__conv2,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ( ^ [Y6: A,Z4: A] : Y6 = Z4
@ A2 ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_2019_singleton__conv,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ^ [X4: A] : X4 = A2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_2020_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N6: nat] : ( ord_less @ nat @ N6 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_2021_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N6: nat] : ( ord_less_eq @ nat @ N6 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_2022_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_2023_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_2024_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_2025_semiring__1__class_Oof__nat__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% semiring_1_class.of_nat_0
thf(fact_2026_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_2027_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_2028_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_2029_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_2030_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_2031_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_2032_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_2033_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X ) @ ( set_ord_lessThan @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% lessThan_subset_iff
thf(fact_2034_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_2035_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta'
thf(fact_2036_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta
thf(fact_2037_foldl__length,axiom,
! [A: $tType,L: list @ A] :
( ( foldl @ nat @ A
@ ^ [I3: nat,X4: A] : ( suc @ I3 )
@ ( zero_zero @ nat )
@ L )
= ( size_size @ ( list @ A ) @ L ) ) ).
% foldl_length
thf(fact_2038_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less_eq @ nat @ I3 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_2039_finite__Collect__subsets,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B7: set @ A] : ( ord_less_eq @ ( set @ A ) @ B7 @ A5 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_2040_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_2041_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_2042_sum_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( G3 @ N ) ) ) ) ).
% sum.lessThan_Suc
thf(fact_2043_single__Diff__lessThan,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A] :
( ( minus_minus @ ( set @ A ) @ ( insert @ A @ K @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K ) )
= ( insert @ A @ K @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% single_Diff_lessThan
thf(fact_2044_Cons__listrel1__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_2045_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Y: B,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_2046_Max__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ B,C3: A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image @ B @ A
@ ^ [Uu3: B] : C3
@ A5 ) )
= C3 ) ) ) ) ).
% Max_const
thf(fact_2047_Min__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ B,C3: A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image @ B @ A
@ ^ [Uu3: B] : C3
@ A5 ) )
= C3 ) ) ) ) ).
% Min_const
thf(fact_2048_foldr__length,axiom,
! [A: $tType,L: list @ A] :
( ( foldr @ A @ nat
@ ^ [X4: A] : suc
@ L
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ A ) @ L ) ) ).
% foldr_length
thf(fact_2049_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_2050_of__nat__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B2: nat,W2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_2051_of__nat__less__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W2: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X ) ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_2052_of__nat__le__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W2: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X ) ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_2053_of__nat__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B2: nat,W2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_2054_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_2055_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A5: set @ nat,C3: nat > A] :
( ( ( ( finite_finite2 @ nat @ A5 )
& ( member @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C3 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) )
@ A5 )
= ( C3 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A5 )
& ( member @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C3 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) )
@ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_2056_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= N )
& ( distinct @ A @ Xs2 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A5 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_2057_vebt__memberi_H__rf__abstr,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] :
( hoare_hoare_triple @ $o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_V854960066525838166emberi @ T2 @ Ti @ X )
@ ^ [R5: $o] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_member @ T2 @ X ) ) ) ) ) ).
% vebt_memberi'_rf_abstr
thf(fact_2058_vebt__succi_H__rf__abstr,axiom,
! [T2: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_VEBT_vebt_succi @ T2 @ Ti @ X )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_succ @ T2 @ X ) ) ) ) ) ) ).
% vebt_succi'_rf_abstr
thf(fact_2059_vebt__pred_H__rf__abstr,axiom,
! [T2: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_VEBT_vebt_predi @ T2 @ Ti @ X )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_pred @ T2 @ X ) ) ) ) ) ) ).
% vebt_pred'_rf_abstr
thf(fact_2060_vebt__inserti_H__rf__abstr,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( hoare_hoare_triple @ vEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T2 @ X ) ) ) ).
% vebt_inserti'_rf_abstr
thf(fact_2061_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R3 ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ S3 ) ) )
= ( R3 = S3 ) ) ).
% pred_equals_eq2
thf(fact_2062_constraint__expand,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Lower: word @ A,Upper: word @ A] :
( ( member @ ( word @ A ) @ X
@ ( collect @ ( word @ A )
@ ^ [Y5: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Lower @ Y5 )
& ( ord_less_eq @ ( word @ A ) @ Y5 @ Upper ) ) ) )
= ( ( ord_less_eq @ ( word @ A ) @ Lower @ X )
& ( ord_less_eq @ ( word @ A ) @ X @ Upper ) ) ) ) ).
% constraint_expand
thf(fact_2063_pred__subset__eq,axiom,
! [A: $tType,R3: set @ A,S3: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R3 )
@ ^ [X4: A] : ( member @ A @ X4 @ S3 ) )
= ( ord_less_eq @ ( set @ A ) @ R3 @ S3 ) ) ).
% pred_subset_eq
thf(fact_2064_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A6 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ).
% less_eq_set_def
thf(fact_2065_Collect__subset,axiom,
! [A: $tType,A5: set @ A,P2: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P2 @ X4 ) ) )
@ A5 ) ).
% Collect_subset
thf(fact_2066_prop__restrict,axiom,
! [A: $tType,X: A,Z7: set @ A,X6: set @ A,P2: A > $o] :
( ( member @ A @ X @ Z7 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z7
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X6 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_2067_Collect__restrict,axiom,
! [A: $tType,X6: set @ A,P2: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X6 )
& ( P2 @ X4 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_2068_subset__CollectI,axiom,
! [A: $tType,B5: set @ A,A5: set @ A,Q2: A > $o,P2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ( ( Q2 @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ B5 )
& ( Q2 @ X4 ) ) )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_2069_subset__Collect__iff,axiom,
! [A: $tType,B5: set @ A,A5: set @ A,P2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B5 )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_2070_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A8: A,B7: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( X4 = A8 )
| ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_2071_insert__Collect,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( insert @ A @ A2 @ ( collect @ A @ P2 ) )
= ( collect @ A
@ ^ [U3: A] :
( ( U3 != A2 )
=> ( P2 @ U3 ) ) ) ) ).
% insert_Collect
thf(fact_2072_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A6: set @ A] :
( collect @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ A6 ) ) ) ) ).
% Compl_eq
thf(fact_2073_Collect__neg__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
~ ( P2 @ X4 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P2 ) ) ) ).
% Collect_neg_eq
thf(fact_2074_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A6: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A6 ) ) ) ) ) ).
% uminus_set_def
thf(fact_2075_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U3: A] :
( collect @ A
@ ^ [X4: A] : ( ord_less @ A @ X4 @ U3 ) ) ) ) ) ).
% lessThan_def
thf(fact_2076_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A6 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ).
% less_set_def
thf(fact_2077_Set_Oempty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $false ) ) ).
% Set.empty_def
thf(fact_2078_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A6 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% minus_set_def
thf(fact_2079_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ~ ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% set_diff_eq
thf(fact_2080_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B,R3: A > B > $o] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ? [Xa: B] :
( ( member @ B @ Xa @ B5 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ B5 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A8: A] :
( ( member @ A @ A8 @ A5 )
& ( R3 @ A8 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_2081_not__finite__existsD,axiom,
! [A: $tType,P2: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
=> ? [X_1: A] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_2082_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A,A5: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A5 ) )
=> ~ ! [X3: B] :
( ( B2
= ( F3 @ X3 ) )
=> ~ ( member @ B @ X3 @ A5 ) ) ) ).
% imageE
thf(fact_2083_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,G3: C > B,A5: set @ C] :
( ( image @ B @ A @ F3 @ ( image @ C @ B @ G3 @ A5 ) )
= ( image @ C @ A
@ ^ [X4: C] : ( F3 @ ( G3 @ X4 ) )
@ A5 ) ) ).
% image_image
thf(fact_2084_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P2: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F3 @ A5 ) )
& ( P2 @ X4 ) ) )
= ( image @ B @ A @ F3
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( P2 @ ( F3 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_2085_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H4: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_2086_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G3 @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_2087_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K4: set @ B,F3: B > A,G3: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ K4 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( G3 @ I2 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ K4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ K4 ) ) ) ) ).
% sum_mono
thf(fact_2088_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ ( image @ A @ B @ F3 @ A5 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A8: A] :
( ( member @ A @ A8 @ A5 )
& ( ( F3 @ A8 )
= ( F3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_2089_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P2: A > $o,F3: A > B,B5: set @ B] :
( ! [X3: A] :
( ( P2 @ X3 )
=> ( member @ B @ ( F3 @ X3 ) @ B5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ ( collect @ A @ P2 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_2090_Collect__conv__if2,axiom,
! [A: $tType,P2: A > $o,A2: A] :
( ( ( P2 @ A2 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A2 = X4 )
& ( P2 @ X4 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A2 = X4 )
& ( P2 @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_2091_Collect__conv__if,axiom,
! [A: $tType,P2: A > $o,A2: A] :
( ( ( P2 @ A2 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A2 )
& ( P2 @ X4 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A2 )
& ( P2 @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_2092_sum_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B5: set @ C,G3: B > C > A,R3: B > C > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ C @ B5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] :
( groups7311177749621191930dd_sum @ C @ A @ ( G3 @ X4 )
@ ( collect @ C
@ ^ [Y5: C] :
( ( member @ C @ Y5 @ B5 )
& ( R3 @ X4 @ Y5 ) ) ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y5: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( G3 @ X4 @ Y5 )
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( R3 @ X4 @ Y5 ) ) ) )
@ B5 ) ) ) ) ) ).
% sum.swap_restrict
thf(fact_2093_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q2: A > nat,P2: A > nat,N: A] :
( ! [X3: A] : ( ord_less_eq @ nat @ ( Q2 @ X3 ) @ ( P2 @ X3 ) )
=> ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P2 @ ( set_ord_lessThan @ A @ N ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q2 @ ( set_ord_lessThan @ A @ N ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( minus_minus @ nat @ ( P2 @ X4 ) @ ( Q2 @ X4 ) )
@ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_2094_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A8: A,B8: A] : ( if @ A @ ( ord_less_eq @ A @ A8 @ B8 ) @ B8 @ A8 ) ) ) ) ).
% max_def_raw
thf(fact_2095_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R3 )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ S3 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S3 ) ) ).
% pred_subset_eq2
thf(fact_2096_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_2097_finite__M__bounded__by__nat,axiom,
! [P2: nat > $o,I: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( P2 @ K3 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_2098_finite__less__ub,axiom,
! [F3: nat > nat,U: nat] :
( ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( F3 @ N2 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N6: nat] : ( ord_less_eq @ nat @ ( F3 @ N6 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_2099_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).
% set_vebt_def
thf(fact_2100_zip__inj,axiom,
! [A: $tType,B: $tType,A2: list @ A,B2: list @ B,A3: list @ A,B3: list @ B] :
( ( ( size_size @ ( list @ A ) @ A2 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B3 ) )
=> ( ( ( zip @ A @ B @ A2 @ B2 )
= ( zip @ A @ B @ A3 @ B3 ) )
=> ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ) ) ).
% zip_inj
thf(fact_2101_pair__list__split,axiom,
! [A: $tType,B: $tType,L: list @ ( product_prod @ A @ B )] :
~ ! [L1: list @ A,L22: list @ B] :
( ( L
= ( zip @ A @ B @ L1 @ L22 ) )
=> ( ( ( size_size @ ( list @ A ) @ L1 )
= ( size_size @ ( list @ B ) @ L22 ) )
=> ( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ L )
!= ( size_size @ ( list @ B ) @ L22 ) ) ) ) ).
% pair_list_split
thf(fact_2102_real__arch__simple,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N2: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% real_arch_simple
thf(fact_2103_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N2: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% reals_Archimedean2
thf(fact_2104_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: nat,Y: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X ) @ Y )
= ( times_times @ A @ Y @ ( semiring_1_of_nat @ A @ X ) ) ) ) ).
% mult_of_nat_commute
thf(fact_2105_lessThan__non__empty,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
( ( set_ord_lessThan @ A @ X )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% lessThan_non_empty
thf(fact_2106_infinite__Iio,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [A2: A] :
~ ( finite_finite2 @ A @ ( set_ord_lessThan @ A @ A2 ) ) ) ).
% infinite_Iio
thf(fact_2107_sum_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G3 @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_2108_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N6: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N6 ) ) @ ( F3 @ N6 ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F3 @ M ) @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_2109_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N6: nat] : ( minus_minus @ A @ ( F3 @ N6 ) @ ( F3 @ ( suc @ N6 ) ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ ( F3 @ M ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_2110_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I6 )
& ( ( X @ I3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I6 )
& ( ( Y @ I3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I6 )
& ( ( plus_plus @ A @ ( X @ I3 ) @ ( Y @ I3 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_2111_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I6 )
& ( ( X @ I3 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I6 )
& ( ( Y @ I3 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I6 )
& ( ( times_times @ A @ ( X @ I3 ) @ ( Y @ I3 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_2112_filter__preserves__multiset,axiom,
! [A: $tType,M7: A > nat,P2: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M7 @ X4 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( P2 @ X4 ) @ ( M7 @ X4 ) @ ( zero_zero @ nat ) ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_2113_sum_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A,P2: B > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( P2 @ X4 ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P2 @ X4 ) @ ( G3 @ X4 ) @ ( zero_zero @ A ) )
@ A5 ) ) ) ) ).
% sum.inter_filter
thf(fact_2114_image__constant,axiom,
! [A: $tType,B: $tType,X: A,A5: set @ A,C3: B] :
( ( member @ A @ X @ A5 )
=> ( ( image @ A @ B
@ ^ [X4: A] : C3
@ A5 )
= ( insert @ B @ C3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_2115_image__constant__conv,axiom,
! [B: $tType,A: $tType,A5: set @ B,C3: A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X4: B] : C3
@ A5 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X4: B] : C3
@ A5 )
= ( insert @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_2116_foldl__length__aux,axiom,
! [A: $tType,A2: nat,L: list @ A] :
( ( foldl @ nat @ A
@ ^ [I3: nat,X4: A] : ( suc @ I3 )
@ A2
@ L )
= ( plus_plus @ nat @ A2 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% foldl_length_aux
thf(fact_2117_sum_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,H2: B > A,G3: B > C] :
( ( finite_finite2 @ B @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y5: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S3 )
& ( ( G3 @ X4 )
= Y5 ) ) ) )
@ ( image @ B @ C @ G3 @ S3 ) ) ) ) ) ).
% sum.image_gen
thf(fact_2118_nat__seg__image__imp__finite,axiom,
! [A: $tType,A5: set @ A,F3: nat > A,N: nat] :
( ( A5
= ( image @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N ) ) ) )
=> ( finite_finite2 @ A @ A5 ) ) ).
% nat_seg_image_imp_finite
thf(fact_2119_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A6: set @ A] :
? [N6: nat,F4: nat > A] :
( A6
= ( image @ nat @ A @ F4
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N6 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_2120_finite__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less_eq @ A @ X4 @ B2 ) ) ) ) ) ).
% finite_int_segment
thf(fact_2121_sum_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G3 @ ( suc @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_2122_listrel1__mono,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R2 ) @ ( listrel1 @ A @ S ) ) ) ).
% listrel1_mono
thf(fact_2123_sum__subtractf__nat,axiom,
! [A: $tType,A5: set @ A,G3: A > nat,F3: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ nat @ ( G3 @ X3 ) @ ( F3 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( minus_minus @ nat @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ A5 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G3 @ A5 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_2124_sum__multicount__gen,axiom,
! [A: $tType,B: $tType,S: set @ A,T2: set @ B,R3: A > B > $o,K: B > nat] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ T2 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ S )
& ( R3 @ I3 @ X3 ) ) ) )
= ( K @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T2 )
& ( R3 @ I3 @ J3 ) ) ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ nat @ K @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_2125_total__onI,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( X3 != Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R2 ) ) ) ) )
=> ( total_on @ A @ A5 @ R2 ) ) ).
% total_onI
thf(fact_2126_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A6: set @ A,R5: set @ ( product_prod @ A @ A )] :
! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ A6 )
=> ( ( X4 != Y5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_2127_finite__if__eq__beyond__finite,axiom,
! [A: $tType,S3: set @ A,S2: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [S8: set @ A] :
( ( minus_minus @ ( set @ A ) @ S8 @ S3 )
= ( minus_minus @ ( set @ A ) @ S2 @ S3 ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_2128_total__on__empty,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R2 ) ).
% total_on_empty
thf(fact_2129_Fpow__def,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A6: set @ A] :
( collect @ ( set @ A )
@ ^ [X8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ X8 @ A6 )
& ( finite_finite2 @ A @ X8 ) ) ) ) ) ).
% Fpow_def
thf(fact_2130_distinct__finite__set,axiom,
! [A: $tType,X: set @ A] :
( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ( set2 @ A @ Ys3 )
= X )
& ( distinct @ A @ Ys3 ) ) ) ) ).
% distinct_finite_set
thf(fact_2131_set__to__map__empty__iff_I2_J,axiom,
! [B: $tType,A: $tType,S3: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A] : ( none @ B ) )
= ( set_to_map @ A @ B @ S3 ) )
= ( S3
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% set_to_map_empty_iff(2)
thf(fact_2132_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N ) ) @ ( power_power @ A @ Y @ ( suc @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P7: nat] : ( times_times @ A @ ( power_power @ A @ X @ P7 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ P7 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_2133_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_2134_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F3: B > A,B5: A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S )
= B5 )
=> ( ( member @ B @ I @ S )
=> ( ord_less_eq @ A @ ( F3 @ I ) @ B5 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_2135_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F3: B > A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I @ S )
=> ( ( F3 @ I )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_2136_add__mset__in__multiset,axiom,
! [A: $tType,M7: A > nat,A2: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M7 @ X4 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( X4 = A2 ) @ ( suc @ ( M7 @ X4 ) ) @ ( M7 @ X4 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_2137_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S3: set @ A,R3: set @ B,G3: A > B,F3: B > C] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ R3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ G3 @ S3 ) @ R3 )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X4: A] : ( F3 @ ( G3 @ X4 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y5: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S3 )
& ( ( G3 @ X4 )
= Y5 ) ) ) ) )
@ ( F3 @ Y5 ) )
@ R3 ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_2138_diff__preserves__multiset,axiom,
! [A: $tType,M7: A > nat,N8: A > nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M7 @ X4 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( M7 @ X4 ) @ ( N8 @ X4 ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_2139_sum_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T4: set @ C,G3: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G3 @ S3 ) @ T4 )
=> ( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y5: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S3 )
& ( ( G3 @ X4 )
= Y5 ) ) ) )
@ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% sum.group
thf(fact_2140_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3
@ ( minus_minus @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [X4: B] :
( ( G3 @ X4 )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_2141_sum__Suc,axiom,
! [A: $tType,F3: A > nat,A5: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( suc @ ( F3 @ X4 ) )
@ A5 )
= ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) @ ( finite_card @ A @ A5 ) ) ) ).
% sum_Suc
thf(fact_2142_sum__multicount,axiom,
! [A: $tType,B: $tType,S3: set @ A,T4: set @ B,R3: A > B > $o,K: nat] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ T4 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ S3 )
& ( R3 @ I3 @ X3 ) ) ) )
= K ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T4 )
& ( R3 @ I3 @ J3 ) ) ) )
@ S3 )
= ( times_times @ nat @ K @ ( finite_card @ B @ T4 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_2143_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_2144_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_2145_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_2146_foldr__length__aux,axiom,
! [A: $tType,L: list @ A,A2: nat] :
( ( foldr @ A @ nat
@ ^ [X4: A] : suc
@ L
@ A2 )
= ( plus_plus @ nat @ A2 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% foldr_length_aux
thf(fact_2147_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ( ordering_top @ A
@ ^ [X4: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X4 )
@ ^ [X4: A,Y5: A] : ( ord_less @ A @ Y5 @ X4 )
@ ( bot_bot @ A ) ) ) ).
% bot.ordering_top_axioms
thf(fact_2148_card__sum__le__nat__sum,axiom,
! [S3: set @ nat] :
( ord_less_eq @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S3 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ S3 ) ) ).
% card_sum_le_nat_sum
thf(fact_2149_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( member @ B @ Y @ ( set2 @ B @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_2150_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_2151_in__set__zipE,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ~ ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( member @ B @ Y @ ( set2 @ B @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_2152_zip__same,axiom,
! [A: $tType,A2: A,B2: A,Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs @ Xs ) ) )
= ( ( member @ A @ A2 @ ( set2 @ A @ Xs ) )
& ( A2 = B2 ) ) ) ).
% zip_same
thf(fact_2153_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_2154_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_2155_zip__update,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I: nat,X: A,Ys: list @ B,Y: B] :
( ( zip @ A @ B @ ( list_update @ A @ Xs @ I @ X ) @ ( list_update @ B @ Ys @ I @ Y ) )
= ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_update
thf(fact_2156_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_2157_semiring__char__0__class_Oof__nat__neq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ N ) )
!= ( zero_zero @ A ) ) ) ).
% semiring_char_0_class.of_nat_neq_0
thf(fact_2158_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_2159_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_2160_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_2161_Iio__eq__empty__iff,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( order_bot @ A ) )
=> ! [N: A] :
( ( ( set_ord_lessThan @ A @ N )
= ( bot_bot @ ( set @ A ) ) )
= ( N
= ( bot_bot @ A ) ) ) ) ).
% Iio_eq_empty_iff
thf(fact_2162_lessThan__atLeast0,axiom,
( ( set_ord_lessThan @ nat )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% lessThan_atLeast0
thf(fact_2163_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_2164_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan @ nat @ N )
= ( bot_bot @ ( set @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_2165_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( insert @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_2166_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ I3 ) ) @ ( F3 @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F3 @ N ) @ ( F3 @ M ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_2167_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S3: set @ B,F3: B > A,K: A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F3 @ X4 ) @ K )
@ S3 ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image @ B @ A @ F3 @ S3 ) ) @ K ) ) ) ) ) ).
% Max_add_commute
thf(fact_2168_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S3: set @ B,F3: B > A,K: A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F3 @ X4 ) @ K )
@ S3 ) )
= ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image @ B @ A @ F3 @ S3 ) ) @ K ) ) ) ) ) ).
% Min_add_commute
thf(fact_2169_sum_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_2170_of__nat__max,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X @ Y ) )
= ( ord_max @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_max
thf(fact_2171_word__coorder_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ordering_top @ ( word @ A )
@ ^ [X4: word @ A,Y5: word @ A] : ( ord_less_eq @ ( word @ A ) @ Y5 @ X4 )
@ ^ [X4: word @ A,Y5: word @ A] : ( ord_less @ ( word @ A ) @ Y5 @ X4 )
@ ( zero_zero @ ( word @ A ) ) ) ) ).
% word_coorder.ordering_top_axioms
thf(fact_2172_finite__lists__length__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_2173_subset__mset_Osum__nonneg__0,axiom,
! [B: $tType,A: $tType,S: set @ B,F3: B > ( multiset @ A ),I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F3 @ I2 ) ) )
=> ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F3 @ S )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( member @ B @ I @ S )
=> ( ( F3 @ I )
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% subset_mset.sum_nonneg_0
thf(fact_2174_subset__mset_Osum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType,S: set @ B,F3: B > ( multiset @ A ),B5: multiset @ A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F3 @ I2 ) ) )
=> ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F3 @ S )
= B5 )
=> ( ( member @ B @ I @ S )
=> ( subseteq_mset @ A @ ( F3 @ I ) @ B5 ) ) ) ) ) ).
% subset_mset.sum_nonneg_leq_bound
thf(fact_2175_distinct__finite__subset,axiom,
! [A: $tType,X: set @ A] :
( ( finite_finite2 @ A @ X )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ X )
& ( distinct @ A @ Ys3 ) ) ) ) ) ).
% distinct_finite_subset
thf(fact_2176_finite__nat__bounded,axiom,
! [S3: set @ nat] :
( ( finite_finite2 @ nat @ S3 )
=> ? [K2: nat] : ( ord_less_eq @ ( set @ nat ) @ S3 @ ( set_ord_lessThan @ nat @ K2 ) ) ) ).
% finite_nat_bounded
thf(fact_2177_finite__nat__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [S6: set @ nat] :
? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S6 @ ( set_ord_lessThan @ nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_2178_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A2: B,B2: B > A,C3: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C3 @ K3 ) )
@ S3 )
= ( plus_plus @ A @ ( B2 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C3 @ K3 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_2179_card__lists__length__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A5 ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_2180_finite__lists__length__le,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A5 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_2181_in__set__impl__in__set__zip2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Y: B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ B @ Y @ ( set2 @ B @ Ys ) )
=> ~ ! [X3: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_2182_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ! [Y3: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_2183_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ X ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_2184_of__nat__gt__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [K: nat] :
( ( ( semiring_1_of_nat @ A @ K )
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) ) ) ).
% of_nat_gt_0
thf(fact_2185_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_2186_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Xy: product_prod @ A @ B,Xys: list @ ( product_prod @ A @ B )] :
( ( ( zip @ A @ B @ Xs @ Ys )
= ( cons @ ( product_prod @ A @ B ) @ Xy @ Xys ) )
=> ~ ! [X3: A,Xs5: list @ A] :
( ( Xs
= ( cons @ A @ X3 @ Xs5 ) )
=> ! [Y3: B,Ys4: list @ B] :
( ( Ys
= ( cons @ B @ Y3 @ Ys4 ) )
=> ( ( Xy
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ( Xys
!= ( zip @ A @ B @ Xs5 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_2187_lenlex__irreflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lenlex @ A @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_2188_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ! [Y4: real] :
? [N2: nat] : ( ord_less @ real @ Y4 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_2189_image__minus__const__atLeastLessThan__nat,axiom,
! [C3: nat,Y: nat,X: nat] :
( ( ( ord_less @ nat @ C3 @ Y )
=> ( ( image @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C3 ) @ ( minus_minus @ nat @ Y @ C3 ) ) ) )
& ( ~ ( ord_less @ nat @ C3 @ Y )
=> ( ( ( ord_less @ nat @ X @ Y )
=> ( ( image @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X @ Y )
=> ( ( image @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_2190_listrel1I1,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Xs ) ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel1I1
thf(fact_2191_Cons__listrel1E1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( ! [Y3: A] :
( ( Ys
= ( cons @ A @ Y3 @ Xs ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 ) )
=> ~ ! [Zs2: list @ A] :
( ( Ys
= ( cons @ A @ X @ Zs2 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_2192_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R2 ) )
=> ( ! [X3: A] :
( ( Xs
= ( cons @ A @ X3 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) )
=> ~ ! [Zs2: list @ A] :
( ( Xs
= ( cons @ A @ Y @ Zs2 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs2 @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_2193_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A5: set @ B,K4: A,F3: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A5 )
=> ( ord_less_eq @ A @ K4 @ ( F3 @ I2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ K4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) ) ) ) ).
% sum_bounded_below
thf(fact_2194_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A5: set @ B,F3: B > A,K4: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ K4 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ K4 ) ) ) ) ).
% sum_bounded_above
thf(fact_2195_nat__less__real__le,axiom,
( ( ord_less @ nat )
= ( ^ [N6: nat,M5: nat] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ N6 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ M5 ) ) ) ) ).
% nat_less_real_le
thf(fact_2196_nat__le__real__less,axiom,
( ( ord_less_eq @ nat )
= ( ^ [N6: nat,M5: nat] : ( ord_less @ real @ ( semiring_1_of_nat @ real @ N6 ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ M5 ) @ ( one_one @ real ) ) ) ) ) ).
% nat_le_real_less
thf(fact_2197_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_2198_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_2199_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C3 )
=> ( ! [M3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M3 )
=> ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M3 ) @ X ) @ C3 ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_2200_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A5: set @ B,F3: B > A,K4: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A5 )
=> ( ord_less @ A @ ( F3 @ I2 ) @ K4 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A5 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ K4 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_2201_linear__plus__1__le__power,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) @ ( one_one @ real ) ) @ ( power_power @ real @ ( plus_plus @ real @ X @ ( one_one @ real ) ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_2202_cnt__cnt__eq,axiom,
( vEBT_VEBT_cnt
= ( ^ [T3: vEBT_VEBT] : ( semiring_1_of_nat @ real @ ( vEBT_VEBT_cnt2 @ T3 ) ) ) ) ).
% cnt_cnt_eq
thf(fact_2203_minNulli__rule,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi] :
( hoare_hoare_triple @ $o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_VEBT_minNulli @ Ti )
@ ^ [R5: $o] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_VEBT_minNull @ T2 ) ) ) ) ) ).
% minNulli_rule
thf(fact_2204_builupi_Hcorr,axiom,
! [N: nat] : ( hoare_hoare_triple @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_V739175172307565963ildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) ) ).
% builupi'corr
thf(fact_2205_builupicorr,axiom,
! [N: nat] : ( hoare_hoare_triple @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) ) ).
% builupicorr
thf(fact_2206_vebt__maxtilist,axiom,
! [I: nat,Ts2: list @ vEBT_VEBT,Tsi: list @ vEBT_VEBTi] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ Ts2 ) )
=> ( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Ts2 @ Tsi ) @ ( vEBT_vebt_maxti @ ( nth @ vEBT_VEBTi @ Tsi @ I ) )
@ ^ [R5: option @ nat] :
( times_times @ assn
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Ts2 @ I ) ) ) )
@ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Ts2 @ Tsi ) ) ) ) ).
% vebt_maxtilist
thf(fact_2207_vebt__mintilist,axiom,
! [I: nat,Ts2: list @ vEBT_VEBT,Tsi: list @ vEBT_VEBTi] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ Ts2 ) )
=> ( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Ts2 @ Tsi ) @ ( vEBT_vebt_minti @ ( nth @ vEBT_VEBTi @ Tsi @ I ) )
@ ^ [R5: option @ nat] :
( times_times @ assn
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ Ts2 @ I ) ) ) )
@ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Ts2 @ Tsi ) ) ) ) ).
% vebt_mintilist
thf(fact_2208_Chebyshev__sum__upper,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: nat > A,B2: nat > A] :
( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( A2 @ I2 ) @ ( A2 @ J2 ) ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( B2 @ J2 ) @ ( B2 @ I2 ) ) ) )
=> ( ord_less_eq @ A
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A2 @ K3 ) @ ( B2 @ K3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ) ).
% Chebyshev_sum_upper
thf(fact_2209_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_2210_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_2211_predicate2I,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Q2: A > B > $o] :
( ! [X3: A,Y3: B] :
( ( P2 @ X3 @ Y3 )
=> ( Q2 @ X3 @ Y3 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q2 ) ) ).
% predicate2I
thf(fact_2212_predicate1I,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq @ ( A > $o ) @ P2 @ Q2 ) ) ).
% predicate1I
thf(fact_2213_vebt__minti__h,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi] :
( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_minti @ Ti )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_mint @ T2 ) ) ) ) ) ).
% vebt_minti_h
thf(fact_2214_vebt__maxti__h,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi] :
( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_maxt @ T2 ) ) ) ) ) ).
% vebt_maxti_h
thf(fact_2215_predicate1D,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P2 @ Q2 )
=> ( ( P2 @ X )
=> ( Q2 @ X ) ) ) ).
% predicate1D
thf(fact_2216_predicate2D,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Q2: A > B > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q2 )
=> ( ( P2 @ X @ Y )
=> ( Q2 @ X @ Y ) ) ) ).
% predicate2D
thf(fact_2217_rev__predicate1D,axiom,
! [A: $tType,P2: A > $o,X: A,Q2: A > $o] :
( ( P2 @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P2 @ Q2 )
=> ( Q2 @ X ) ) ) ).
% rev_predicate1D
thf(fact_2218_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,X: A,Y: B,Q2: A > B > $o] :
( ( P2 @ X @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q2 )
=> ( Q2 @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_2219_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W3: int,Z2: int] :
? [N6: nat] :
( Z2
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ ( suc @ N6 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_2220_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_2221_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_2222_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_2223_negD,axiom,
! [X: int] :
( ( ord_less @ int @ X @ ( zero_zero @ int ) )
=> ? [N2: nat] :
( X
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_2224_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_2225_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_2226_int__of__nat__induct,axiom,
! [P2: int > $o,Z: int] :
( ! [N2: nat] : ( P2 @ ( semiring_1_of_nat @ int @ N2 ) )
=> ( ! [N2: nat] : ( P2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) )
=> ( P2 @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_2227_int__ops_I1_J,axiom,
( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ) ).
% int_ops(1)
thf(fact_2228_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_2229_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A8: nat,B8: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A8 ) @ ( semiring_1_of_nat @ int @ B8 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_2230_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_2231_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A8: nat,B8: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A8 ) @ ( semiring_1_of_nat @ int @ B8 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_2232_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_2233_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A8: nat,B8: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A8 ) @ ( semiring_1_of_nat @ int @ B8 ) ) ) ) ).
% nat_less_as_int
thf(fact_2234_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A8: nat,B8: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A8 ) @ ( semiring_1_of_nat @ int @ B8 ) ) ) ) ).
% nat_leq_as_int
thf(fact_2235_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_2236_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_2237_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_2238_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_2239_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_2240_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_2241_Abs__fnat__hom__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( zero_zero @ ( word @ A ) )
= ( semiring_1_of_nat @ ( word @ A ) @ ( zero_zero @ nat ) ) ) ) ).
% Abs_fnat_hom_0
thf(fact_2242_word__le__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ? [N2: nat] :
( Y
= ( plus_plus @ ( word @ A ) @ X @ ( semiring_1_of_nat @ ( word @ A ) @ N2 ) ) ) ) ) ).
% word_le_add
thf(fact_2243_Abs__fnat__hom__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( one_one @ ( word @ A ) )
= ( semiring_1_of_nat @ ( word @ A ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% Abs_fnat_hom_1
thf(fact_2244_zdiff__int__split,axiom,
! [P2: int > $o,X: nat,Y: nat] :
( ( P2 @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X @ Y ) ) )
= ( ( ( ord_less_eq @ nat @ Y @ X )
=> ( P2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y ) ) ) )
& ( ( ord_less @ nat @ X @ Y )
=> ( P2 @ ( zero_zero @ int ) ) ) ) ) ).
% zdiff_int_split
thf(fact_2245_VEBT__internal_Ocnt_H_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_VEBT_cnt2 @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( one_one @ nat ) ) ).
% VEBT_internal.cnt'.simps(1)
thf(fact_2246_word__1__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,X: nat] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ ( one_one @ ( word @ A ) ) ) @ B2 )
=> ( ( ord_less @ ( word @ A ) @ A2 @ ( semiring_1_of_nat @ ( word @ A ) @ X ) )
=> ( ord_less @ ( word @ A ) @ A2 @ B2 ) ) ) ) ).
% word_1_0
thf(fact_2247_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top @ nat
@ ^ [X4: nat,Y5: nat] : ( ord_less_eq @ nat @ Y5 @ X4 )
@ ^ [X4: nat,Y5: nat] : ( ord_less @ nat @ Y5 @ X4 )
@ ( zero_zero @ nat ) ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_2248_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A2: nat > nat,B2: nat > nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( A2 @ I2 ) @ ( A2 @ J2 ) ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( B2 @ J2 ) @ ( B2 @ I2 ) ) ) )
=> ( ord_less_eq @ nat
@ ( times_times @ nat @ N
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( times_times @ nat @ ( A2 @ I3 ) @ ( B2 @ I3 ) )
@ ( 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_2249_minNrulli__ruleT,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi] :
( time_htt @ $o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_VEBT_minNulli @ Ti )
@ ^ [R5: $o] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_VEBT_minNull @ T2 ) ) ) )
@ ( one_one @ nat ) ) ).
% minNrulli_ruleT
thf(fact_2250_T__vebt__buildupi,axiom,
! [N: nat,H2: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ vEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ N ) @ H2 ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).
% T_vebt_buildupi
thf(fact_2251_vebt__maxti__hT,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi] :
( time_htt @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_maxt @ T2 ) ) ) )
@ ( one_one @ nat ) ) ).
% vebt_maxti_hT
thf(fact_2252_vebt__minti__hT,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi] :
( time_htt @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_minti @ Ti )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_mint @ T2 ) ) ) )
@ ( one_one @ nat ) ) ).
% vebt_minti_hT
thf(fact_2253_htt__vebt__buildupi_H,axiom,
! [N: nat] : ( time_htt @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_V739175172307565963ildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).
% htt_vebt_buildupi'
thf(fact_2254_htt__vebt__buildupi,axiom,
! [N: nat] : ( time_htt @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).
% htt_vebt_buildupi
thf(fact_2255_finite__interval__int1,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less_eq @ int @ A2 @ I3 )
& ( ord_less_eq @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int1
thf(fact_2256_finite__interval__int4,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less @ int @ A2 @ I3 )
& ( ord_less @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int4
thf(fact_2257_finite__interval__int3,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less @ int @ A2 @ I3 )
& ( ord_less_eq @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int3
thf(fact_2258_finite__interval__int2,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less_eq @ int @ A2 @ I3 )
& ( ord_less @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int2
thf(fact_2259_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_2260_zle__add1__eq__le,axiom,
! [W2: int,Z: int] :
( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z @ ( one_one @ int ) ) )
= ( ord_less_eq @ int @ W2 @ Z ) ) ).
% zle_add1_eq_le
thf(fact_2261_zle__diff1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq @ int @ W2 @ ( minus_minus @ int @ Z @ ( one_one @ int ) ) )
= ( ord_less @ int @ W2 @ Z ) ) ).
% zle_diff1_eq
thf(fact_2262_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_2263_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_2264_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_2265_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_2266_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W3: int,Z2: int] :
? [N6: nat] :
( Z2
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ N6 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_2267_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_2268_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_2269_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_2270_less__int__code_I1_J,axiom,
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% less_int_code(1)
thf(fact_2271_zless__add1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z @ ( one_one @ int ) ) )
= ( ( ord_less @ int @ W2 @ Z )
| ( W2 = Z ) ) ) ).
% zless_add1_eq
thf(fact_2272_int__gr__induct,axiom,
! [K: int,I: int,P2: int > $o] :
( ( ord_less @ int @ K @ I )
=> ( ( P2 @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I2: int] :
( ( ord_less @ int @ K @ I2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_gr_induct
thf(fact_2273_int__less__induct,axiom,
! [I: int,K: int,P2: int > $o] :
( ( ord_less @ int @ I @ K )
=> ( ( P2 @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I2: int] :
( ( ord_less @ int @ I2 @ K )
=> ( ( P2 @ I2 )
=> ( P2 @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_less_induct
thf(fact_2274_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_2275_plusinfinity,axiom,
! [D2: int,P6: int > $o,P2: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P6 @ X3 )
= ( P6 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ Z5 @ X3 )
=> ( ( P2 @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [X_12: int] : ( P6 @ X_12 )
=> ? [X_1: int] : ( P2 @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_2276_minusinfinity,axiom,
! [D2: int,P1: int > $o,P2: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z5 )
=> ( ( P2 @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P2 @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_2277_verit__la__generic,axiom,
! [A2: int,X: int] :
( ( ord_less_eq @ int @ A2 @ X )
| ( A2 = X )
| ( ord_less_eq @ int @ X @ A2 ) ) ).
% verit_la_generic
thf(fact_2278_zless__imp__add1__zle,axiom,
! [W2: int,Z: int] :
( ( ord_less @ int @ W2 @ Z )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_2279_add1__zle__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z )
= ( ord_less @ int @ W2 @ Z ) ) ).
% add1_zle_eq
thf(fact_2280_int__ge__induct,axiom,
! [K: int,I: int,P2: int > $o] :
( ( ord_less_eq @ int @ K @ I )
=> ( ( P2 @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ K @ I2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_ge_induct
thf(fact_2281_int__le__induct,axiom,
! [I: int,K: int,P2: int > $o] :
( ( ord_less_eq @ int @ I @ K )
=> ( ( P2 @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ I2 @ K )
=> ( ( P2 @ I2 )
=> ( P2 @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_le_induct
thf(fact_2282_int__induct,axiom,
! [P2: int > $o,K: int,I: int] :
( ( P2 @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ K @ I2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ I2 @ K )
=> ( ( P2 @ I2 )
=> ( P2 @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_induct
thf(fact_2283_unique__quotient__lemma__neg,axiom,
! [B2: int,Q7: int,R6: int,Q3: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q7 ) @ R6 ) @ ( 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 @ R6 )
=> ( ord_less_eq @ int @ Q3 @ Q7 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_2284_unique__quotient__lemma,axiom,
! [B2: int,Q7: int,R6: int,Q3: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q7 ) @ R6 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R6 )
=> ( ( ord_less @ int @ R6 @ B2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ord_less_eq @ int @ Q7 @ Q3 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_2285_zdiv__mono2__neg__lemma,axiom,
! [B2: int,Q3: int,R2: int,B3: int,Q7: int,R6: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q7 ) @ R6 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q7 ) @ R6 ) @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R6 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ B3 @ B2 )
=> ( ord_less_eq @ int @ Q7 @ Q3 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_2286_zdiv__mono2__lemma,axiom,
! [B2: int,Q3: int,R2: int,B3: int,Q7: int,R6: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q7 ) @ R6 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q7 ) @ R6 ) )
=> ( ( ord_less @ int @ R6 @ 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 @ Q7 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_2287_q__pos__lemma,axiom,
! [B3: int,Q7: int,R6: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q7 ) @ R6 ) )
=> ( ( ord_less @ int @ R6 @ B3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q7 ) ) ) ) ).
% q_pos_lemma
thf(fact_2288_incr__mult__lemma,axiom,
! [D2: int,P2: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int] :
( ( P2 @ X3 )
=> ( P2 @ ( plus_plus @ int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X5: int] :
( ( P2 @ X5 )
=> ( P2 @ ( plus_plus @ int @ X5 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_2289_pos__mult__pos__ge,axiom,
! [X: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ ( times_times @ int @ N @ ( one_one @ int ) ) @ ( times_times @ int @ N @ X ) ) ) ) ).
% pos_mult_pos_ge
thf(fact_2290_decr__mult__lemma,axiom,
! [D2: int,P2: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int] :
( ( P2 @ X3 )
=> ( P2 @ ( minus_minus @ int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X5: int] :
( ( P2 @ X5 )
=> ( P2 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_2291_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_2292_less__1__helper,axiom,
! [N: int,M: int] :
( ( ord_less_eq @ int @ N @ M )
=> ( ord_less @ int @ ( minus_minus @ int @ N @ ( one_one @ int ) ) @ M ) ) ).
% less_1_helper
thf(fact_2293_imp__le__cong,axiom,
! [X: int,X9: int,P2: $o,P6: $o] :
( ( X = X9 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X9 )
=> ( P2 = P6 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> P2 )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X9 )
=> P6 ) ) ) ) ).
% imp_le_cong
thf(fact_2294_conj__le__cong,axiom,
! [X: int,X9: int,P2: $o,P6: $o] :
( ( X = X9 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X9 )
=> ( P2 = P6 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& P2 )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X9 )
& P6 ) ) ) ) ).
% conj_le_cong
thf(fact_2295_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_2296_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_2297_Gcd__int__greater__eq__0,axiom,
! [K4: set @ int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_Gcd @ int @ K4 ) ) ).
% Gcd_int_greater_eq_0
thf(fact_2298_one__integer_Orsp,axiom,
( ( one_one @ int )
= ( one_one @ int ) ) ).
% one_integer.rsp
thf(fact_2299_time__replicate,axiom,
! [A: $tType,X: heap_Time_Heap @ A,C3: nat,N: nat,H2: heap_ext @ product_unit] :
( ! [H: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ A @ X @ H ) @ C3 )
=> ( ord_less_eq @ nat @ ( time_time @ ( list @ A ) @ ( vEBT_VEBT_replicatei @ A @ N @ X ) @ H2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ C3 ) @ N ) ) ) ) ).
% time_replicate
thf(fact_2300_real__sum__nat__ivl__bounded2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,F3: nat > A,K4: A,K: nat] :
( ! [P8: nat] :
( ( ord_less @ nat @ P8 @ N )
=> ( ord_less_eq @ A @ ( F3 @ P8 ) @ K4 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ K4 )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ K ) ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ K4 ) ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_2301_rel__of__empty,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o] :
( ( rel_of @ A @ B
@ ^ [X4: A] : ( none @ B )
@ P2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% rel_of_empty
thf(fact_2302_htt__def,axiom,
! [A: $tType] :
( ( time_htt @ A )
= ( ^ [P5: assn,C7: heap_Time_Heap @ A,Q6: A > assn,T3: nat] :
( ( hoare_hoare_triple @ A @ P5 @ C7 @ Q6 )
& ! [H4: heap_ext @ product_unit,As2: set @ nat] :
( ( rep_assn @ P5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) )
=> ( ord_less_eq @ nat @ ( time_time @ A @ C7 @ H4 ) @ T3 ) ) ) ) ) ).
% htt_def
thf(fact_2303_httI,axiom,
! [A: $tType,P2: assn,C3: heap_Time_Heap @ A,Q2: A > assn,T2: nat] :
( ( hoare_hoare_triple @ A @ P2 @ C3 @ Q2 )
=> ( ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( rep_assn @ P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( ord_less_eq @ nat @ ( time_time @ A @ C3 @ H ) @ T2 ) )
=> ( time_htt @ A @ P2 @ C3 @ Q2 @ T2 ) ) ) ).
% httI
thf(fact_2304_Tbuildupi__buildupi_H,axiom,
! [N: nat] :
( ( semiring_1_of_nat @ int @ ( vEBT_V441764108873111860ildupi @ N ) )
= ( vEBT_V9176841429113362141ildupi @ N ) ) ).
% Tbuildupi_buildupi'
thf(fact_2305_VEBT__internal_OT__vebt__buildupi_H_Osimps_I1_J,axiom,
( ( vEBT_V9176841429113362141ildupi @ ( zero_zero @ nat ) )
= ( one_one @ int ) ) ).
% VEBT_internal.T_vebt_buildupi'.simps(1)
thf(fact_2306_VEBT__internal_OT__vebt__buildupi_H_Osimps_I2_J,axiom,
( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ int ) ) ).
% VEBT_internal.T_vebt_buildupi'.simps(2)
thf(fact_2307_htt__cons__rule,axiom,
! [A: $tType,P6: assn,C3: heap_Time_Heap @ A,Q4: A > assn,T8: nat,P2: assn,Q2: A > assn,T2: nat] :
( ( time_htt @ A @ P6 @ C3 @ Q4 @ T8 )
=> ( ( entails @ P2 @ P6 )
=> ( ! [X3: A] : ( entails @ ( Q4 @ X3 ) @ ( Q2 @ X3 ) )
=> ( ( ord_less_eq @ nat @ T8 @ T2 )
=> ( time_htt @ A @ P2 @ C3 @ Q2 @ T2 ) ) ) ) ) ).
% htt_cons_rule
thf(fact_2308_finite__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z2: A] :
( ( power_power @ A @ Z2 @ N )
= ( one_one @ A ) ) ) ) ) ) ).
% finite_roots_unity
thf(fact_2309_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
@ ^ [Z2: A] :
( ( power_power @ A @ Z2 @ N )
= ( one_one @ A ) ) ) )
@ N ) ) ) ).
% card_roots_unity
thf(fact_2310_arcosh__1,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% arcosh_1
thf(fact_2311_finite__nth__roots,axiom,
! [N: nat,C3: complex] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite2 @ complex
@ ( collect @ complex
@ ^ [Z2: complex] :
( ( power_power @ complex @ Z2 @ N )
= C3 ) ) ) ) ).
% finite_nth_roots
thf(fact_2312_foldr__same,axiom,
! [Xs: list @ real,Y: real] :
( ! [X3: real,Y3: real] :
( ( member @ real @ X3 @ ( set2 @ real @ Xs ) )
=> ( ( member @ real @ Y3 @ ( set2 @ real @ Xs ) )
=> ( X3 = Y3 ) ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set2 @ real @ Xs ) )
=> ( X3 = Y ) )
=> ( ( foldr @ real @ real @ ( plus_plus @ real ) @ Xs @ ( zero_zero @ real ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( list @ real ) @ Xs ) ) @ Y ) ) ) ) ).
% foldr_same
thf(fact_2313_list__every__elemnt__bound__sum__bound,axiom,
! [A: $tType,Xs: list @ A,F3: A > nat,Bound: nat,I: nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( F3 @ X3 ) @ Bound ) )
=> ( ord_less_eq @ nat @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ A @ nat @ F3 @ Xs ) @ I ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ Bound ) @ I ) ) ) ).
% list_every_elemnt_bound_sum_bound
thf(fact_2314_vebt__maxti__rule,axiom,
! [N: nat,S: set @ nat,Ti: vEBT_VEBTi,Y: nat] :
( time_htt @ ( option @ nat ) @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti )
@ ( pure_assn
@ ( ( R5
= ( some @ nat @ Y ) )
= ( vEBT_VEBT_max_in_set @ S @ Y ) ) ) )
@ ( one_one @ nat ) ) ).
% vebt_maxti_rule
thf(fact_2315_foldr0,axiom,
! [Xs: list @ real,C3: real,D2: real] :
( ( foldr @ real @ real @ ( plus_plus @ real ) @ Xs @ ( plus_plus @ real @ C3 @ D2 ) )
= ( plus_plus @ real @ ( foldr @ real @ real @ ( plus_plus @ real ) @ Xs @ D2 ) @ C3 ) ) ).
% foldr0
thf(fact_2316_nth__map,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,F3: A > B] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ B @ ( map @ A @ B @ F3 @ Xs ) @ N )
= ( F3 @ ( nth @ A @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_2317_map__consI_I1_J,axiom,
! [A: $tType,B: $tType,W2: list @ A,F3: B > A,Ww: list @ B,A2: B] :
( ( W2
= ( map @ B @ A @ F3 @ Ww ) )
=> ( ( cons @ A @ ( F3 @ A2 ) @ W2 )
= ( map @ B @ A @ F3 @ ( cons @ B @ A2 @ Ww ) ) ) ) ).
% map_consI(1)
thf(fact_2318_map__eq__consE,axiom,
! [B: $tType,A: $tType,F3: B > A,Ls: list @ B,Fa: A,Fl: list @ A] :
( ( ( map @ B @ A @ F3 @ Ls )
= ( cons @ A @ Fa @ Fl ) )
=> ~ ! [A4: B,L2: list @ B] :
( ( Ls
= ( cons @ B @ A4 @ L2 ) )
=> ( ( ( F3 @ A4 )
= Fa )
=> ( ( map @ B @ A @ F3 @ L2 )
!= Fl ) ) ) ) ).
% map_eq_consE
thf(fact_2319_map__eq__nth__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,L: list @ B,L3: list @ B,I: nat] :
( ( ( map @ B @ A @ F3 @ L )
= ( map @ B @ A @ F3 @ L3 ) )
=> ( ( F3 @ ( nth @ B @ L @ I ) )
= ( F3 @ ( nth @ B @ L3 @ I ) ) ) ) ).
% map_eq_nth_eq
thf(fact_2320_distinct__mapI,axiom,
! [A: $tType,B: $tType,F3: B > A,L: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F3 @ L ) )
=> ( distinct @ B @ L ) ) ).
% distinct_mapI
thf(fact_2321_distinct__map__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,L: list @ B,X: B,Y: B] :
( ( distinct @ A @ ( map @ B @ A @ F3 @ L ) )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
=> ( ( member @ B @ X @ ( set2 @ B @ L ) )
=> ( ( member @ B @ Y @ ( set2 @ B @ L ) )
=> ( X = Y ) ) ) ) ) ).
% distinct_map_eq
thf(fact_2322_sorted__map,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X4: B,Y5: B] : ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y5 ) )
@ Xs ) ) ) ).
% sorted_map
thf(fact_2323_sorted__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ).
% sorted_map_remove1
thf(fact_2324_distinct__idx,axiom,
! [B: $tType,A: $tType,F3: B > A,L: list @ B,I: nat,J: nat] :
( ( distinct @ A @ ( map @ B @ A @ F3 @ L ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ L ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ B ) @ L ) )
=> ( ( ( F3 @ ( nth @ B @ L @ I ) )
= ( F3 @ ( nth @ B @ L @ J ) ) )
=> ( I = J ) ) ) ) ) ).
% distinct_idx
thf(fact_2325_map__upd__eq,axiom,
! [B: $tType,A: $tType,I: nat,L: list @ A,F3: A > B,X: A] :
( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F3 @ ( nth @ A @ L @ I ) )
= ( F3 @ X ) ) )
=> ( ( map @ A @ B @ F3 @ ( list_update @ A @ L @ I @ X ) )
= ( map @ A @ B @ F3 @ L ) ) ) ).
% map_upd_eq
thf(fact_2326_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_VEBT_cnt2 @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_cnt2 @ Summary ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_cnt2 @ TreeList ) @ ( zero_zero @ nat ) ) ) ) ).
% VEBT_internal.cnt'.simps(2)
thf(fact_2327_VEBT__internal_Ocnt_H_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_cnt2 @ X )
= Y )
=> ( ( ? [A4: $o,B4: $o] :
( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_cnt2 @ Summary2 ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_cnt2 @ TreeList2 ) @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% VEBT_internal.cnt'.elims
thf(fact_2328_card__nth__roots,axiom,
! [C3: complex,N: nat] :
( ( C3
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z2: complex] :
( ( power_power @ complex @ Z2 @ N )
= C3 ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_2329_card__roots__unity__eq,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z2: complex] :
( ( power_power @ complex @ Z2 @ N )
= ( one_one @ complex ) ) ) )
= N ) ) ).
% card_roots_unity_eq
thf(fact_2330_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X4: complex] : X4
@ ( collect @ complex
@ ^ [Z2: complex] :
( ( power_power @ complex @ Z2 @ N )
= ( one_one @ complex ) ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_roots_unity
thf(fact_2331_sum__nth__roots,axiom,
! [N: nat,C3: complex] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X4: complex] : X4
@ ( collect @ complex
@ ^ [Z2: complex] :
( ( power_power @ complex @ Z2 @ N )
= C3 ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_nth_roots
thf(fact_2332_vebt__buildupi__rule__basic,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( hoare_hoare_triple @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_Intf_vebt_assn @ N @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% vebt_buildupi_rule_basic
thf(fact_2333_vebt__minti__rule,axiom,
! [N: nat,S: set @ nat,Ti: vEBT_VEBTi,Y: nat] :
( time_htt @ ( option @ nat ) @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti ) @ ( vEBT_vebt_minti @ Ti )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti )
@ ( pure_assn
@ ( ( R5
= ( some @ nat @ Y ) )
= ( vEBT_VEBT_min_in_set @ S @ Y ) ) ) )
@ ( one_one @ nat ) ) ).
% vebt_minti_rule
thf(fact_2334_VEBT__internal_Ocnt_H_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_cnt2 @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel @ X )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_cnt2 @ Summary2 ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_cnt2 @ TreeList2 ) @ ( zero_zero @ nat ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% VEBT_internal.cnt'.pelims
thf(fact_2335_list__every__elemnt__bound__sum__bound__real,axiom,
! [A: $tType,Xs: list @ A,F3: A > real,Bound: real,I: real] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ real @ ( F3 @ X3 ) @ Bound ) )
=> ( ord_less_eq @ real @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ F3 @ Xs ) @ I ) @ ( plus_plus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( list @ A ) @ Xs ) ) @ Bound ) @ I ) ) ) ).
% list_every_elemnt_bound_sum_bound_real
thf(fact_2336_real__nat__list,axiom,
! [A: $tType,F3: A > nat,Xs: list @ A,C3: nat] :
( ( semiring_1_of_nat @ real @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ A @ nat @ F3 @ Xs ) @ C3 ) )
= ( foldr @ real @ real @ ( plus_plus @ real )
@ ( map @ A @ real
@ ^ [X4: A] : ( semiring_1_of_nat @ real @ ( F3 @ X4 ) )
@ Xs )
@ ( semiring_1_of_nat @ real @ C3 ) ) ) ).
% real_nat_list
thf(fact_2337_f__g__map__foldr__bound,axiom,
! [A: $tType,Xs: list @ A,F3: A > real,C3: real,G3: A > real,D2: real] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ real @ ( F3 @ X3 ) @ ( times_times @ real @ C3 @ ( G3 @ X3 ) ) ) )
=> ( ord_less_eq @ real @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ F3 @ Xs ) @ D2 ) @ ( plus_plus @ real @ ( times_times @ real @ C3 @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ G3 @ Xs ) @ ( zero_zero @ real ) ) ) @ D2 ) ) ) ).
% f_g_map_foldr_bound
thf(fact_2338_artanh__0,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% artanh_0
thf(fact_2339_arsinh__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% arsinh_0
thf(fact_2340_listsum__bound,axiom,
! [A: $tType,Xs: list @ A,F3: A > real,Y: real] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ real @ Y @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ F3 @ Xs ) @ Y ) ) ) ).
% listsum_bound
thf(fact_2341_zip__same__conv__map,axiom,
! [A: $tType,Xs: list @ A] :
( ( zip @ A @ A @ Xs @ Xs )
= ( map @ A @ ( product_prod @ A @ A )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Xs ) ) ).
% zip_same_conv_map
thf(fact_2342_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_VEBT_cnt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
= ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( vEBT_VEBT_cnt @ Summary ) ) @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ vEBT_VEBT @ real @ vEBT_VEBT_cnt @ TreeList ) @ ( zero_zero @ real ) ) ) ) ).
% VEBT_internal.cnt.simps(2)
thf(fact_2343_VEBT__internal_Ocnt_Oelims,axiom,
! [X: vEBT_VEBT,Y: real] :
( ( ( vEBT_VEBT_cnt @ X )
= Y )
=> ( ( ? [A4: $o,B4: $o] :
( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
!= ( one_one @ real ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( vEBT_VEBT_cnt @ Summary2 ) ) @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ vEBT_VEBT @ real @ vEBT_VEBT_cnt @ TreeList2 ) @ ( zero_zero @ real ) ) ) ) ) ) ) ).
% VEBT_internal.cnt.elims
thf(fact_2344_VEBT__internal_Ocnt_Opelims,axiom,
! [X: vEBT_VEBT,Y: real] :
( ( ( vEBT_VEBT_cnt @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ X )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( one_one @ real ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( vEBT_VEBT_cnt @ Summary2 ) ) @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ vEBT_VEBT @ real @ vEBT_VEBT_cnt @ TreeList2 ) @ ( zero_zero @ real ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% VEBT_internal.cnt.pelims
thf(fact_2345_nth__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,Xs: list @ A,A2: array @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( hoare_hoare_triple @ A @ ( snga_assn @ A @ A2 @ Xs ) @ ( array_nth @ A @ A2 @ I )
@ ^ [R5: A] :
( times_times @ assn @ ( snga_assn @ A @ A2 @ Xs )
@ ( pure_assn
@ ( R5
= ( nth @ A @ Xs @ I ) ) ) ) ) ) ) ).
% nth_rule
thf(fact_2346_same__fst__trancl,axiom,
! [B: $tType,A: $tType,P2: A > $o,R3: A > ( set @ ( product_prod @ B @ B ) )] :
( ( transitive_trancl @ ( product_prod @ A @ B ) @ ( same_fst @ A @ B @ P2 @ R3 ) )
= ( same_fst @ A @ B @ P2
@ ^ [X4: A] : ( transitive_trancl @ B @ ( R3 @ X4 ) ) ) ) ).
% same_fst_trancl
thf(fact_2347_accp__subset,axiom,
! [A: $tType,R12: A > A > $o,R23: A > A > $o] :
( ( ord_less_eq @ ( A > A > $o ) @ R12 @ R23 )
=> ( ord_less_eq @ ( A > $o ) @ ( accp @ A @ R23 ) @ ( accp @ A @ R12 ) ) ) ).
% accp_subset
thf(fact_2348_repli__emp,axiom,
! [A: $tType,B: $tType,X: heap_Time_Heap @ A,A5: B > A > assn,Y: B,N: nat] :
( ( hoare_hoare_triple @ A @ ( one_one @ assn ) @ X @ ( A5 @ Y ) )
=> ( hoare_hoare_triple @ ( list @ A ) @ ( one_one @ assn ) @ ( vEBT_VEBT_replicatei @ A @ N @ X ) @ ( vEBT_List_list_assn @ B @ A @ A5 @ ( replicate @ B @ N @ Y ) ) ) ) ).
% repli_emp
thf(fact_2349_product__code,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( product_product @ A @ B @ ( set2 @ A @ Xs ) @ ( set2 @ B @ Ys ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X4: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 ) @ Ys )
@ Xs ) ) ) ) ).
% product_code
thf(fact_2350_intind,axiom,
! [A: $tType,I: nat,N: nat,P2: A > $o,X: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( P2 @ X )
=> ( P2 @ ( nth @ A @ ( replicate @ A @ N @ X ) @ I ) ) ) ) ).
% intind
thf(fact_2351_repli__cons__repl,axiom,
! [B: $tType,A: $tType,Q2: assn,X: heap_Time_Heap @ A,A5: B > A > assn,Y: B,N: nat] :
( ( hoare_hoare_triple @ A @ Q2 @ X
@ ^ [R5: A] : ( times_times @ assn @ Q2 @ ( A5 @ Y @ R5 ) ) )
=> ( hoare_hoare_triple @ ( list @ A ) @ Q2 @ ( vEBT_VEBT_replicatei @ A @ N @ X )
@ ^ [R5: list @ A] : ( times_times @ assn @ Q2 @ ( vEBT_List_list_assn @ B @ A @ A5 @ ( replicate @ B @ N @ Y ) @ R5 ) ) ) ) ).
% repli_cons_repl
thf(fact_2352_replicate__eq__replicate,axiom,
! [A: $tType,M: nat,X: A,N: nat,Y: A] :
( ( ( replicate @ A @ M @ X )
= ( replicate @ A @ N @ Y ) )
= ( ( M = N )
& ( ( M
!= ( zero_zero @ nat ) )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_2353_Ball__set__replicate,axiom,
! [A: $tType,N: nat,A2: A,P2: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ ( replicate @ A @ N @ A2 ) ) )
=> ( P2 @ X4 ) ) )
= ( ( P2 @ A2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_2354_Bex__set__replicate,axiom,
! [A: $tType,N: nat,A2: A,P2: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ ( replicate @ A @ N @ A2 ) ) )
& ( P2 @ X4 ) ) )
= ( ( P2 @ A2 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_2355_in__set__replicate,axiom,
! [A: $tType,X: A,N: nat,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N @ Y ) ) )
= ( ( X = Y )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% in_set_replicate
thf(fact_2356_nth__replicate,axiom,
! [A: $tType,I: nat,N: nat,X: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( replicate @ A @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_2357_last__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( replicate @ A @ N @ X ) )
= X ) ) ).
% last_replicate
thf(fact_2358_set__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_2359_replicate__Suc,axiom,
! [A: $tType,N: nat,X: A] :
( ( replicate @ A @ ( suc @ N ) @ X )
= ( cons @ A @ X @ ( replicate @ A @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_2360_sorted__replicate,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N: nat,X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( replicate @ A @ N @ X ) ) ) ).
% sorted_replicate
thf(fact_2361_zip__replicate1,axiom,
! [A: $tType,B: $tType,N: nat,X: A,Ys: list @ B] :
( ( zip @ A @ B @ ( replicate @ A @ N @ X ) @ Ys )
= ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ ( take @ B @ N @ Ys ) ) ) ).
% zip_replicate1
thf(fact_2362_map__zip2,axiom,
! [A: $tType,B: $tType,K: A,L: list @ B] :
( ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K ) @ L )
= ( zip @ A @ B @ ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) @ L ) ) ).
% map_zip2
thf(fact_2363_map__zip1,axiom,
! [A: $tType,B: $tType,K: B,L: list @ A] :
( ( map @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ K )
@ L )
= ( zip @ A @ B @ L @ ( replicate @ B @ ( size_size @ ( list @ A ) @ L ) @ K ) ) ) ).
% map_zip1
thf(fact_2364_zip__replicate2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,N: nat,Y: B] :
( ( zip @ A @ B @ Xs @ ( replicate @ B @ N @ Y ) )
= ( map @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ Y )
@ ( take @ A @ N @ Xs ) ) ) ).
% zip_replicate2
thf(fact_2365_set__replicate__Suc,axiom,
! [A: $tType,N: nat,X: A] :
( ( set2 @ A @ ( replicate @ A @ ( suc @ N ) @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_replicate_Suc
thf(fact_2366_set__replicate__conv__if,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_2367_Cons__replicate__eq,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat,Y: A] :
( ( ( cons @ A @ X @ Xs )
= ( replicate @ A @ N @ Y ) )
= ( ( X = Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( Xs
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_2368_accp__subset__induct,axiom,
! [A: $tType,D5: A > $o,R3: A > A > $o,X: A,P2: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ D5 @ ( accp @ A @ R3 ) )
=> ( ! [X3: A,Z3: A] :
( ( D5 @ X3 )
=> ( ( R3 @ Z3 @ X3 )
=> ( D5 @ Z3 ) ) )
=> ( ( D5 @ X )
=> ( ! [X3: A] :
( ( D5 @ X3 )
=> ( ! [Z5: A] :
( ( R3 @ Z5 @ X3 )
=> ( P2 @ Z5 ) )
=> ( P2 @ X3 ) ) )
=> ( P2 @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_2369_same__fstI,axiom,
! [B: $tType,A: $tType,P2: A > $o,X: A,Y8: B,Y: B,R3: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P2 @ X )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y ) @ ( R3 @ X ) )
=> ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y8 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P2 @ R3 ) ) ) ) ).
% same_fstI
thf(fact_2370_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq @ nat @ Xa2 @ X )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa2 @ ( minus_minus @ nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa2 @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_2371_in__measure,axiom,
! [A: $tType,X: A,Y: A,F3: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measure @ A @ F3 ) )
= ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ).
% in_measure
thf(fact_2372_in__finite__psubset,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A5 @ B5 ) @ ( finite_psubset @ A ) )
= ( ( ord_less @ ( set @ A ) @ A5 @ B5 )
& ( finite_finite2 @ A @ B5 ) ) ) ).
% in_finite_psubset
thf(fact_2373_vebt__buildupi__refines,axiom,
! [N: nat] : ( refine_Imp_refines @ vEBT_VEBTi @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_V739175172307565963ildupi @ N ) ) ).
% vebt_buildupi_refines
thf(fact_2374_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N6: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I3: A] : ( plus_plus @ A @ I3 @ ( one_one @ A ) )
@ N6
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_2375_refines__replicate,axiom,
! [A: $tType,F3: heap_Time_Heap @ A,F8: heap_Time_Heap @ A,N: nat] :
( ( refine_Imp_refines @ A @ F3 @ F8 )
=> ( refine_Imp_refines @ ( list @ A ) @ ( vEBT_VEBT_replicatei @ A @ N @ F3 ) @ ( vEBT_VEBT_replicatei @ A @ N @ F8 ) ) ) ).
% refines_replicate
thf(fact_2376_vebt__deletei__refines,axiom,
! [Ti: vEBT_VEBTi,X: nat,T2: vEBT_VEBT] : ( refine_Imp_refines @ vEBT_VEBTi @ ( vEBT_vebt_deletei @ Ti @ X ) @ ( vEBT_V1365221501068881998eletei @ T2 @ Ti @ X ) ) ).
% vebt_deletei_refines
thf(fact_2377_vebt__inserti__refines,axiom,
! [Ti: vEBT_VEBTi,X: nat,T2: vEBT_VEBT] : ( refine_Imp_refines @ vEBT_VEBTi @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X ) ) ).
% vebt_inserti_refines
thf(fact_2378_refines__case__VEBTi,axiom,
! [A: $tType,Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: $o > $o > ( heap_Time_Heap @ A ),F12: $o > $o > ( heap_Time_Heap @ A ),F22: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > ( heap_Time_Heap @ A ),F23: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > ( heap_Time_Heap @ A )] :
( ( Ti = Ti2 )
=> ( ! [A4: $o,B4: $o] : ( refine_Imp_refines @ A @ ( F1 @ A4 @ B4 ) @ ( F12 @ A4 @ B4 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeArray: array @ vEBT_VEBTi,Summary2: vEBT_VEBTi] : ( refine_Imp_refines @ A @ ( F22 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( F23 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
=> ( refine_Imp_refines @ A @ ( vEBT_case_VEBTi @ ( heap_Time_Heap @ A ) @ F22 @ F1 @ Ti ) @ ( vEBT_case_VEBTi @ ( heap_Time_Heap @ A ) @ F23 @ F12 @ Ti2 ) ) ) ) ) ).
% refines_case_VEBTi
thf(fact_2379_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_2380_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_2381_upd__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,Xs: list @ A,A2: array @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( hoare_hoare_triple @ ( array @ A ) @ ( snga_assn @ A @ A2 @ Xs ) @ ( array_upd @ A @ I @ X @ A2 )
@ ^ [R5: array @ A] : ( times_times @ assn @ ( snga_assn @ A @ A2 @ ( list_update @ A @ Xs @ I @ X ) ) @ ( pure_assn @ ( R5 = A2 ) ) ) ) ) ) ).
% upd_rule
thf(fact_2382_product__concat__map,axiom,
! [B: $tType,A: $tType] :
( ( product @ A @ B )
= ( ^ [Xs2: list @ A,Ys3: list @ B] :
( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X4: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 ) @ Ys3 )
@ Xs2 ) ) ) ) ).
% product_concat_map
thf(fact_2383_VEBTi_Osize_I3_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: array @ vEBT_VEBTi,X14: vEBT_VEBTi] :
( ( size_size @ vEBT_VEBTi @ ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_array @ vEBT_VEBTi @ ( size_size @ vEBT_VEBTi ) @ X13 ) @ ( size_size @ vEBT_VEBTi @ X14 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% VEBTi.size(3)
thf(fact_2384_vebt__succi__refines,axiom,
! [Ti: vEBT_VEBTi,X: nat,T2: vEBT_VEBT] : ( refine_Imp_refines @ ( option @ nat ) @ ( vEBT_vebt_succi @ Ti @ X ) @ ( vEBT_VEBT_vebt_succi @ T2 @ Ti @ X ) ) ).
% vebt_succi_refines
thf(fact_2385_vebt__predi__refines,axiom,
! [Ti: vEBT_VEBTi,X: nat,T2: vEBT_VEBT] : ( refine_Imp_refines @ ( option @ nat ) @ ( vEBT_vebt_predi @ Ti @ X ) @ ( vEBT_VEBT_vebt_predi @ T2 @ Ti @ X ) ) ).
% vebt_predi_refines
thf(fact_2386_vebt__memberi__refines,axiom,
! [Ti: vEBT_VEBTi,X: nat,T2: vEBT_VEBT] : ( refine_Imp_refines @ $o @ ( vEBT_vebt_memberi @ Ti @ X ) @ ( vEBT_V854960066525838166emberi @ T2 @ Ti @ X ) ) ).
% vebt_memberi_refines
thf(fact_2387_VEBTi_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H2: A > B,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > A,F22: $o > $o > A,VEBTi: vEBT_VEBTi] :
( ( H2 @ ( vEBT_case_VEBTi @ A @ F1 @ F22 @ VEBTi ) )
= ( vEBT_case_VEBTi @ B
@ ^ [X16: option @ ( product_prod @ nat @ nat ),X24: nat,X32: array @ vEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X16 @ X24 @ X32 @ X42 ) )
@ ^ [X16: $o,X24: $o] : ( H2 @ ( F22 @ X16 @ X24 ) )
@ VEBTi ) ) ).
% VEBTi.case_distrib
thf(fact_2388_VEBTi_Osimps_I5_J,axiom,
! [A: $tType,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > A,F22: $o > $o > A,X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: array @ vEBT_VEBTi,X14: vEBT_VEBTi] :
( ( vEBT_case_VEBTi @ A @ F1 @ F22 @ ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 ) )
= ( F1 @ X11 @ X12 @ X13 @ X14 ) ) ).
% VEBTi.simps(5)
thf(fact_2389_VEBTi_Osize__gen_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: array @ vEBT_VEBTi,X14: vEBT_VEBTi] :
( ( vEBT_size_VEBTi @ ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_array @ vEBT_VEBTi @ vEBT_size_VEBTi @ X13 ) @ ( vEBT_size_VEBTi @ X14 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% VEBTi.size_gen(1)
thf(fact_2390_TBOUND__minNull,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] :
( ( vEBT_VEBT_minNull @ T2 )
=> ( time_TBOUND @ vEBT_VEBTi @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X ) @ ( one_one @ nat ) ) ) ).
% TBOUND_minNull
thf(fact_2391_listI__assn__def,axiom,
! [A: $tType,B: $tType] :
( ( vEBT_List_listI_assn @ A @ B )
= ( ^ [I8: set @ nat,A6: A > B > assn,Xs2: list @ A,Xsi3: list @ B] :
( times_times @ assn
@ ( pure_assn
@ ( ( ( size_size @ ( list @ B ) @ Xsi3 )
= ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ord_less_eq @ ( set @ nat ) @ I8 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) )
@ ( finite_fold @ nat @ assn
@ ^ [I3: nat,A8: assn] : ( times_times @ assn @ A8 @ ( A6 @ ( nth @ A @ Xs2 @ I3 ) @ ( nth @ B @ Xsi3 @ I3 ) ) )
@ ( one_one @ assn )
@ I8 ) ) ) ) ).
% listI_assn_def
thf(fact_2392_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S3 )
=> ( ( finite_finite2 @ B @ A5 )
=> ~ ! [L2: list @ B] :
( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ L2 ) )
=> ( ( ( set2 @ B @ L2 )
= A5 )
=> ( ( size_size @ ( list @ B ) @ L2 )
!= ( finite_card @ B @ A5 ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_2393_card__lists__length__le,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A5 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A5 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_2394_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A: $tType,A0: nat > A > A,A12: nat,A23: nat,A32: A,P2: ( nat > A > A ) > nat > nat > A > $o] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ A0 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A12 @ ( product_Pair @ nat @ A @ A23 @ A32 ) ) ) )
=> ( ! [F2: nat > A > A,A4: nat,B4: nat,Acc: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A4 @ ( product_Pair @ nat @ A @ B4 @ Acc ) ) ) )
=> ( ( ~ ( ord_less @ nat @ B4 @ A4 )
=> ( P2 @ F2 @ ( plus_plus @ nat @ A4 @ ( one_one @ nat ) ) @ B4 @ ( F2 @ A4 @ Acc ) ) )
=> ( P2 @ F2 @ A4 @ B4 @ Acc ) ) )
=> ( P2 @ A0 @ A12 @ A23 @ A32 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_2395_TBOUND__vebt__buildupi,axiom,
! [N: nat] : ( time_TBOUND @ vEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ N ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).
% TBOUND_vebt_buildupi
thf(fact_2396_TBOUND__replicate,axiom,
! [A: $tType,X: heap_Time_Heap @ A,C3: nat,N: nat] :
( ( time_TBOUND @ A @ X @ C3 )
=> ( time_TBOUND @ ( list @ A ) @ ( vEBT_VEBT_replicatei @ A @ N @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ C3 ) @ N ) ) ) ) ).
% TBOUND_replicate
thf(fact_2397_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_2398_fold__empty,axiom,
! [B: $tType,A: $tType,F3: B > A > A,Z: A] :
( ( finite_fold @ B @ A @ F3 @ Z @ ( bot_bot @ ( set @ B ) ) )
= Z ) ).
% fold_empty
thf(fact_2399_fold__infinite,axiom,
! [A: $tType,B: $tType,A5: set @ A,F3: A > B > B,Z: B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ A5 )
= Z ) ) ).
% fold_infinite
thf(fact_2400_finite__atMost,axiom,
! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_atMost @ nat @ K ) ) ).
% finite_atMost
thf(fact_2401_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X ) @ ( set_ord_atMost @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% atMost_subset_iff
thf(fact_2402_card__atMost,axiom,
! [U: nat] :
( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_2403_sum_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ).
% sum.atMost_Suc
thf(fact_2404_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_2405_fold__closed__eq,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B,F3: A > B > B,G3: A > B > B,Z: B] :
( ! [A4: A,B4: B] :
( ( member @ A @ A4 @ A5 )
=> ( ( member @ B @ B4 @ B5 )
=> ( ( F3 @ A4 @ B4 )
= ( G3 @ A4 @ B4 ) ) ) )
=> ( ! [A4: A,B4: B] :
( ( member @ A @ A4 @ A5 )
=> ( ( member @ B @ B4 @ B5 )
=> ( member @ B @ ( G3 @ A4 @ B4 ) @ B5 ) ) )
=> ( ( member @ B @ Z @ B5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ A5 )
= ( finite_fold @ A @ B @ G3 @ Z @ A5 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_2406_TBOUND__mono,axiom,
! [A: $tType,C3: heap_Time_Heap @ A,T2: nat,T8: nat] :
( ( time_TBOUND @ A @ C3 @ T2 )
=> ( ( ord_less_eq @ nat @ T2 @ T8 )
=> ( time_TBOUND @ A @ C3 @ T8 ) ) ) ).
% TBOUND_mono
thf(fact_2407_not__empty__eq__Iic__eq__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [H2: A] :
( ( bot_bot @ ( set @ A ) )
!= ( set_ord_atMost @ A @ H2 ) ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_2408_infinite__Iic,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [A2: A] :
~ ( finite_finite2 @ A @ ( set_ord_atMost @ A @ A2 ) ) ) ).
% infinite_Iic
thf(fact_2409_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U3: A] :
( collect @ A
@ ^ [X4: A] : ( ord_less_eq @ A @ X4 @ U3 ) ) ) ) ) ).
% atMost_def
thf(fact_2410_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( set_ord_atMost @ nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_2411_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost @ nat @ ( suc @ K ) )
= ( insert @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).
% atMost_Suc
thf(fact_2412_finite__nat__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [S6: set @ nat] :
? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S6 @ ( set_ord_atMost @ nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_2413_TBOUND__def,axiom,
! [A: $tType] :
( ( time_TBOUND @ A )
= ( ^ [M5: heap_Time_Heap @ A,T3: nat] :
! [H4: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ A @ M5 @ H4 ) @ T3 ) ) ) ).
% TBOUND_def
thf(fact_2414_TBOUNDI,axiom,
! [A: $tType,M: heap_Time_Heap @ A,T2: nat] :
( ! [H: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ A @ M @ H ) @ T2 )
=> ( time_TBOUND @ A @ M @ T2 ) ) ).
% TBOUNDI
thf(fact_2415_TBOUNDD,axiom,
! [A: $tType,M: heap_Time_Heap @ A,T2: nat,H2: heap_ext @ product_unit] :
( ( time_TBOUND @ A @ M @ T2 )
=> ( ord_less_eq @ nat @ ( time_time @ A @ M @ H2 ) @ T2 ) ) ).
% TBOUNDD
thf(fact_2416_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_2417_sum_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G3 @ ( suc @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_2418_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,I: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) )
@ ( set_ord_atMost @ nat @ I ) )
= ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ ( F3 @ ( suc @ I ) ) ) ) ) ).
% sum_telescope
thf(fact_2419_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_2420_polyfun__eq__coeffs,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat,D2: nat > A] :
( ( ! [X4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C3 @ I3 ) @ ( power_power @ A @ X4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( D2 @ I3 ) @ ( power_power @ A @ X4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
=> ( ( C3 @ I3 )
= ( D2 @ I3 ) ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_2421_Max_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A5 ) )
= ( finite_fold @ A @ A @ ( ord_max @ A ) @ X @ A5 ) ) ) ) ).
% Max.eq_fold
thf(fact_2422_polyfun__eq__0,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat] :
( ( ! [X4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C3 @ I3 ) @ ( power_power @ A @ X4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
=> ( ( C3 @ I3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_0
thf(fact_2423_zero__polynom__imp__zero__coeffs,axiom,
! [A: $tType] :
( ( ( ab_semigroup_mult @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [C3: nat > A,N: nat,K: nat] :
( ! [W: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C3 @ I3 ) @ ( power_power @ A @ W @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( C3 @ K )
= ( zero_zero @ A ) ) ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_2424_sum_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G3 @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_2425_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_basic
thf(fact_2426_polyfun__finite__roots,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C3 @ I3 ) @ ( power_power @ A @ X4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
= ( ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
& ( ( C3 @ I3 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_finite_roots
thf(fact_2427_polyfun__roots__finite,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,K: nat,N: nat] :
( ( ( C3 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C3 @ I3 ) @ ( power_power @ A @ Z2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_2428_polyfun__roots__card,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,K: nat,N: nat] :
( ( ( C3 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C3 @ I3 ) @ ( power_power @ A @ Z2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ).
% polyfun_roots_card
thf(fact_2429_polyfun__linear__factor__root,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C3: nat > A,A2: A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C3 @ I3 ) @ ( power_power @ A @ A2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ~ ! [B4: nat > A] :
~ ! [Z5: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C3 @ I3 ) @ ( power_power @ A @ Z5 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ Z5 @ A2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( B4 @ I3 ) @ ( power_power @ A @ Z5 @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_2430_polynomial__product,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [M: nat,A2: nat > A,N: nat,B2: nat > A,X: A] :
( ! [I2: nat] :
( ( ord_less @ nat @ M @ I2 )
=> ( ( A2 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B2 @ J2 )
= ( zero_zero @ A ) ) )
=> ( ( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( times_times @ A @ ( B2 @ J3 ) @ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [R5: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A2 @ K3 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ A @ X @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ) ).
% polynomial_product
thf(fact_2431_polyfun__rootbound,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,K: nat,N: nat] :
( ( ( C3 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C3 @ I3 ) @ ( power_power @ A @ Z2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
& ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C3 @ I3 ) @ ( power_power @ A @ Z2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ) ).
% polyfun_rootbound
thf(fact_2432_polynomial__product__nat,axiom,
! [M: nat,A2: nat > nat,N: nat,B2: nat > nat,X: nat] :
( ! [I2: nat] :
( ( ord_less @ nat @ M @ I2 )
=> ( ( A2 @ I2 )
= ( zero_zero @ nat ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B2 @ J2 )
= ( zero_zero @ nat ) ) )
=> ( ( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( times_times @ nat @ ( A2 @ I3 ) @ ( power_power @ nat @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( times_times @ nat @ ( B2 @ J3 ) @ ( power_power @ nat @ X @ J3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [R5: nat] :
( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( A2 @ K3 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ nat @ X @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_2433_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P3: nat,K: nat,G3: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P3 )
=> ( ( ord_less_eq @ nat @ K @ P3 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G3 @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( zero_zero @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P3 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G3 @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_2434_aux,axiom,
! [B: $tType,A: $tType,P2: A > B > assn,A2: A,As3: list @ A,C3: B,Cs: list @ B] :
( ( finite_fold @ nat @ assn
@ ^ [I3: nat,Aa3: assn] : ( times_times @ assn @ Aa3 @ ( P2 @ ( nth @ A @ ( cons @ A @ A2 @ As3 ) @ I3 ) @ ( nth @ B @ ( cons @ B @ C3 @ Cs ) @ I3 ) ) )
@ ( one_one @ assn )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ ( size_size @ ( list @ A ) @ As3 ) ) ) )
= ( times_times @ assn @ ( P2 @ A2 @ C3 )
@ ( finite_fold @ nat @ assn
@ ^ [I3: nat,Aa3: assn] : ( times_times @ assn @ Aa3 @ ( P2 @ ( nth @ A @ As3 @ I3 ) @ ( nth @ B @ Cs @ I3 ) ) )
@ ( one_one @ assn )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ As3 ) ) ) ) ) ).
% aux
thf(fact_2435_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
@ ^ [I3: nat] :
( times_times @ A
@ ( if @ A
@ ( I3
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ A2 )
@ ( if @ A @ ( I3 = N ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) )
@ ( power_power @ A @ Z @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% root_polyfun
thf(fact_2436_polyfun__diff__alt,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A2: nat > A,X: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ Y @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( A2 @ ( plus_plus @ nat @ ( plus_plus @ nat @ J3 @ K3 ) @ ( one_one @ nat ) ) ) @ ( power_power @ A @ Y @ K3 ) ) @ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ J3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_2437_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X: nat > A > A,Xa2: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa2 @ Xb @ Xc )
= Y )
=> ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa2 @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less @ nat @ Xb @ Xa2 )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa2 )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa2 @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa2 @ Xc ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa2 @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_2438_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F3: nat > A > A,A2: nat,B2: nat,Acc2: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F3 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A2 @ ( product_Pair @ nat @ A @ B2 @ Acc2 ) ) ) )
=> ( ( ( ord_less @ nat @ B2 @ A2 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F3 @ A2 @ B2 @ Acc2 )
= Acc2 ) )
& ( ~ ( ord_less @ nat @ B2 @ A2 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F3 @ A2 @ B2 @ Acc2 )
= ( set_fo6178422350223883121st_nat @ A @ F3 @ ( plus_plus @ nat @ A2 @ ( one_one @ nat ) ) @ B2 @ ( F3 @ A2 @ Acc2 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_2439_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A5: set @ B,L: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S3 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ L ) )
& ( ( set2 @ B @ L )
= A5 )
& ( ( size_size @ ( list @ B ) @ L )
= ( finite_card @ B @ A5 ) ) )
= ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_2440_polyfun__extremal__lemma,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [E3: real,C3: nat > A,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M11: real] :
! [Z5: A] :
( ( ord_less_eq @ real @ M11 @ ( real_V7770717601297561774m_norm @ A @ Z5 ) )
=> ( ord_less_eq @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C3 @ I3 ) @ ( power_power @ A @ Z5 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
@ ( times_times @ real @ E3 @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ Z5 ) @ ( suc @ N ) ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_2441_polyfun__diff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A2: nat > A,X: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ Y @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ J3 ) @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff
thf(fact_2442_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,X: B,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X @ A5 ) @ S3 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( remove1 @ B @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_remove
thf(fact_2443_TBOUND__vebt__minti,axiom,
! [T2: vEBT_VEBTi] : ( time_TBOUND @ ( option @ nat ) @ ( vEBT_vebt_minti @ T2 ) @ ( one_one @ nat ) ) ).
% TBOUND_vebt_minti
thf(fact_2444_TBOUND__vebt__maxti,axiom,
! [T2: vEBT_VEBTi] : ( time_TBOUND @ ( option @ nat ) @ ( vEBT_vebt_maxti @ T2 ) @ ( one_one @ nat ) ) ).
% TBOUND_vebt_maxti
thf(fact_2445_TBOUND__minNulli,axiom,
! [T2: vEBT_VEBTi] : ( time_TBOUND @ $o @ ( vEBT_VEBT_minNulli @ T2 ) @ ( one_one @ nat ) ) ).
% TBOUND_minNulli
thf(fact_2446_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_2447_Icc__eq__Icc,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,H2: A,L3: A,H3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ L @ H2 )
= ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) )
= ( ( ( L = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq @ A @ L @ H2 )
& ~ ( ord_less_eq @ A @ L3 @ H3 ) ) ) ) ) ).
% Icc_eq_Icc
thf(fact_2448_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_2449_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_2450_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_2451_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_2452_infinite__Icc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Icc_iff
thf(fact_2453_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D2 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_2454_atLeastAtMost__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
( ( set_or1337092689740270186AtMost @ A @ A2 @ A2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastAtMost_singleton
thf(fact_2455_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( insert @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( B2 = C3 ) ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_2456_image__Suc__atLeastAtMost,axiom,
! [I: nat,J: nat] :
( ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
= ( set_or1337092689740270186AtMost @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_2457_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H2: A,H3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atMost @ A @ H3 ) )
= ( ~ ( ord_less_eq @ A @ L @ H2 )
| ( ord_less_eq @ A @ H2 @ H3 ) ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_2458_card__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) )
= ( minus_minus @ nat @ ( suc @ U ) @ L ) ) ).
% card_atLeastAtMost
thf(fact_2459_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image @ A @ A @ ( times_times @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D2 @ A2 ) @ ( times_times @ A @ D2 @ B2 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_2460_sum_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G3: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_2461_infinite__Icc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Icc
thf(fact_2462_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
=> ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_2463_sum__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,A2: nat,B2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A8: nat] : ( plus_plus @ A @ ( F3 @ A8 ) )
@ A2
@ B2
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_2464_not__Iic__le__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H2: A,L3: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H2 ) @ ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) ) ) ).
% not_Iic_le_Icc
thf(fact_2465_all__nat__less,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_eq @ nat @ M5 @ N )
=> ( P2 @ M5 ) ) )
= ( ! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P2 @ X4 ) ) ) ) ).
% all_nat_less
thf(fact_2466_ex__nat__less,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_eq @ nat @ M5 @ N )
& ( P2 @ M5 ) ) )
= ( ? [X4: nat] :
( ( member @ nat @ X4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P2 @ X4 ) ) ) ) ).
% ex_nat_less
thf(fact_2467_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) )
= ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_2468_atMost__atLeast0,axiom,
( ( set_ord_atMost @ nat )
= ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% atMost_atLeast0
thf(fact_2469_complex__mod__minus__le__complex__mod,axiom,
! [X: complex] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( real_V7770717601297561774m_norm @ complex @ X ) ) @ ( real_V7770717601297561774m_norm @ complex @ X ) ) ).
% complex_mod_minus_le_complex_mod
thf(fact_2470_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_2471_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G3 @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_2472_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D2 ) )
= ( ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 )
& ( ( ord_less @ A @ C3 @ A2 )
| ( ord_less @ A @ B2 @ D2 ) ) ) )
& ( ord_less_eq @ A @ C3 @ D2 ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_2473_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_2474_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_2475_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_2476_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_2477_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu3: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_2478_subset__eq__atLeast0__atMost__finite,axiom,
! [N8: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N8 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N8 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_2479_minus__fold__remove,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( minus_minus @ ( set @ A ) @ B5 @ A5 )
= ( finite_fold @ A @ ( set @ A ) @ ( remove @ A ) @ B5 @ A5 ) ) ) ).
% minus_fold_remove
thf(fact_2480_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_2481_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_2482_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A8: A,B8: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A8 @ B8 ) @ ( insert @ A @ B8 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_2483_sum__shift__lb__Suc0__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,K: nat] :
( ( ( F3 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_2484_sum_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_2485_sum_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G3 @ ( suc @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_2486_sum_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G3 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_2487_sum_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G3 @ N ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% sum.last_plus
thf(fact_2488_image__Suc__lessThan,axiom,
! [N: nat] :
( ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ).
% image_Suc_lessThan
thf(fact_2489_image__Suc__atMost,axiom,
! [N: nat] :
( ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N ) ) ) ).
% image_Suc_atMost
thf(fact_2490_image__fold__insert,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( image @ A @ B @ F3 @ A5 )
= ( finite_fold @ A @ ( set @ B )
@ ^ [K3: A] : ( insert @ B @ ( F3 @ K3 ) )
@ ( bot_bot @ ( set @ B ) )
@ A5 ) ) ) ).
% image_fold_insert
thf(fact_2491_sum_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G3 @ M )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G3 @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_2492_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_2493_sum__Suc__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ I3 ) ) @ ( F3 @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F3 @ ( suc @ N ) ) @ ( F3 @ M ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_2494_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( A2 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sum.nested_swap
thf(fact_2495_sum_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( G3 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_2496_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( A2 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nested_swap'
thf(fact_2497_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A5: set @ B,B5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ S3 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 )
= ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ B5 ) )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( A5 = B5 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_inject
thf(fact_2498_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X: nat > A > A,Xa2: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa2 @ Xb @ Xc )
= Y )
=> ( ( ( ord_less @ nat @ Xb @ Xa2 )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa2 )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa2 @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa2 @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_2499_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F4: nat > A > A,A8: nat,B8: nat,Acc3: A] : ( if @ A @ ( ord_less @ nat @ B8 @ A8 ) @ Acc3 @ ( set_fo6178422350223883121st_nat @ A @ F4 @ ( plus_plus @ nat @ A8 @ ( one_one @ nat ) ) @ B8 @ ( F4 @ A8 @ Acc3 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_2500_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G3: nat > A,P3: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P3 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P3 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_2501_sum_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G3: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G3 @ N ) ) ) ) ) ) ).
% sum.head_if
thf(fact_2502_lemma__NBseq__def2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X6: A > B] :
( ( ? [K5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K5 )
& ! [N6: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X6 @ N6 ) ) @ K5 ) ) )
= ( ? [N7: nat] :
! [N6: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X6 @ N6 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N7 ) ) ) ) ) ) ).
% lemma_NBseq_def2
thf(fact_2503_lemma__NBseq__def,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X6: A > B] :
( ( ? [K5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K5 )
& ! [N6: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X6 @ N6 ) ) @ K5 ) ) )
= ( ? [N7: nat] :
! [N6: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X6 @ N6 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N7 ) ) ) ) ) ) ).
% lemma_NBseq_def
thf(fact_2504_sum__up__index__split,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_atMost @ nat @ M ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ).
% sum_up_index_split
thf(fact_2505_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_2506_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,X: A,Y: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C3 @ X ) @ ( times_times @ A @ C3 @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( image @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C3 @ Y ) @ ( times_times @ A @ C3 @ X ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_2507_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,C3: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C3 ) @ ( times_times @ A @ Y @ C3 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C3 ) @ ( times_times @ A @ X @ C3 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_2508_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X4 ) @ C3 )
@ ( 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
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X4 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X4 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_2509_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X4 ) @ C3 )
@ ( 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
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X4 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C3 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X4 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C3 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_2510_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S3 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( set2 @ B @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) )
= A5 ) ) ) ) ).
% folding_insort_key.set_sorted_key_list_of_set
thf(fact_2511_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S3 )
=> ( ( size_size @ ( list @ B ) @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) )
= ( finite_card @ B @ A5 ) ) ) ) ).
% folding_insort_key.length_sorted_key_list_of_set
thf(fact_2512_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S3 )
=> ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_2513_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S3 )
=> ( sorted_wrt @ A @ Less_eq @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_2514_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S3 )
=> ( distinct @ A @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_2515_sum__natinterval__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F3: nat > A] :
( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F3 @ M ) @ ( F3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_2516_sum__telescope_H_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
= ( minus_minus @ A @ ( F3 @ N ) @ ( F3 @ M ) ) ) ) ) ).
% sum_telescope''
thf(fact_2517_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_2518_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_2519_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_2520_polyfun__eq__const,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat,K: A] :
( ( ! [X4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C3 @ I3 ) @ ( power_power @ A @ X4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= K ) )
= ( ( ( C3 @ ( zero_zero @ nat ) )
= K )
& ! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) )
=> ( ( C3 @ X4 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_const
thf(fact_2521_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,Xs: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( set2 @ B @ Xs ) @ S3 )
=> ( ( sorted_wrt @ A @ Less_eq @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( distinct @ B @ Xs )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( set2 @ B @ Xs ) )
= Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_2522_norm__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( zero_zero @ real ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% norm_le_zero_iff
thf(fact_2523_zero__less__norm__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
= ( X
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_norm_iff
thf(fact_2524_norm__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( zero_zero @ A ) )
= ( zero_zero @ real ) ) ) ).
% norm_zero
thf(fact_2525_norm__eq__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ( real_V7770717601297561774m_norm @ A @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% norm_eq_zero
thf(fact_2526_norm__power__diff,axiom,
! [A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z: A,W2: A,M: nat] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ W2 ) @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( power_power @ A @ Z @ M ) @ ( power_power @ A @ W2 @ M ) ) ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Z @ W2 ) ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_2527_sum__norm__bound,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [S3: set @ B,F3: B > A,K4: real] :
( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ X3 ) ) @ K4 ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S3 ) ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( finite_card @ B @ S3 ) ) @ K4 ) ) ) ) ).
% sum_norm_bound
thf(fact_2528_range__subset__card,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,C3: word @ A,D2: word @ A] :
( ( ord_less_eq @ ( set @ ( word @ A ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ C3 @ D2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( word @ A ) @ C3 @ D2 )
& ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) @ ( minus_minus @ ( word @ A ) @ D2 @ C3 ) ) ) ) ) ) ).
% range_subset_card
thf(fact_2529_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_2530_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I3: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I3 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I3 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_2531_aset_I7_J,axiom,
! [D5: int,A5: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ! [X5: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa4 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X5 )
=> ( ord_less @ int @ T2 @ ( plus_plus @ int @ X5 @ D5 ) ) ) ) ) ).
% aset(7)
thf(fact_2532_aset_I5_J,axiom,
! [D5: int,T2: int,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ T2 @ A5 )
=> ! [X5: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa4 ) ) ) )
=> ( ( ord_less @ int @ X5 @ T2 )
=> ( ord_less @ int @ ( plus_plus @ int @ X5 @ D5 ) @ T2 ) ) ) ) ) ).
% aset(5)
thf(fact_2533_aset_I4_J,axiom,
! [D5: int,T2: int,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ T2 @ A5 )
=> ! [X5: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa4 ) ) ) )
=> ( ( X5 != T2 )
=> ( ( plus_plus @ int @ X5 @ D5 )
!= T2 ) ) ) ) ) ).
% aset(4)
thf(fact_2534_aset_I3_J,axiom,
! [D5: int,T2: int,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A5 )
=> ! [X5: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa4 ) ) ) )
=> ( ( X5 = T2 )
=> ( ( plus_plus @ int @ X5 @ D5 )
= T2 ) ) ) ) ) ).
% aset(3)
thf(fact_2535_bset_I7_J,axiom,
! [D5: int,T2: int,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ T2 @ B5 )
=> ! [X5: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa4 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X5 )
=> ( ord_less @ int @ T2 @ ( minus_minus @ int @ X5 @ D5 ) ) ) ) ) ) ).
% bset(7)
thf(fact_2536_bset_I5_J,axiom,
! [D5: int,B5: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ! [X5: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa4 ) ) ) )
=> ( ( ord_less @ int @ X5 @ T2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X5 @ D5 ) @ T2 ) ) ) ) ).
% bset(5)
thf(fact_2537_bset_I4_J,axiom,
! [D5: int,T2: int,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ T2 @ B5 )
=> ! [X5: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa4 ) ) ) )
=> ( ( X5 != T2 )
=> ( ( minus_minus @ int @ X5 @ D5 )
!= T2 ) ) ) ) ) ).
% bset(4)
thf(fact_2538_bset_I3_J,axiom,
! [D5: int,T2: int,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B5 )
=> ! [X5: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa4 ) ) ) )
=> ( ( X5 = T2 )
=> ( ( minus_minus @ int @ X5 @ D5 )
= T2 ) ) ) ) ) ).
% bset(3)
thf(fact_2539_periodic__finite__ex,axiom,
! [D2: int,P2: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ( ? [X8: int] : ( P2 @ X8 ) )
= ( ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
& ( P2 @ X4 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_2540_word__subset__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,R2: word @ A,Y: word @ A,S: word @ A] :
( ( ord_less_eq @ ( set @ ( word @ A ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ R2 ) @ ( one_one @ ( word @ A ) ) ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ Y @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y @ S ) @ ( one_one @ ( word @ A ) ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ R2 ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y @ S ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( S
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less_eq @ ( word @ A ) @ R2 @ S ) ) ) ) ) ) ).
% word_subset_less
thf(fact_2541_aset_I8_J,axiom,
! [D5: int,A5: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ! [X5: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa4 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X5 )
=> ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X5 @ D5 ) ) ) ) ) ).
% aset(8)
thf(fact_2542_aset_I6_J,axiom,
! [D5: int,T2: int,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A5 )
=> ! [X5: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X5
!= ( minus_minus @ int @ Xb3 @ Xa4 ) ) ) )
=> ( ( ord_less_eq @ int @ X5 @ T2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X5 @ D5 ) @ T2 ) ) ) ) ) ).
% aset(6)
thf(fact_2543_bset_I8_J,axiom,
! [D5: int,T2: int,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B5 )
=> ! [X5: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa4 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X5 )
=> ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X5 @ D5 ) ) ) ) ) ) ).
% bset(8)
thf(fact_2544_bset_I6_J,axiom,
! [D5: int,B5: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ! [X5: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X5
!= ( plus_plus @ int @ Xb3 @ Xa4 ) ) ) )
=> ( ( ord_less_eq @ int @ X5 @ T2 )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X5 @ D5 ) @ T2 ) ) ) ) ).
% bset(6)
thf(fact_2545_cpmi,axiom,
! [D5: int,P2: int > $o,P6: int > $o,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z5 )
=> ( ( P2 @ X3 )
= ( P6 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ B5 )
=> ( X3
!= ( plus_plus @ int @ Xb4 @ Xa ) ) ) )
=> ( ( P2 @ X3 )
=> ( P2 @ ( minus_minus @ int @ X3 @ D5 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P6 @ X3 )
= ( P6 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D5 ) ) ) )
=> ( ( ? [X8: int] : ( P2 @ X8 ) )
= ( ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
& ( P6 @ X4 ) )
| ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
& ? [Y5: int] :
( ( member @ int @ Y5 @ B5 )
& ( P2 @ ( plus_plus @ int @ Y5 @ X4 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_2546_cppi,axiom,
! [D5: int,P2: int > $o,P6: int > $o,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ Z5 @ X3 )
=> ( ( P2 @ X3 )
= ( P6 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ A5 )
=> ( X3
!= ( minus_minus @ int @ Xb4 @ Xa ) ) ) )
=> ( ( P2 @ X3 )
=> ( P2 @ ( plus_plus @ int @ X3 @ D5 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P6 @ X3 )
= ( P6 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D5 ) ) ) )
=> ( ( ? [X8: int] : ( P2 @ X8 ) )
= ( ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
& ( P6 @ X4 ) )
| ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
& ? [Y5: int] :
( ( member @ int @ Y5 @ A5 )
& ( P2 @ ( minus_minus @ int @ Y5 @ X4 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_2547_norm__not__less__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
~ ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( zero_zero @ real ) ) ) ).
% norm_not_less_zero
thf(fact_2548_norm__ge__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) ) ) ).
% norm_ge_zero
thf(fact_2549_sum__norm__le,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [S3: set @ B,F3: B > A,G3: B > real] :
( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ X3 ) ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S3 ) ) @ ( groups7311177749621191930dd_sum @ B @ real @ G3 @ S3 ) ) ) ) ).
% sum_norm_le
thf(fact_2550_norm__sum,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,A5: set @ B] :
( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) )
@ ( groups7311177749621191930dd_sum @ B @ real
@ ^ [I3: B] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ I3 ) )
@ A5 ) ) ) ).
% norm_sum
thf(fact_2551_power__eq__imp__eq__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W2: A,N: nat,Z: A] :
( ( ( power_power @ A @ W2 @ N )
= ( power_power @ A @ Z @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( real_V7770717601297561774m_norm @ A @ W2 )
= ( real_V7770717601297561774m_norm @ A @ Z ) ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_2552_norm__mult__less,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X: A,R2: real,Y: A,S: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ R2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X @ Y ) ) @ ( times_times @ real @ R2 @ S ) ) ) ) ) ).
% norm_mult_less
thf(fact_2553_norm__mult__ineq,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X @ Y ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) ) ) ).
% norm_mult_ineq
thf(fact_2554_norm__triangle__lt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E3: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ E3 ) ) ) ).
% norm_triangle_lt
thf(fact_2555_norm__add__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,R2: real,Y: A,S: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ R2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ real @ R2 @ S ) ) ) ) ) ).
% norm_add_less
thf(fact_2556_norm__add__leD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A,C3: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ C3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ C3 ) ) ) ) ).
% norm_add_leD
thf(fact_2557_norm__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E3: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ E3 ) ) ) ).
% norm_triangle_le
thf(fact_2558_norm__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) ) ) ).
% norm_triangle_ineq
thf(fact_2559_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_2560_norm__diff__triangle__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E1: real,Z: A,E22: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y ) ) @ E1 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ Z ) ) @ E22 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Z ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_2561_norm__power__ineq,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: A,N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X @ N ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ N ) ) ) ).
% norm_power_ineq
thf(fact_2562_norm__triangle__sub,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y ) ) ) ) ) ).
% norm_triangle_sub
thf(fact_2563_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_2564_norm__diff__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E1: real,Z: A,E22: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y ) ) @ E1 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ Z ) ) @ E22 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Z ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_2565_norm__triangle__le__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E3: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y ) ) @ E3 ) ) ) ).
% norm_triangle_le_diff
thf(fact_2566_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_2567_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_2568_power__eq__1__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W2: A,N: nat] :
( ( ( power_power @ A @ W2 @ N )
= ( one_one @ A ) )
=> ( ( ( real_V7770717601297561774m_norm @ A @ W2 )
= ( one_one @ real ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% power_eq_1_iff
thf(fact_2569_norm__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C3 @ D2 ) ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ C3 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_2570_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,Mm: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( F3 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ Mm ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_2571_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,X: B,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X @ A5 ) @ S3 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( insert @ B @ X @ A5 ) )
= ( insort_key @ A @ B @ Less_eq @ F3 @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert_remove
thf(fact_2572_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N: nat,M: nat,X: A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_2573_card__lists__distinct__length__eq,axiom,
! [A: $tType,A5: set @ A,K: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A5 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K )
& ( distinct @ A @ Xs2 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A5 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A5 ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_2574_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K: nat,A5: set @ A] :
( ( ord_less @ nat @ K @ ( finite_card @ A @ A5 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K )
& ( distinct @ A @ Xs2 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A5 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A5 ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_2575_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,X: B,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X @ A5 ) @ S3 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ~ ( member @ B @ X @ A5 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( insert @ B @ X @ A5 ) )
= ( insort_key @ A @ B @ Less_eq @ F3 @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert
thf(fact_2576_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_2577_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ C3 )
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% divide_cancel_right
thf(fact_2578_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ( divide_divide @ A @ C3 @ A2 )
= ( divide_divide @ A @ C3 @ B2 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% divide_cancel_left
thf(fact_2579_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_2580_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_2581_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_2582_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_2583_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_2584_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_2585_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_2586_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_2587_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_2588_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_2589_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_2590_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_2591_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_2592_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_2593_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_2594_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_2595_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_2596_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_2597_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_2598_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ( C3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_2599_prod__zero__iff,axiom,
! [A: $tType,B: $tType] :
( ( semidom @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 )
= ( zero_zero @ A ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( ( F3 @ X4 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% prod_zero_iff
thf(fact_2600_prod_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: B > A] :
( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty
thf(fact_2601_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G3: B > A] :
( ~ ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 )
= ( one_one @ A ) ) ) ) ).
% prod.infinite
thf(fact_2602_prod__eq__1__iff,axiom,
! [A: $tType,A5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( groups7121269368397514597t_prod @ A @ nat @ F3 @ A5 )
= ( one_one @ nat ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ( F3 @ X4 )
= ( one_one @ nat ) ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_2603_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_2604_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_2605_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_2606_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_2607_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_2608_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_2609_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_2610_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_2611_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_2612_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_2613_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_2614_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_2615_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ~ ( member @ B @ X @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( times_times @ A @ ( G3 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) ) ) ) ) ).
% prod.insert
thf(fact_2616_prod_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( G3 @ N ) ) ) ) ).
% prod.lessThan_Suc
thf(fact_2617_prod_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ).
% prod.atMost_Suc
thf(fact_2618_prod__pos__nat__iff,axiom,
! [A: $tType,A5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F3 @ A5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F3 @ X4 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_2619_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_2620_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_2621_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_2622_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_2623_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image @ A @ A
@ ^ [C7: A] : ( divide_divide @ A @ C7 @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A2 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_2624_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G3: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G3 @ N ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_2625_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G3: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_2626_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A5: set @ nat,C3: nat > A,D2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A5 )
& ( member @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C3 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) @ ( D2 @ I3 ) )
@ A5 )
= ( divide_divide @ A @ ( C3 @ ( zero_zero @ nat ) ) @ ( D2 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A5 )
& ( member @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C3 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) @ ( D2 @ I3 ) )
@ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_2627_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_2628_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_2629_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_2630_norm__prod__le,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [F3: B > A,A5: set @ B] :
( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) )
@ ( groups7121269368397514597t_prod @ B @ real
@ ^ [A8: B] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ A8 ) )
@ A5 ) ) ) ).
% norm_prod_le
thf(fact_2631_word__div__mult__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ A2 @ B2 ) @ B2 ) @ A2 ) ) ).
% word_div_mult_le
thf(fact_2632_div__to__mult__word__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( divide_divide @ ( word @ A ) @ Y @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ X @ Z ) @ Y ) ) ) ).
% div_to_mult_word_lt
thf(fact_2633_prod_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B5: set @ C,G3: B > C > A,R3: B > C > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ C @ B5 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] :
( groups7121269368397514597t_prod @ C @ A @ ( G3 @ X4 )
@ ( collect @ C
@ ^ [Y5: C] :
( ( member @ C @ Y5 @ B5 )
& ( R3 @ X4 @ Y5 ) ) ) )
@ A5 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y5: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( G3 @ X4 @ Y5 )
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( R3 @ X4 @ Y5 ) ) ) )
@ B5 ) ) ) ) ) ).
% prod.swap_restrict
thf(fact_2634_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) ) ) ) ).
% prod_nonneg
thf(fact_2635_prod__mono,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F3: B > A,G3: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) )
& ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( G3 @ I2 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) ) ) ).
% prod_mono
thf(fact_2636_prod__pos,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) ) ) ) ).
% prod_pos
thf(fact_2637_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A5: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) ) ) ) ).
% prod_ge_1
thf(fact_2638_prod__zero,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ? [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( ( F3 @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% prod_zero
thf(fact_2639_divide__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( divide_divide @ A @ A2 @ C3 ) ) ) ) ) ).
% divide_right_mono_neg
thf(fact_2640_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_2641_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_nonneg
thf(fact_2642_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_nonpos
thf(fact_2643_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_2644_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_2645_divide__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% divide_right_mono
thf(fact_2646_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_2647_divide__neg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_neg_neg
thf(fact_2648_divide__neg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_neg_pos
thf(fact_2649_divide__pos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_pos_neg
thf(fact_2650_divide__pos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_pos_pos
thf(fact_2651_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_2652_divide__less__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) )
& ( C3
!= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_cancel
thf(fact_2653_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_2654_divide__strict__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( divide_divide @ A @ A2 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% divide_strict_right_mono
thf(fact_2655_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ A2 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_2656_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_2657_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( A2
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( times_times @ A @ A2 @ C3 )
= B2 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_2658_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ C3 )
= A2 )
= ( B2
= ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_2659_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C3 )
= B2 )
=> ( A2
= ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_2660_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( B2
= ( times_times @ A @ A2 @ C3 ) )
=> ( ( divide_divide @ A @ B2 @ C3 )
= A2 ) ) ) ) ).
% divide_eq_imp
thf(fact_2661_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ C3 )
= B2 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_2662_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ( divide_divide @ A @ B2 @ C3 )
= A2 )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_2663_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X @ Y )
= ( divide_divide @ A @ W2 @ Z ) )
= ( ( times_times @ A @ X @ Z )
= ( times_times @ A @ W2 @ Y ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2664_real__of__nat__div2,axiom,
! [N: nat,X: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_2665_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_2666_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_2667_real__of__nat__div3,axiom,
! [N: nat,X: nat] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) ) @ ( one_one @ real ) ) ).
% real_of_nat_div3
thf(fact_2668_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_2669_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_2670_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_2671_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G3: B > A,P2: B > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( P2 @ X4 ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P2 @ X4 ) @ ( G3 @ X4 ) @ ( one_one @ A ) )
@ A5 ) ) ) ) ).
% prod.inter_filter
thf(fact_2672_prod_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,H2: B > A,G3: B > C] :
( ( finite_finite2 @ B @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y5: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S3 )
& ( ( G3 @ X4 )
= Y5 ) ) ) )
@ ( image @ B @ C @ G3 @ S3 ) ) ) ) ) ).
% prod.image_gen
thf(fact_2673_More__Word_Oword__div__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [C3: word @ A,A2: word @ A,B2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ C3 )
=> ( ( ord_less @ ( word @ A ) @ A2 @ ( times_times @ ( word @ A ) @ B2 @ C3 ) )
=> ( ord_less @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ A2 @ C3 ) @ B2 ) ) ) ) ).
% More_Word.word_div_mult
thf(fact_2674_div__lt__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X ) )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
=> ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ X ) @ K ) ) ) ) ).
% div_lt_mult
thf(fact_2675_word__div__lt__eq__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ( divide_divide @ ( word @ A ) @ X @ Y )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_div_lt_eq_0
thf(fact_2676_word__less__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( divide_divide @ ( word @ A ) @ X @ Y )
= ( zero_zero @ ( word @ A ) ) )
=> ( ( Y
= ( zero_zero @ ( word @ A ) ) )
| ( ord_less @ ( word @ A ) @ X @ Y ) ) ) ) ).
% word_less_div
thf(fact_2677_word__div__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,V3: word @ A] :
( ( ord_less @ ( word @ A ) @ W2 @ V3 )
=> ( ( divide_divide @ ( word @ A ) @ W2 @ V3 )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_div_less
thf(fact_2678_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A5: set @ B,F3: B > A,A2: B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( F3 @ A2 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A2 @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( F3 @ A2 ) ) ) )
& ( ~ ( member @ B @ A2 @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_2679_prod_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G3 @ ( suc @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_2680_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G3 @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_2681_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A5: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) )
& ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_2682_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R3: A > A > $o,S3: set @ B,H2: B > A,G3: B > A] :
( ( R3 @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X15: A,Y15: A,X23: A,Y23: A] :
( ( ( R3 @ X15 @ X23 )
& ( R3 @ Y15 @ Y23 ) )
=> ( R3 @ ( times_times @ A @ X15 @ Y15 ) @ ( times_times @ A @ X23 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( R3 @ ( H2 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( R3 @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ S3 ) ) ) ) ) ) ) ).
% prod.related
thf(fact_2683_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A2: B,C3: B,B2: B,D2: B,G3: B > A,H2: B > A] :
( ( A2 = C3 )
=> ( ( B2 = D2 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C3 @ X3 )
=> ( ( ord_less @ B @ X3 @ D2 )
=> ( ( G3 @ X3 )
= ( H2 @ X3 ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C3 @ D2 ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_2684_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( member @ B @ X @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) )
& ( ~ ( member @ B @ X @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( times_times @ A @ ( G3 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_2685_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S7: set @ B,T6: set @ C,S3: set @ B,I: C > B,J: B > C,T4: set @ C,G3: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S7 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) )
=> ( member @ C @ ( J @ A4 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( member @ B @ ( I @ B4 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S7 )
=> ( ( G3 @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T6 )
=> ( ( H2 @ B4 )
= ( one_one @ A ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S3 )
=> ( ( H2 @ ( J @ A4 ) )
= ( G3 @ A4 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ S3 )
= ( groups7121269368397514597t_prod @ C @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_2686_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,P3: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P3 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ P3 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ P3 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_2687_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,W2: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_le
thf(fact_2688_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,W2: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_less
thf(fact_2689_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,W2: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less @ A @ W2 @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_2690_divide__le__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% divide_le_cancel
thf(fact_2691_divide__nonneg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_neg
thf(fact_2692_divide__nonneg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonneg_pos
thf(fact_2693_divide__nonpos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonpos_neg
thf(fact_2694_divide__nonpos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_pos
thf(fact_2695_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_2696_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_2697_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2698_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_2699_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_2700_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_2701_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_2702_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_2703_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_2704_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A2 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_2705_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_2706_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ X @ ( times_times @ A @ Z @ Y ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ Z ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_2707_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ ( times_times @ A @ Z @ Y ) @ X )
=> ( ord_less @ A @ Z @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_2708_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C3 @ A2 ) @ ( divide_divide @ A @ C3 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_2709_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C3 @ A2 ) @ ( divide_divide @ A @ C3 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_2710_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% divide_add_eq_iff
thf(fact_2711_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z ) @ Y ) @ Z ) ) ) ) ).
% add_divide_eq_iff
thf(fact_2712_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z @ ( divide_divide @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z @ Y ) ) @ Y ) ) ) ) ).
% add_num_frac
thf(fact_2713_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,X: A,Z: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ Z )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z @ Y ) ) @ Y ) ) ) ) ).
% add_frac_num
thf(fact_2714_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_2715_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_2716_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_2717_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_2718_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_2719_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Z ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ X @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_2720_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X @ ( divide_divide @ A @ Y @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ Y ) @ Z ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_2721_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_2722_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_2723_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_2724_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( C3
= ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) )
= ( ( times_times @ A @ C3 @ B2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_2725_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= C3 )
= ( ( uminus_uminus @ A @ A2 )
= ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_2726_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) )
= A2 )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B2 )
= ( times_times @ A @ A2 @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_2727_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2
= ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ C3 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_2728_prod_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T4: set @ C,G3: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G3 @ S3 ) @ T4 )
=> ( ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y5: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S3 )
& ( ( G3 @ X4 )
= Y5 ) ) ) )
@ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% prod.group
thf(fact_2729_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3
@ ( minus_minus @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [X4: B] :
( ( G3 @ X4 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_2730_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_2731_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_2732_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_2733_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_2734_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_2735_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_2736_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_2737_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_2738_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_2739_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_2740_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_2741_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_2742_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_2743_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_2744_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ ( one_one @ int ) @ K )
=> ( ord_less @ int @ ( divide_divide @ int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_2745_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G3 @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_2746_div__less__dividend__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: word @ A] :
( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( N
!= ( one_one @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ N ) @ X ) ) ) ) ).
% div_less_dividend_word
thf(fact_2747_div__le__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X ) )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
=> ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ X ) @ K ) ) ) ) ).
% div_le_mult
thf(fact_2748_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,I: A,F3: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( member @ A @ I @ I6 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F3 @ I ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F3 @ I2 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ I6 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_2749_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F3 @ I2 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ I6 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_2750_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S3: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G3 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_2751_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S3: set @ B,H2: B > A,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( H2 @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G3 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_2752_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S3: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ G3 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_2753_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S3: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ G3 @ T4 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_2754_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C4: set @ B,A5: set @ B,B5: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C4 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C4 @ A5 ) )
=> ( ( G3 @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C4 @ B5 ) )
=> ( ( H2 @ B4 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ C4 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C4 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B5 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_2755_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C4: set @ B,A5: set @ B,B5: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C4 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C4 @ A5 ) )
=> ( ( G3 @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C4 @ B5 ) )
=> ( ( H2 @ B4 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B5 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ C4 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C4 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_2756_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B5: set @ B,A5: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B5 @ A5 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ B5 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B5 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_2757_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G3 @ N ) ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_2758_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G3: nat > A] :
( ( ord_less @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( times_times @ A @ ( G3 @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_2759_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_2760_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat,B2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G3 @ B2 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_2761_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( G3 @ ( suc @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_2762_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G3 @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_2763_prod_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G3 @ N ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% prod.last_plus
thf(fact_2764_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_2765_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_2766_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C3 @ A2 ) @ ( divide_divide @ A @ C3 @ B2 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_2767_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ Y ) @ X )
=> ( ord_less_eq @ A @ Z @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_2768_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X @ ( times_times @ A @ Z @ Y ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ Z ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_2769_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_2770_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_2771_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_2772_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_2773_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C3 @ A2 ) @ ( divide_divide @ A @ C3 @ B2 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_2774_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_2775_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_2776_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_2777_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_2778_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_2779_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_2780_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_2781_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_2782_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_2783_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_2784_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_2785_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z ) ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_2786_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_2787_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_2788_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z ) ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_2789_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_2790_norm__prod__diff,axiom,
! [A: $tType,I5: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [I6: set @ I5,Z: I5 > A,W2: I5 > A] :
( ! [I2: I5] :
( ( member @ I5 @ I2 @ I6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( Z @ I2 ) ) @ ( one_one @ real ) ) )
=> ( ! [I2: I5] :
( ( member @ I5 @ I2 @ I6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( W2 @ I2 ) ) @ ( one_one @ real ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( groups7121269368397514597t_prod @ I5 @ A @ Z @ I6 ) @ ( groups7121269368397514597t_prod @ I5 @ A @ W2 @ I6 ) ) )
@ ( groups7311177749621191930dd_sum @ I5 @ real
@ ^ [I3: I5] : ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( Z @ I3 ) @ ( W2 @ I3 ) ) )
@ I6 ) ) ) ) ) ).
% norm_prod_diff
thf(fact_2791_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G3 @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_2792_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) )
= ( times_times @ A @ ( G3 @ M )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G3 @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_2793_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G3 @ ( suc @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_2794_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_2795_prod_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_2796_msrevs_I1_J,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N ) @ M ) @ N )
= ( plus_plus @ nat @ ( divide_divide @ nat @ M @ N ) @ K ) ) ) ).
% msrevs(1)
thf(fact_2797_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( A2 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% prod.nested_swap
thf(fact_2798_prod_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( G3 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_2799_verit__less__mono__div__int2,axiom,
! [A5: int,B5: int,N: int] :
( ( ord_less_eq @ int @ A5 @ B5 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ B5 @ N ) @ ( divide_divide @ int @ A5 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_2800_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_2801_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( A2 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nested_swap'
thf(fact_2802_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F3: B > A,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) )
& ( ord_less @ A @ ( F3 @ I2 ) @ ( G3 @ I2 ) ) ) )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_2803_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( member @ B @ X @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 )
= ( times_times @ A @ ( G3 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_2804_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G3: B > A,X: B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( times_times @ A @ ( G3 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_2805_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G3: nat > A,P3: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P3 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P3 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_2806_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G3: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G3 @ N ) ) ) ) ) ) ).
% prod.head_if
thf(fact_2807_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A2: B,B2: B > A,C3: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C3 @ K3 ) )
@ S3 )
= ( times_times @ A @ ( B2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C3 @ K3 ) )
@ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_2808_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V3: A,R2: A,S: A] :
( ( ord_less_eq @ A @ U @ V3 )
=> ( ( 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 @ V3 @ U ) ) @ S ) ) @ V3 ) ) ) ) ) ).
% scaling_mono
thf(fact_2809_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_2810_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_2811_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_2812_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_2813_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_2814_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_2815_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
=> ~ ! [N2: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) ) @ E3 ) ) ) ).
% nat_approx_posE
thf(fact_2816_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_2817_prod_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G3 @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_2818_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_2819_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_2820_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_2821_split__zdiv,axiom,
! [P2: int > $o,N: int,K: int] :
( ( P2 @ ( divide_divide @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P2 @ ( zero_zero @ int ) ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I3: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P2 @ I3 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I3: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P2 @ I3 ) ) ) ) ) ).
% split_zdiv
thf(fact_2822_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F3: B > A,N: A,K: nat] :
( ! [I2: B] :
( ( member @ B @ I2 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) )
& ( ord_less_eq @ A @ ( F3 @ I2 ) @ N ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A5 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( power_power @ A @ N @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_2823_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B5: set @ A,A5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ ( minus_minus @ ( set @ A ) @ B5 @ A5 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F3 @ B4 ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ A4 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ B5 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_2824_word__div__sub,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ X )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y )
=> ( ( divide_divide @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ Y )
= ( minus_minus @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ Y ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_div_sub
thf(fact_2825_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A2: B,B2: B > A,C3: A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ C3 )
@ S3 )
= ( times_times @ A @ ( B2 @ A2 ) @ ( power_power @ A @ C3 @ ( minus_minus @ nat @ ( finite_card @ B @ S3 ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ C3 )
@ S3 )
= ( power_power @ A @ C3 @ ( finite_card @ B @ S3 ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_2826_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A5: set @ B,F3: B > A,K4: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( divide_divide @ A @ K4 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) ) ) )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ K4 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_2827_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_2828_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_2829_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M ) @ C3 )
@ ( 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
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C3 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C3 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_2830_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M ) @ C3 )
@ ( 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
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_2831_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P3: nat,K: nat,G3: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P3 )
=> ( ( ord_less_eq @ nat @ K @ P3 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G3 @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P3 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G3 @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_2832_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp0
thf(fact_2833_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_2834_int__div__minus__is__minus1,axiom,
! [A2: int,B2: int] :
( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ( divide_divide @ int @ A2 @ B2 )
= ( uminus_uminus @ int @ A2 ) )
= ( B2
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% int_div_minus_is_minus1
thf(fact_2835_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C3 ) @ A2 ) @ B2 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self4
thf(fact_2836_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C3 @ B2 ) @ A2 ) @ B2 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self3
thf(fact_2837_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) ) @ B2 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self2
thf(fact_2838_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C3 @ B2 ) ) @ B2 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self1
thf(fact_2839_int__div__same__is__1,axiom,
! [A2: int,B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ( divide_divide @ int @ A2 @ B2 )
= A2 )
= ( B2
= ( one_one @ int ) ) ) ) ).
% int_div_same_is_1
thf(fact_2840_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% div_mult_mult1
thf(fact_2841_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% div_mult_mult2
thf(fact_2842_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ( C3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_2843_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= M ) ).
% div_by_Suc_0
thf(fact_2844_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_2845_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_2846_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_2847_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_2848_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_2849_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_2850_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_2851_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_2852_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_2853_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_2854_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_2855_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_2856_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_2857_div__mult__le,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A2 @ B2 ) @ B2 ) @ A2 ) ).
% div_mult_le
thf(fact_2858_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_2859_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_2860_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_2861_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_2862_td__gal__lt,axiom,
! [C3: nat,A2: nat,B2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C3 )
=> ( ( ord_less @ nat @ A2 @ ( times_times @ nat @ B2 @ C3 ) )
= ( ord_less @ nat @ ( divide_divide @ nat @ A2 @ C3 ) @ B2 ) ) ) ).
% td_gal_lt
thf(fact_2863_zdiv__le__dividend,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ A2 ) ) ) ).
% zdiv_le_dividend
thf(fact_2864_zdiv__zmult2__eq,axiom,
! [C3: int,A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C3 )
=> ( ( divide_divide @ int @ A2 @ ( times_times @ int @ B2 @ C3 ) )
= ( divide_divide @ int @ ( divide_divide @ int @ A2 @ B2 ) @ C3 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_2865_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_2866_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_2867_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M5: nat,N6: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M5 @ N6 )
| ( N6
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M5 @ N6 ) @ N6 ) ) ) ) ) ).
% div_if
thf(fact_2868_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_2869_split__div,axiom,
! [P2: nat > $o,M: nat,N: nat] :
( ( P2 @ ( divide_divide @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P2 @ ( zero_zero @ nat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I3: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I3 ) @ J3 ) )
=> ( P2 @ I3 ) ) ) ) ) ) ).
% split_div
thf(fact_2870_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_2871_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_2872_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_2873_td__gal,axiom,
! [C3: nat,B2: nat,A2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C3 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ C3 ) @ A2 )
= ( ord_less_eq @ nat @ B2 @ ( divide_divide @ nat @ A2 @ C3 ) ) ) ) ).
% td_gal
thf(fact_2874_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M5: nat,N6: nat] :
( if @ nat
@ ( N6
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K3: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K3 @ N6 ) @ M5 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_2875_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_2876_split__div_H,axiom,
! [P2: nat > $o,M: nat,N: nat] :
( ( P2 @ ( divide_divide @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
& ( P2 @ ( zero_zero @ nat ) ) )
| ? [Q8: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q8 ) @ M )
& ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q8 ) ) )
& ( P2 @ Q8 ) ) ) ) ).
% split_div'
thf(fact_2877_power__sub,axiom,
! [N: nat,M: nat,A2: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 )
=> ( ( power_power @ nat @ A2 @ ( minus_minus @ nat @ M @ N ) )
= ( divide_divide @ nat @ ( power_power @ nat @ A2 @ M ) @ ( power_power @ nat @ A2 @ N ) ) ) ) ) ).
% power_sub
thf(fact_2878_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_2879_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_2880_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_2881_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_2882_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_2883_set__n__lists,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) )
= ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys3 )
= N )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_2884_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) @ N )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth @ B @ Ys @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_2885_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_2886_mod__self,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_self
thf(fact_2887_mod__by__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% mod_by_0
thf(fact_2888_mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_0
thf(fact_2889_nat__mod__eq_H,axiom,
! [A2: nat,N: nat] :
( ( ord_less @ nat @ A2 @ N )
=> ( ( modulo_modulo @ nat @ A2 @ N )
= A2 ) ) ).
% nat_mod_eq'
thf(fact_2890_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( modulo_modulo @ nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_2891_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_2892_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_2893_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_2894_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_2895_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_2896_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_2897_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_2898_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_2899_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ ( zero_zero @ nat ) ) )
= N ) ).
% binomial_1
thf(fact_2900_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) )
= ( zero_zero @ nat ) ) ).
% binomial_0_Suc
thf(fact_2901_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_2902_binomial__Suc__Suc,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_2903_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_2904_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_2905_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ K @ N ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_2906_Suc__mod__mult__self2,axiom,
! [M: nat,N: nat,K: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ N @ K ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_2907_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N ) @ M ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_2908_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N @ K ) @ M ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_2909_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_2910_mod__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( modulo_modulo @ nat @ M @ N ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% mod_Suc_eq
thf(fact_2911_mod__Suc__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( modulo_modulo @ nat @ M @ N ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_2912_nat__mod__eq,axiom,
! [B2: nat,N: nat,A2: nat] :
( ( ord_less @ nat @ B2 @ N )
=> ( ( ( modulo_modulo @ nat @ A2 @ N )
= ( modulo_modulo @ nat @ B2 @ N ) )
=> ( ( modulo_modulo @ nat @ A2 @ N )
= B2 ) ) ) ).
% nat_mod_eq
thf(fact_2913_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_2914_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( ( binomial @ N @ K )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_2915_Suc__times__binomial,axiom,
! [K: nat,N: nat] :
( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
= ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% Suc_times_binomial
thf(fact_2916_Suc__times__binomial__eq,axiom,
! [N: nat,K: nat] :
( ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
= ( times_times @ nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% Suc_times_binomial_eq
thf(fact_2917_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_2918_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_2919_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_2920_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_2921_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_2922_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_2923_mod__induct,axiom,
! [P2: nat > $o,N: nat,P3: nat,M: nat] :
( ( P2 @ N )
=> ( ( ord_less @ nat @ N @ P3 )
=> ( ( ord_less @ nat @ M @ P3 )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ P3 )
=> ( ( P2 @ N2 )
=> ( P2 @ ( modulo_modulo @ nat @ ( suc @ N2 ) @ P3 ) ) ) )
=> ( P2 @ M ) ) ) ) ) ).
% mod_induct
thf(fact_2924_gcd__nat__induct,axiom,
! [P2: nat > nat > $o,M: nat,N: nat] :
( ! [M3: nat] : ( P2 @ M3 @ ( zero_zero @ nat ) )
=> ( ! [M3: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( P2 @ N2 @ ( modulo_modulo @ nat @ M3 @ N2 ) )
=> ( P2 @ M3 @ N2 ) ) )
=> ( P2 @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_2925_nat__mod__lem,axiom,
! [N: nat,B2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ B2 @ N )
= ( ( modulo_modulo @ nat @ B2 @ N )
= B2 ) ) ) ).
% nat_mod_lem
thf(fact_2926_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_2927_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_2928_word__rot__lem,axiom,
! [L: nat,K: nat,D2: nat,N: nat] :
( ( ( plus_plus @ nat @ L @ K )
= ( plus_plus @ nat @ D2 @ ( modulo_modulo @ nat @ K @ L ) ) )
=> ( ( ord_less @ nat @ N @ L )
=> ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ D2 @ N ) @ L )
= N ) ) ) ).
% word_rot_lem
thf(fact_2929_nat__minus__mod,axiom,
! [N: nat,M: nat] :
( ( modulo_modulo @ nat @ ( minus_minus @ nat @ N @ ( modulo_modulo @ nat @ N @ M ) ) @ M )
= ( zero_zero @ nat ) ) ).
% nat_minus_mod
thf(fact_2930_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_2931_mod__if,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M5: nat,N6: nat] : ( if @ nat @ ( ord_less @ nat @ M5 @ N6 ) @ M5 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M5 @ N6 ) @ N6 ) ) ) ) ).
% mod_if
thf(fact_2932_mod__nat__sub,axiom,
! [X: nat,Z: nat,Y: nat] :
( ( ord_less @ nat @ X @ Z )
=> ( ( modulo_modulo @ nat @ ( minus_minus @ nat @ X @ Y ) @ Z )
= ( minus_minus @ nat @ X @ Y ) ) ) ).
% mod_nat_sub
thf(fact_2933_mod__eq__0D,axiom,
! [M: nat,D2: nat] :
( ( ( modulo_modulo @ nat @ M @ D2 )
= ( zero_zero @ nat ) )
=> ? [Q5: nat] :
( M
= ( times_times @ nat @ D2 @ Q5 ) ) ) ).
% mod_eq_0D
thf(fact_2934_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_2935_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ N )
= ( modulo_modulo @ nat @ Y @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus @ nat @ X @ ( times_times @ nat @ N @ Q1 ) )
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_2936_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_2937_Suc__times__binomial__add,axiom,
! [A2: nat,B2: nat] :
( ( times_times @ nat @ ( suc @ A2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A2 @ B2 ) ) @ ( suc @ A2 ) ) )
= ( times_times @ nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A2 @ B2 ) ) @ A2 ) ) ) ).
% Suc_times_binomial_add
thf(fact_2938_binomial__Suc__Suc__eq__times,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% binomial_Suc_Suc_eq_times
thf(fact_2939_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_2940_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_2941_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_2942_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_2943_div__less__mono,axiom,
! [A5: nat,B5: nat,N: nat] :
( ( ord_less @ nat @ A5 @ B5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( modulo_modulo @ nat @ A5 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B5 @ N )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A5 @ N ) @ ( divide_divide @ nat @ B5 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_2944_mod__nat__add,axiom,
! [X: nat,Z: nat,Y: nat] :
( ( ord_less @ nat @ X @ Z )
=> ( ( ord_less @ nat @ Y @ Z )
=> ( ( ( ord_less @ nat @ ( plus_plus @ nat @ X @ Y ) @ Z )
=> ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ X @ Y ) @ Z )
= ( plus_plus @ nat @ X @ Y ) ) )
& ( ~ ( ord_less @ nat @ ( plus_plus @ nat @ X @ Y ) @ Z )
=> ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ X @ Y ) @ Z )
= ( minus_minus @ nat @ ( plus_plus @ nat @ X @ Y ) @ Z ) ) ) ) ) ) ).
% mod_nat_add
thf(fact_2945_nat__mod__eq__lemma,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ N )
= ( modulo_modulo @ nat @ Y @ N ) )
=> ( ( ord_less_eq @ nat @ Y @ X )
=> ? [Q5: nat] :
( X
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q5 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_2946_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 )
=> ~ ! [S4: nat] :
( N
!= ( plus_plus @ nat @ M @ ( times_times @ nat @ Q3 @ S4 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_2947_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 )
=> ~ ! [S4: nat] :
( M
!= ( plus_plus @ nat @ N @ ( times_times @ nat @ Q3 @ S4 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_2948_n__subsets,axiom,
! [A: $tType,A5: set @ A,K: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ A5 )
& ( ( finite_card @ A @ B7 )
= K ) ) ) )
= ( binomial @ ( finite_card @ A @ A5 ) @ K ) ) ) ).
% n_subsets
thf(fact_2949_sum__choose__upper,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ K3 @ M )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% sum_choose_upper
thf(fact_2950_binomial__absorption,axiom,
! [K: nat,N: nat] :
( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% binomial_absorption
thf(fact_2951_split__mod,axiom,
! [P2: nat > $o,M: nat,N: nat] :
( ( P2 @ ( modulo_modulo @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P2 @ M ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I3: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I3 ) @ J3 ) )
=> ( P2 @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_2952_mod__lemma,axiom,
! [C3: nat,R2: nat,B2: nat,Q3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C3 )
=> ( ( ord_less @ nat @ R2 @ B2 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ B2 @ ( modulo_modulo @ nat @ Q3 @ C3 ) ) @ R2 ) @ ( times_times @ nat @ B2 @ C3 ) ) ) ) ).
% mod_lemma
thf(fact_2953_sum__choose__lower,axiom,
! [R2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( plus_plus @ nat @ R2 @ K3 ) @ K3 )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ ( plus_plus @ nat @ R2 @ N ) ) @ N ) ) ).
% sum_choose_lower
thf(fact_2954_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_2955_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_2956_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_2957_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C3 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2958_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_2959_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_2960_nth__rotate1,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rotate1 @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_2961_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_2962_verit__le__mono__div,axiom,
! [A5: nat,B5: nat,N: nat] :
( ( ord_less @ nat @ A5 @ B5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A5 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B5 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B5 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_2963_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
@ ^ [I3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_2964_n__lists_Osimps_I2_J,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( n_lists @ A @ ( suc @ N ) @ Xs )
= ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ^ [Ys3: list @ A] :
( map @ A @ ( list @ A )
@ ^ [Y5: A] : ( cons @ A @ Y5 @ Ys3 )
@ Xs )
@ ( n_lists @ A @ N @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_2965_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( N
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_2966_ran__nth__set__encoding__conv,axiom,
! [A: $tType,L: list @ A] :
( ( ran @ nat @ A
@ ^ [I3: nat] : ( if @ ( option @ A ) @ ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) ) @ ( some @ A @ ( nth @ A @ L @ I3 ) ) @ ( none @ A ) ) )
= ( set2 @ A @ L ) ) ).
% ran_nth_set_encoding_conv
thf(fact_2967_vebt__assn__raw_Osimps_I2_J,axiom,
! [Mmo: option @ ( product_prod @ nat @ nat ),Deg: nat,Tree_list: list @ vEBT_VEBT,Summary: vEBT_VEBT,Mmoi: option @ ( product_prod @ nat @ nat ),Degi: nat,Tree_array: array @ vEBT_VEBTi,Summaryi: vEBT_VEBTi] :
( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ Mmo @ Deg @ Tree_list @ Summary ) @ ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) )
= ( times_times @ assn
@ ( times_times @ assn
@ ( pure_assn
@ ( ( Mmoi = Mmo )
& ( Degi = Deg ) ) )
@ ( vEBT_vebt_assn_raw @ Summary @ Summaryi ) )
@ ( ex_assn @ ( list @ vEBT_VEBTi )
@ ^ [Tree_is2: list @ vEBT_VEBTi] : ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ Tree_array @ Tree_is2 ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Tree_list @ Tree_is2 ) ) ) ) ) ).
% vebt_assn_raw.simps(2)
thf(fact_2968_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A8: A,N6: nat] :
( if @ A
@ ( N6
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A8 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N6 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_2969_Set__filter__fold,axiom,
! [A: $tType,A5: set @ A,P2: A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( filter3 @ A @ P2 @ A5 )
= ( finite_fold @ A @ ( set @ A )
@ ^ [X4: A,A11: set @ A] : ( if @ ( set @ A ) @ ( P2 @ X4 ) @ ( insert @ A @ X4 @ A11 ) @ A11 )
@ ( bot_bot @ ( set @ A ) )
@ A5 ) ) ) ).
% Set_filter_fold
thf(fact_2970_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J3: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) @ ( cons @ int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_2971_member__filter,axiom,
! [A: $tType,X: A,P2: A > $o,A5: set @ A] :
( ( member @ A @ X @ ( filter3 @ A @ P2 @ A5 ) )
= ( ( member @ A @ X @ A5 )
& ( P2 @ X ) ) ) ).
% member_filter
thf(fact_2972_ex__assn__const,axiom,
! [A: $tType,C3: assn] :
( ( ex_assn @ A
@ ^ [X4: A] : C3 )
= C3 ) ).
% ex_assn_const
thf(fact_2973_triv__exI,axiom,
! [A: $tType,Q2: A > assn,X: A] : ( entails @ ( Q2 @ X ) @ ( ex_assn @ A @ Q2 ) ) ).
% triv_exI
thf(fact_2974_mod__ex__dist,axiom,
! [A: $tType,P2: A > assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( ex_assn @ A @ P2 ) @ H2 )
= ( ? [X4: A] : ( rep_assn @ ( P2 @ X4 ) @ H2 ) ) ) ).
% mod_ex_dist
thf(fact_2975_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_2976_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_2977_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_2978_pochhammer__Suc0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% pochhammer_Suc0
thf(fact_2979_map__update__eta__repair_I2_J,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),K: A,V3: B] :
( ( ( M @ K )
= ( none @ B ) )
=> ( ( ran @ A @ B
@ ^ [X4: A] : ( if @ ( option @ B ) @ ( X4 = K ) @ ( some @ B @ V3 ) @ ( M @ X4 ) ) )
= ( insert @ B @ V3 @ ( ran @ A @ B @ M ) ) ) ) ).
% map_update_eta_repair(2)
thf(fact_2980_mod__h__bot__iff_I8_J,axiom,
! [C: $tType,R3: C > assn,H2: heap_ext @ product_unit] :
( ( rep_assn @ ( ex_assn @ C @ R3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ? [X4: C] : ( rep_assn @ ( R3 @ X4 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(8)
thf(fact_2981_mod__word__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,V3: word @ A] :
( ( ord_less @ ( word @ A ) @ W2 @ V3 )
=> ( ( modulo_modulo @ ( word @ A ) @ W2 @ V3 )
= W2 ) ) ) ).
% mod_word_less
thf(fact_2982_Set_Ofilter__def,axiom,
! [A: $tType] :
( ( filter3 @ A )
= ( ^ [P5: A > $o,A6: set @ A] :
( collect @ A
@ ^ [A8: A] :
( ( member @ A @ A8 @ A6 )
& ( P5 @ A8 ) ) ) ) ) ).
% Set.filter_def
thf(fact_2983_ex__distrib__star,axiom,
! [A: $tType,P2: A > assn,Q2: assn] :
( ( ex_assn @ A
@ ^ [X4: A] : ( times_times @ assn @ ( P2 @ X4 ) @ Q2 ) )
= ( times_times @ assn @ ( ex_assn @ A @ P2 ) @ Q2 ) ) ).
% ex_distrib_star
thf(fact_2984_ent__ex__preI,axiom,
! [A: $tType,P2: A > assn,Q2: assn] :
( ! [X3: A] : ( entails @ ( P2 @ X3 ) @ Q2 )
=> ( entails @ ( ex_assn @ A @ P2 ) @ Q2 ) ) ).
% ent_ex_preI
thf(fact_2985_ent__ex__postI,axiom,
! [A: $tType,P2: assn,Q2: A > assn,X: A] :
( ( entails @ P2 @ ( Q2 @ X ) )
=> ( entails @ P2 @ ( ex_assn @ A @ Q2 ) ) ) ).
% ent_ex_postI
thf(fact_2986_ex__one__point__gen,axiom,
! [A: $tType,P2: A > assn,V3: A] :
( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),X3: A] :
( ( rep_assn @ ( P2 @ X3 ) @ H )
=> ( X3 = V3 ) )
=> ( ( ex_assn @ A @ P2 )
= ( P2 @ V3 ) ) ) ).
% ex_one_point_gen
thf(fact_2987_mod__exI,axiom,
! [A: $tType,P2: A > assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ? [X5: A] : ( rep_assn @ ( P2 @ X5 ) @ H2 )
=> ( rep_assn @ ( ex_assn @ A @ P2 ) @ H2 ) ) ).
% mod_exI
thf(fact_2988_mod__exE,axiom,
! [A: $tType,P2: A > assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( ex_assn @ A @ P2 ) @ H2 )
=> ~ ! [X3: A] :
~ ( rep_assn @ ( P2 @ X3 ) @ H2 ) ) ).
% mod_exE
thf(fact_2989_pochhammer__pos,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N ) ) ) ) ).
% pochhammer_pos
thf(fact_2990_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_2991_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_2992_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_2993_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_2994_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_2995_zmod__eq__0__iff,axiom,
! [M: int,D2: int] :
( ( ( modulo_modulo @ int @ M @ D2 )
= ( zero_zero @ int ) )
= ( ? [Q8: int] :
( M
= ( times_times @ int @ D2 @ Q8 ) ) ) ) ).
% zmod_eq_0_iff
thf(fact_2996_zmod__eq__0D,axiom,
! [M: int,D2: int] :
( ( ( modulo_modulo @ int @ M @ D2 )
= ( zero_zero @ int ) )
=> ? [Q5: int] :
( M
= ( times_times @ int @ D2 @ Q5 ) ) ) ).
% zmod_eq_0D
thf(fact_2997_word__mod__less__divisor,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,M: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N )
=> ( ord_less @ ( word @ A ) @ ( modulo_modulo @ ( word @ A ) @ M @ N ) @ N ) ) ) ).
% word_mod_less_divisor
thf(fact_2998_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_2999_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_3000_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_3001_finite__filter,axiom,
! [A: $tType,S3: set @ A,P2: A > $o] :
( ( finite_finite2 @ A @ S3 )
=> ( finite_finite2 @ A @ ( filter3 @ A @ P2 @ S3 ) ) ) ).
% finite_filter
thf(fact_3002_pochhammer__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N ) ) ) ) ).
% pochhammer_nonneg
thf(fact_3003_int__mod__eq,axiom,
! [B2: int,N: int,A2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ int @ B2 @ N )
=> ( ( ( modulo_modulo @ int @ A2 @ N )
= ( modulo_modulo @ int @ B2 @ N ) )
=> ( ( modulo_modulo @ int @ A2 @ N )
= B2 ) ) ) ) ).
% int_mod_eq
thf(fact_3004_int__mod__lem,axiom,
! [N: int,B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
& ( ord_less @ int @ B2 @ N ) )
= ( ( modulo_modulo @ int @ B2 @ N )
= B2 ) ) ) ).
% int_mod_lem
thf(fact_3005_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_3006_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_3007_int__mod__ge,axiom,
! [A2: int,N: int] :
( ( ord_less @ int @ A2 @ N )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ A2 @ ( modulo_modulo @ int @ A2 @ N ) ) ) ) ).
% int_mod_ge
thf(fact_3008_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_3009_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_3010_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_3011_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_3012_int__mod__le_H,axiom,
! [B2: int,N: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ B2 @ N ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ B2 @ N ) @ ( minus_minus @ int @ B2 @ N ) ) ) ).
% int_mod_le'
thf(fact_3013_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_3014_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_3015_nonneg__mod__div,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A2 @ B2 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ).
% nonneg_mod_div
thf(fact_3016_zdiv__mono__strict,axiom,
! [A5: int,B5: int,N: int] :
( ( ord_less @ int @ A5 @ B5 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ( modulo_modulo @ int @ A5 @ N )
= ( zero_zero @ int ) )
=> ( ( ( modulo_modulo @ int @ B5 @ N )
= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( divide_divide @ int @ A5 @ N ) @ ( divide_divide @ int @ B5 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_3017_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( times_times @ A @ A2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% pochhammer_rec
thf(fact_3018_int__mod__ge_H,axiom,
! [B2: int,N: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ B2 @ N ) @ ( modulo_modulo @ int @ B2 @ N ) ) ) ) ).
% int_mod_ge'
thf(fact_3019_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_3020_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_3021_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_3022_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_3023_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_3024_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_3025_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_3026_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_3027_mod__power__lem,axiom,
! [A2: int,M: nat,N: nat] :
( ( ord_less @ int @ ( one_one @ int ) @ A2 )
=> ( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ int @ ( power_power @ int @ A2 @ N ) @ ( power_power @ int @ A2 @ M ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ int @ ( power_power @ int @ A2 @ N ) @ ( power_power @ int @ A2 @ M ) )
= ( power_power @ int @ A2 @ N ) ) ) ) ) ).
% mod_power_lem
thf(fact_3028_split__zmod,axiom,
! [P2: int > $o,N: int,K: int] :
( ( P2 @ ( modulo_modulo @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P2 @ N ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I3: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P2 @ J3 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I3: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P2 @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_3029_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_3030_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_3031_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_3032_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_3033_mod__add__if__z,axiom,
! [X: int,Z: int,Y: int] :
( ( ord_less @ int @ X @ Z )
=> ( ( ord_less @ int @ Y @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ( ord_less @ int @ ( plus_plus @ int @ X @ Y ) @ Z )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ X @ Y ) @ Z )
= ( plus_plus @ int @ X @ Y ) ) )
& ( ~ ( ord_less @ int @ ( plus_plus @ int @ X @ Y ) @ Z )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ X @ Y ) @ Z )
= ( minus_minus @ int @ ( plus_plus @ int @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ).
% mod_add_if_z
thf(fact_3034_mod__sub__if__z,axiom,
! [X: int,Z: int,Y: int] :
( ( ord_less @ int @ X @ Z )
=> ( ( ord_less @ int @ Y @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ( ord_less_eq @ int @ Y @ X )
=> ( ( modulo_modulo @ int @ ( minus_minus @ int @ X @ Y ) @ Z )
= ( minus_minus @ int @ X @ Y ) ) )
& ( ~ ( ord_less_eq @ int @ Y @ X )
=> ( ( modulo_modulo @ int @ ( minus_minus @ int @ X @ Y ) @ Z )
= ( plus_plus @ int @ ( minus_minus @ int @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ).
% mod_sub_if_z
thf(fact_3035_zmod__zmult2__eq,axiom,
! [C3: int,A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C3 )
=> ( ( modulo_modulo @ int @ A2 @ ( times_times @ int @ B2 @ C3 ) )
= ( plus_plus @ int @ ( times_times @ int @ B2 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A2 @ B2 ) @ C3 ) ) @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_3036_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_3037_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_3038_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_3039_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A8: A,N6: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A8 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N6 ) ) ) ) ) ).
% pochhammer_prod
thf(fact_3040_verit__le__mono__div__int,axiom,
! [A5: int,B5: int,N: int] :
( ( ord_less @ int @ A5 @ B5 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A5 @ N )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B5 @ N )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B5 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_3041_split__neg__lemma,axiom,
! [K: int,P2: int > int > $o,N: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ( P2 @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I3: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P2 @ I3 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_3042_split__pos__lemma,axiom,
! [K: int,P2: int > int > $o,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( P2 @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I3: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P2 @ I3 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_3043_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_3044_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
@ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_3045_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X4: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_3046_vebt__assn__raw_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: vEBT_VEBTi,Y: assn] :
( ( ( vEBT_vebt_assn_raw @ X @ Xa2 )
= Y )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ! [Ai: $o,Bi: $o] :
( ( Xa2
= ( vEBT_Leafi @ Ai @ Bi ) )
=> ( Y
!= ( pure_assn
@ ( ( Ai = A4 )
& ( Bi = B4 ) ) ) ) ) )
=> ( ! [Mmo2: option @ ( product_prod @ nat @ nat ),Deg2: nat,Tree_list2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) )
=> ! [Mmoi2: option @ ( product_prod @ nat @ nat ),Degi2: nat,Tree_array2: array @ vEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
( ( Xa2
= ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) )
=> ( Y
!= ( times_times @ assn
@ ( times_times @ assn
@ ( pure_assn
@ ( ( Mmoi2 = Mmo2 )
& ( Degi2 = Deg2 ) ) )
@ ( vEBT_vebt_assn_raw @ Summary2 @ Summaryi2 ) )
@ ( ex_assn @ ( list @ vEBT_VEBTi )
@ ^ [Tree_is2: list @ vEBT_VEBTi] : ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ Tree_array2 @ Tree_is2 ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Tree_list2 @ Tree_is2 ) ) ) ) ) ) )
=> ( ( ? [V: option @ ( product_prod @ nat @ nat ),Va: nat,Vb3: list @ vEBT_VEBT,Vc3: vEBT_VEBT] :
( X
= ( vEBT_Node @ V @ Va @ Vb3 @ Vc3 ) )
=> ( ? [Vd3: $o,Ve3: $o] :
( Xa2
= ( vEBT_Leafi @ Vd3 @ Ve3 ) )
=> ( Y
!= ( bot_bot @ assn ) ) ) )
=> ~ ( ? [Vd3: $o,Ve3: $o] :
( X
= ( vEBT_Leaf @ Vd3 @ Ve3 ) )
=> ( ? [V: option @ ( product_prod @ nat @ nat ),Va: nat,Vb3: array @ vEBT_VEBTi,Vc3: vEBT_VEBTi] :
( Xa2
= ( vEBT_Nodei @ V @ Va @ Vb3 @ Vc3 ) )
=> ( Y
!= ( bot_bot @ assn ) ) ) ) ) ) ) ) ).
% vebt_assn_raw.elims
thf(fact_3047_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_3048_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ ( uminus_uminus @ real @ X ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_3049_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N: nat,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N ) ) @ X )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ X ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_3050_TBOUND__VEBT__case,axiom,
! [A: $tType,Ti: vEBT_VEBTi,F3: $o > $o > ( heap_Time_Heap @ A ),Bnd: $o > $o > nat,F8: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > ( heap_Time_Heap @ A ),Bnd2: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > nat] :
( ! [A4: $o,B4: $o] :
( ( Ti
= ( vEBT_Leafi @ A4 @ B4 ) )
=> ( time_TBOUND @ A @ ( F3 @ A4 @ B4 ) @ ( Bnd @ A4 @ B4 ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeArray: array @ vEBT_VEBTi,Summary2: vEBT_VEBTi] :
( ( Ti
= ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
=> ( time_TBOUND @ A @ ( F8 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) ) )
=> ( time_TBOUND @ A @ ( vEBT_case_VEBTi @ ( heap_Time_Heap @ A ) @ F8 @ F3 @ Ti ) @ ( vEBT_case_VEBTi @ nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).
% TBOUND_VEBT_case
thf(fact_3051_VEBTi_Oinject_I2_J,axiom,
! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
( ( ( vEBT_Leafi @ X21 @ X22 )
= ( vEBT_Leafi @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% VEBTi.inject(2)
thf(fact_3052_exp__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) ) ) ).
% exp_less_mono
thf(fact_3053_exp__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% exp_less_cancel_iff
thf(fact_3054_exp__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% exp_le_cancel_iff
thf(fact_3055_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_3056_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_3057_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_3058_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_3059_one__less__exp__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% one_less_exp_iff
thf(fact_3060_exp__less__one__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% exp_less_one_iff
thf(fact_3061_one__le__exp__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( exp @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% one_le_exp_iff
thf(fact_3062_exp__le__one__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% exp_le_one_iff
thf(fact_3063_exp__not__eq__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( exp @ A @ X )
!= ( zero_zero @ A ) ) ) ).
% exp_not_eq_zero
thf(fact_3064_exp__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) )
=> ( ord_less @ real @ X @ Y ) ) ).
% exp_less_cancel
thf(fact_3065_norm__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ X ) ) @ ( exp @ real @ ( real_V7770717601297561774m_norm @ A @ X ) ) ) ) ).
% norm_exp
thf(fact_3066_VEBTi_Odistinct_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: array @ vEBT_VEBTi,X14: vEBT_VEBTi,X21: $o,X22: $o] :
( ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 )
!= ( vEBT_Leafi @ X21 @ X22 ) ) ).
% VEBTi.distinct(1)
thf(fact_3067_VEBTi_Oexhaust,axiom,
! [Y: vEBT_VEBTi] :
( ! [X112: option @ ( product_prod @ nat @ nat ),X122: nat,X132: array @ vEBT_VEBTi,X142: vEBT_VEBTi] :
( Y
!= ( vEBT_Nodei @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X212: $o,X222: $o] :
( Y
!= ( vEBT_Leafi @ X212 @ X222 ) ) ) ).
% VEBTi.exhaust
thf(fact_3068_VEBTi_Osimps_I6_J,axiom,
! [A: $tType,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > A,F22: $o > $o > A,X21: $o,X22: $o] :
( ( vEBT_case_VEBTi @ A @ F1 @ F22 @ ( vEBT_Leafi @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% VEBTi.simps(6)
thf(fact_3069_not__exp__less__zero,axiom,
! [X: real] :
~ ( ord_less @ real @ ( exp @ real @ X ) @ ( zero_zero @ real ) ) ).
% not_exp_less_zero
thf(fact_3070_exp__gt__zero,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( exp @ real @ X ) ) ).
% exp_gt_zero
thf(fact_3071_exp__total,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ? [X3: real] :
( ( exp @ real @ X3 )
= Y ) ) ).
% exp_total
thf(fact_3072_exp__ge__zero,axiom,
! [X: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( exp @ real @ X ) ) ).
% exp_ge_zero
thf(fact_3073_not__exp__le__zero,axiom,
! [X: real] :
~ ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( zero_zero @ real ) ) ).
% not_exp_le_zero
thf(fact_3074_VEBTi_Osize_I4_J,axiom,
! [X21: $o,X22: $o] :
( ( size_size @ vEBT_VEBTi @ ( vEBT_Leafi @ X21 @ X22 ) )
= ( zero_zero @ nat ) ) ).
% VEBTi.size(4)
thf(fact_3075_VEBTi_Osize__gen_I2_J,axiom,
! [X21: $o,X22: $o] :
( ( vEBT_size_VEBTi @ ( vEBT_Leafi @ X21 @ X22 ) )
= ( zero_zero @ nat ) ) ).
% VEBTi.size_gen(2)
thf(fact_3076_exp__gt__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X ) ) ) ).
% exp_gt_one
thf(fact_3077_exp__ge__add__one__self,axiom,
! [X: real] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( exp @ real @ X ) ) ).
% exp_ge_add_one_self
thf(fact_3078_vebt__assn__raw_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ vEBT_VEBTi] :
( ! [A4: $o,B4: $o,Ai: $o,Bi: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ A4 @ B4 ) @ ( vEBT_Leafi @ Ai @ Bi ) ) )
=> ( ! [Mmo2: option @ ( product_prod @ nat @ nat ),Deg2: nat,Tree_list2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Mmoi2: option @ ( product_prod @ nat @ nat ),Degi2: nat,Tree_array2: array @ vEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
( X
!= ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) @ ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) ) )
=> ( ! [V: option @ ( product_prod @ nat @ nat ),Va: nat,Vb3: list @ vEBT_VEBT,Vc3: vEBT_VEBT,Vd3: $o,Ve3: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Node @ V @ Va @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) ) )
=> ~ ! [Vd3: $o,Ve3: $o,V: option @ ( product_prod @ nat @ nat ),Va: nat,Vb3: array @ vEBT_VEBTi,Vc3: vEBT_VEBTi] :
( X
!= ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V @ Va @ Vb3 @ Vc3 ) ) ) ) ) ) ).
% vebt_assn_raw.cases
thf(fact_3079_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ A2 @ K ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_3080_VEBT__internal_OminNulli_Ocases,axiom,
! [X: vEBT_VEBTi] :
( ( X
!= ( vEBT_Leafi @ $false @ $false ) )
=> ( ! [Uv: $o] :
( X
!= ( vEBT_Leafi @ $true @ Uv ) )
=> ( ! [Uu: $o] :
( X
!= ( vEBT_Leafi @ Uu @ $true ) )
=> ( ! [Uw: nat,Ux: array @ vEBT_VEBTi,Uy: vEBT_VEBTi] :
( X
!= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: array @ vEBT_VEBTi,Vc2: vEBT_VEBTi] :
( X
!= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% VEBT_internal.minNulli.cases
thf(fact_3081_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_3082_vebt__assn__raw_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,Ai2: $o,Bi2: $o] :
( ( vEBT_vebt_assn_raw @ ( vEBT_Leaf @ A2 @ B2 ) @ ( vEBT_Leafi @ Ai2 @ Bi2 ) )
= ( pure_assn
@ ( ( Ai2 = A2 )
& ( Bi2 = B2 ) ) ) ) ).
% vebt_assn_raw.simps(1)
thf(fact_3083_vebt__assn__raw_Osimps_I3_J,axiom,
! [V3: option @ ( product_prod @ nat @ nat ),Va2: nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,Vd2: $o,Ve2: $o] :
( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ V3 @ Va2 @ Vb @ Vc ) @ ( vEBT_Leafi @ Vd2 @ Ve2 ) )
= ( bot_bot @ assn ) ) ).
% vebt_assn_raw.simps(3)
thf(fact_3084_exp__sum,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ B )
& ( real_Vector_banach @ B )
& ( real_V2822296259951069270ebra_1 @ B ) )
=> ! [I6: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( exp @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ I6 ) )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X4: A] : ( exp @ B @ ( F3 @ X4 ) )
@ I6 ) ) ) ) ).
% exp_sum
thf(fact_3085_exp__ge__add__one__self__aux,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( exp @ real @ X ) ) ) ).
% exp_ge_add_one_self_aux
thf(fact_3086_lemma__exp__total,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ Y )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( minus_minus @ real @ Y @ ( one_one @ real ) ) )
& ( ( exp @ real @ X3 )
= Y ) ) ) ).
% lemma_exp_total
thf(fact_3087_gbinomial__addition__formula,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ A2 @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_addition_formula
thf(fact_3088_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_3089_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_3090_vebt__minti_Ocases,axiom,
! [X: vEBT_VEBTi] :
( ! [A4: $o,B4: $o] :
( X
!= ( vEBT_Leafi @ A4 @ B4 ) )
=> ( ! [Uu: nat,Uv: array @ vEBT_VEBTi,Uw: vEBT_VEBTi] :
( X
!= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
( X
!= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ).
% vebt_minti.cases
thf(fact_3091_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_3092_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_3093_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_3094_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_3095_exp__divide__power__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N: nat,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( power_power @ A @ ( exp @ A @ ( divide_divide @ A @ X @ ( semiring_1_of_nat @ A @ N ) ) ) @ N )
= ( exp @ A @ X ) ) ) ) ).
% exp_divide_power_eq
thf(fact_3096_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_3097_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_3098_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_3099_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A8: A,K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A8 @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K3 @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_3100_vebt__assn__def,axiom,
( vEBT_Intf_vebt_assn
= ( ^ [N6: nat,S8: set @ nat,Ti3: vEBT_VEBTi] :
( ex_assn @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T3 @ Ti3 )
@ ( pure_assn
@ ( ( S8
= ( vEBT_set_vebt @ T3 ) )
& ( vEBT_invar_vebt @ T3 @ N6 ) ) ) ) ) ) ) ).
% vebt_assn_def
thf(fact_3101_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_3102_vebt__assn__raw_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: vEBT_VEBTi,Y: assn] :
( ( ( vEBT_vebt_assn_raw @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ! [Ai: $o,Bi: $o] :
( ( Xa2
= ( vEBT_Leafi @ Ai @ Bi ) )
=> ( ( Y
= ( pure_assn
@ ( ( Ai = A4 )
& ( Bi = B4 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ A4 @ B4 ) @ ( vEBT_Leafi @ Ai @ Bi ) ) ) ) ) )
=> ( ! [Mmo2: option @ ( product_prod @ nat @ nat ),Deg2: nat,Tree_list2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) )
=> ! [Mmoi2: option @ ( product_prod @ nat @ nat ),Degi2: nat,Tree_array2: array @ vEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
( ( Xa2
= ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) )
=> ( ( Y
= ( times_times @ assn
@ ( times_times @ assn
@ ( pure_assn
@ ( ( Mmoi2 = Mmo2 )
& ( Degi2 = Deg2 ) ) )
@ ( vEBT_vebt_assn_raw @ Summary2 @ Summaryi2 ) )
@ ( ex_assn @ ( list @ vEBT_VEBTi )
@ ^ [Tree_is2: list @ vEBT_VEBTi] : ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ Tree_array2 @ Tree_is2 ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Tree_list2 @ Tree_is2 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) @ ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) ) ) ) ) )
=> ( ! [V: option @ ( product_prod @ nat @ nat ),Va: nat,Vb3: list @ vEBT_VEBT,Vc3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ V @ Va @ Vb3 @ Vc3 ) )
=> ! [Vd3: $o,Ve3: $o] :
( ( Xa2
= ( vEBT_Leafi @ Vd3 @ Ve3 ) )
=> ( ( Y
= ( bot_bot @ assn ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Node @ V @ Va @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) ) ) ) ) )
=> ~ ! [Vd3: $o,Ve3: $o] :
( ( X
= ( vEBT_Leaf @ Vd3 @ Ve3 ) )
=> ! [V: option @ ( product_prod @ nat @ nat ),Va: nat,Vb3: array @ vEBT_VEBTi,Vc3: vEBT_VEBTi] :
( ( Xa2
= ( vEBT_Nodei @ V @ Va @ Vb3 @ Vc3 ) )
=> ( ( Y
= ( bot_bot @ assn ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V @ Va @ Vb3 @ Vc3 ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_assn_raw.pelims
thf(fact_3103_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A8: A,K3: nat] :
( if @ A
@ ( K3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L4: nat] : ( times_times @ A @ ( minus_minus @ A @ A8 @ ( semiring_1_of_nat @ A @ L4 ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_3104_vebt__minti_Oelims,axiom,
! [X: vEBT_VEBTi,Y: heap_Time_Heap @ ( option @ nat )] :
( ( ( vEBT_vebt_minti @ X )
= Y )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leafi @ A4 @ B4 ) )
=> ~ ( ( A4
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
& ( ~ A4
=> ( ( B4
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
& ( ~ B4
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: array @ vEBT_VEBTi,Uw: vEBT_VEBTi] :
( X
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
=> ~ ! [Mi2: nat] :
( ? [Ma2: nat,Ux: nat,Uy: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
( X
= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
=> ( Y
!= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Mi2 ) ) ) ) ) ) ) ).
% vebt_minti.elims
thf(fact_3105_vebt__maxti_Oelims,axiom,
! [X: vEBT_VEBTi,Y: heap_Time_Heap @ ( option @ nat )] :
( ( ( vEBT_vebt_maxti @ X )
= Y )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leafi @ A4 @ B4 ) )
=> ~ ( ( B4
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
& ( ~ B4
=> ( ( A4
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
& ( ~ A4
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: array @ vEBT_VEBTi,Uw: vEBT_VEBTi] :
( X
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat] :
( ? [Ux: nat,Uy: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
( X
= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
=> ( Y
!= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Ma2 ) ) ) ) ) ) ) ).
% vebt_maxti.elims
thf(fact_3106_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
@ ^ [I3: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_3107_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_3108_fact__Suc__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ A ) ) ) ).
% fact_Suc_0
thf(fact_3109_fact__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ ( suc @ N ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_Suc
thf(fact_3110_fact__ge__self,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( semiring_char_0_fact @ nat @ N ) ) ).
% fact_ge_self
thf(fact_3111_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_3112_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_3113_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_3114_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_3115_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_3116_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_3117_fact__ge__1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% fact_ge_1
thf(fact_3118_fact__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [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_3119_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_3120_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_3121_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_3122_fact__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ N @ N ) ) ) ) ).
% fact_le_power
thf(fact_3123_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_3124_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_3125_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_3126_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_3127_vebt__minti_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: array @ vEBT_VEBTi,Uw2: vEBT_VEBTi] :
( ( vEBT_vebt_minti @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ).
% vebt_minti.simps(2)
thf(fact_3128_vebt__maxti_Osimps_I2_J,axiom,
! [Uu2: nat,Uv2: array @ vEBT_VEBTi,Uw2: vEBT_VEBTi] :
( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ).
% vebt_maxti.simps(2)
thf(fact_3129_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_3130_vebt__minti_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array @ vEBT_VEBTi,Uz: vEBT_VEBTi] :
( ( vEBT_vebt_minti @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz ) )
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Mi ) ) ) ).
% vebt_minti.simps(3)
thf(fact_3131_vebt__maxti_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array @ vEBT_VEBTi,Uz: vEBT_VEBTi] :
( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz ) )
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Ma ) ) ) ).
% vebt_maxti.simps(3)
thf(fact_3132_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
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_3133_fact__prod__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N6: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N6 ) ) ) ) ) ) ).
% fact_prod_Suc
thf(fact_3134_fact__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N6: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N6 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N6 ) ) ) ) ) ) ).
% fact_prod_rev
thf(fact_3135_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M5: nat] :
( if @ A
@ ( M5
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M5 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_3136_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_3137_vebt__maxti_Osimps_I1_J,axiom,
! [B2: $o,A2: $o] :
( ( B2
=> ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A2 @ B2 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
& ( ~ B2
=> ( ( A2
=> ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A2 @ B2 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
& ( ~ A2
=> ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A2 @ B2 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) ) ).
% vebt_maxti.simps(1)
thf(fact_3138_vebt__minti_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( A2
=> ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A2 @ B2 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
& ( ~ A2
=> ( ( B2
=> ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A2 @ B2 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
& ( ~ B2
=> ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A2 @ B2 ) )
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) ) ).
% vebt_minti.simps(1)
thf(fact_3139_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_3140_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_3141_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
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_3142_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_3143_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
@ ^ [I3: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_3144_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
@ ^ [I3: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_3145_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ( ( gbinomial @ A )
= ( ^ [A8: A,K3: nat] :
( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ A8 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_3146_vebt__maxti_Opelims,axiom,
! [X: vEBT_VEBTi,Y: heap_Time_Heap @ ( option @ nat )] :
( ( ( vEBT_vebt_maxti @ X )
= Y )
=> ( ( accp @ vEBT_VEBTi @ vEBT_vebt_maxti_rel @ X )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leafi @ A4 @ B4 ) )
=> ( ( ( B4
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
& ( ~ B4
=> ( ( A4
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
& ( ~ A4
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Leafi @ A4 @ B4 ) ) ) )
=> ( ! [Uu: nat,Uv: array @ vEBT_VEBTi,Uw: vEBT_VEBTi] :
( ( X
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
( ( X
= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
=> ( ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Ma2 ) ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_maxti.pelims
thf(fact_3147_vebt__minti_Opelims,axiom,
! [X: vEBT_VEBTi,Y: heap_Time_Heap @ ( option @ nat )] :
( ( ( vEBT_vebt_minti @ X )
= Y )
=> ( ( accp @ vEBT_VEBTi @ vEBT_vebt_minti_rel @ X )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leafi @ A4 @ B4 ) )
=> ( ( ( A4
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
& ( ~ A4
=> ( ( B4
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
& ( ~ B4
=> ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Leafi @ A4 @ B4 ) ) ) )
=> ( ! [Uu: nat,Uv: array @ vEBT_VEBTi,Uw: vEBT_VEBTi] :
( ( X
= ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
( ( X
= ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
=> ( ( Y
= ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Mi2 ) ) )
=> ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_minti.pelims
thf(fact_3148_Maclaurin__lemma,axiom,
! [H2: real,F3: real > real,J: nat > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ? [B9: real] :
( ( F3 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( J @ M5 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ B9 @ ( divide_divide @ real @ ( power_power @ real @ H2 @ N ) @ ( semiring_char_0_fact @ real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_3149_Maclaurin__zero,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: real,N: nat,Diff: nat > A > real] :
( ( X
= ( zero_zero @ real ) )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% Maclaurin_zero
thf(fact_3150_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_3151_sumr__cos__zero__one,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ ( zero_zero @ real ) @ M5 ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( one_one @ real ) ) ).
% sumr_cos_zero_one
thf(fact_3152_Maclaurin__exp__lt,axiom,
! [X: real,N: nat] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T7 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( exp @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M5 ) @ ( semiring_char_0_fact @ real @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_3153_Maclaurin__exp__le,axiom,
! [X: real,N: nat] :
? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( exp @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M5 ) @ ( semiring_char_0_fact @ real @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_exp_le
thf(fact_3154_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F5: set @ A,I6: set @ A,F3: A > B,I: A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ I6 )
& ( ( F3 @ I3 )
!= ( zero_zero @ B ) ) ) )
@ F5 )
=> ( ( ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I6 ) @ ( F3 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I6 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_3155_finite__enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ( finite_finite2 @ A @ S3 )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( finite_card @ A @ S3 ) )
=> ( ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S8: A] :
( ( member @ A @ S8 @ S3 )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ N ) @ S8 ) ) ) ) ) ) ) ).
% finite_enumerate_Suc''
thf(fact_3156_Least__eq__0,axiom,
! [P2: nat > $o] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P2 )
= ( zero_zero @ nat ) ) ) ).
% Least_eq_0
thf(fact_3157_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_3158_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_3159_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_3160_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_3161_abs__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_0
thf(fact_3162_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_3163_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_3164_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_3165_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_3166_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_3167_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_3168_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_3169_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P3: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P3 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_3170_sum_Oeq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,P3: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P3 @ I6 )
= ( groups7311177749621191930dd_sum @ B @ A @ P3 @ I6 ) ) ) ) ).
% sum.eq_sum
thf(fact_3171_cos__coeff__0,axiom,
( ( cos_coeff @ ( zero_zero @ nat ) )
= ( one_one @ real ) ) ).
% cos_coeff_0
thf(fact_3172_sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F3: A > B,A5: set @ A] :
( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A5 ) )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( abs_abs @ B @ ( F3 @ I3 ) )
@ A5 ) ) ) ).
% sum_abs
thf(fact_3173_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_3174_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_3175_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_3176_artanh__minus__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( artanh @ real @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( artanh @ real @ X ) ) ) ) ).
% artanh_minus_real
thf(fact_3177_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F3: A > B,A5: set @ A] :
( ord_less_eq @ B @ ( zero_zero @ B )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( abs_abs @ B @ ( F3 @ I3 ) )
@ A5 ) ) ) ).
% sum_abs_ge_zero
thf(fact_3178_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_3179_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,P3: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I6 )
& ( ( P3 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P3 @ ( insert @ B @ I @ I6 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P3 @ I6 ) ) )
& ( ~ ( member @ B @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P3 @ ( insert @ B @ I @ I6 ) )
= ( plus_plus @ A @ ( P3 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P3 @ I6 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_3180_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_3181_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_3182_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_3183_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_3184_Least1I,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P2: A > $o] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ A @ X5 @ Y3 ) )
& ! [Y3: A] :
( ( ( P2 @ Y3 )
& ! [Ya: A] :
( ( P2 @ Ya )
=> ( ord_less_eq @ A @ Y3 @ Ya ) ) )
=> ( Y3 = X5 ) ) )
=> ( P2 @ ( ord_Least @ A @ P2 ) ) ) ) ).
% Least1I
thf(fact_3185_Least1__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P2: A > $o,Z: A] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ A @ X5 @ Y3 ) )
& ! [Y3: A] :
( ( ( P2 @ Y3 )
& ! [Ya: A] :
( ( P2 @ Ya )
=> ( ord_less_eq @ A @ Y3 @ Ya ) ) )
=> ( Y3 = X5 ) ) )
=> ( ( P2 @ Z )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P2 ) @ Z ) ) ) ) ).
% Least1_le
thf(fact_3186_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P2: A > $o,X: A,Q2: A > $o] :
( ( P2 @ X )
=> ( ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( ! [Y4: A] :
( ( P2 @ Y4 )
=> ( ord_less_eq @ A @ X3 @ Y4 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( ord_Least @ A @ P2 ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_3187_Least__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P2: A > $o,X: A] :
( ( P2 @ X )
=> ( ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( ord_Least @ A @ P2 )
= X ) ) ) ) ).
% Least_equality
thf(fact_3188_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P2: A > $o,A2: A,Q2: A > $o] :
( ( P2 @ A2 )
=> ( ! [A4: A] :
( ( P2 @ A4 )
=> ( ! [B10: A] :
( ( P2 @ B10 )
=> ( ord_less_eq @ A @ A4 @ B10 ) )
=> ( Q2 @ A4 ) ) )
=> ( Q2 @ ( ord_Least @ A @ P2 ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_3189_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P2: A > $o,Q2: A > $o] :
( ? [X_12: A] : ( P2 @ X_12 )
=> ( ! [A4: A] :
( ( P2 @ A4 )
=> ( ! [B10: A] :
( ( P2 @ B10 )
=> ( ord_less_eq @ A @ A4 @ B10 ) )
=> ( Q2 @ A4 ) ) )
=> ( Q2 @ ( ord_Least @ A @ P2 ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_3190_sum_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > A,I6: set @ B] :
( ( groups1027152243600224163dd_sum @ B @ A @ G3
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I6 )
& ( ( G3 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G3 @ I6 ) ) ) ).
% sum.non_neutral'
thf(fact_3191_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_3192_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_3193_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_3194_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_3195_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C3: A,B2: A,D2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ C3 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D2 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C3 @ D2 ) ) ) ) ) ).
% abs_mult_less
thf(fact_3196_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_3197_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_3198_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_3199_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_3200_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_3201_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_3202_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_3203_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_3204_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_3205_Least__le,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P2: A > $o,K: A] :
( ( P2 @ K )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P2 ) @ K ) ) ) ).
% Least_le
thf(fact_3206_not__less__Least,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [K: A,P2: A > $o] :
( ( ord_less @ A @ K @ ( ord_Least @ A @ P2 ) )
=> ~ ( P2 @ K ) ) ) ).
% not_less_Least
thf(fact_3207_finite__image__absD,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ ( image @ A @ A @ ( abs_abs @ A ) @ S3 ) )
=> ( finite_finite2 @ A @ S3 ) ) ) ).
% finite_image_absD
thf(fact_3208_Least__Suc2,axiom,
! [P2: nat > $o,N: nat,Q2: nat > $o,M: nat] :
( ( P2 @ N )
=> ( ( Q2 @ M )
=> ( ~ ( P2 @ ( zero_zero @ nat ) )
=> ( ! [K2: nat] :
( ( P2 @ ( suc @ K2 ) )
= ( Q2 @ K2 ) )
=> ( ( ord_Least @ nat @ P2 )
= ( suc @ ( ord_Least @ nat @ Q2 ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_3209_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ E2 ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_3210_Least__Suc,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P2 )
= ( suc
@ ( ord_Least @ nat
@ ^ [M5: nat] : ( P2 @ ( suc @ M5 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_3211_abs__mult__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( times_times @ A @ ( abs_abs @ A @ Y ) @ X )
= ( abs_abs @ A @ ( times_times @ A @ Y @ X ) ) ) ) ) ).
% abs_mult_pos
thf(fact_3212_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_3213_abs__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( divide_divide @ A @ ( abs_abs @ A @ X ) @ Y )
= ( abs_abs @ A @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% abs_div_pos
thf(fact_3214_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_3215_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_3216_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_3217_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_3218_abs__if__raw,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A8: A] : ( if @ A @ ( ord_less @ A @ A8 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A8 ) @ A8 ) ) ) ) ).
% abs_if_raw
thf(fact_3219_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_3220_abs__if,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A8: A] : ( if @ A @ ( ord_less @ A @ A8 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A8 ) @ A8 ) ) ) ) ).
% abs_if
thf(fact_3221_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C3 @ D2 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ C3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_3222_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_3223_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A2: A,R2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A2 ) ) @ R2 )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ R2 ) @ X )
& ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A2 @ R2 ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_3224_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A2: A,R2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A2 ) ) @ R2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ A2 @ R2 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ A2 @ R2 ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_3225_sum_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I3: B] : ( plus_plus @ A @ ( G3 @ I3 ) @ ( H2 @ I3 ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G3 @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ).
% sum.distrib_triv'
thf(fact_3226_abs__real__def,axiom,
( ( abs_abs @ real )
= ( ^ [A8: real] : ( if @ real @ ( ord_less @ real @ A8 @ ( zero_zero @ real ) ) @ ( uminus_uminus @ real @ A8 ) @ A8 ) ) ) ).
% abs_real_def
thf(fact_3227_sin__bound__lemma,axiom,
! [X: real,Y: real,U: real,V3: real] :
( ( X = Y )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ U ) @ V3 )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( plus_plus @ real @ X @ U ) @ Y ) ) @ V3 ) ) ) ).
% sin_bound_lemma
thf(fact_3228_finite__abs__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [K3: A] :
( ( member @ A @ K3 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( abs_abs @ A @ K3 ) @ A2 ) ) ) ) ) ).
% finite_abs_int_segment
thf(fact_3229_Least__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
=> ( ? [X_12: A] : ( P2 @ X_12 )
=> ( ( ord_Least @ A @ P2 )
= ( lattic643756798350308766er_Min @ A @ ( collect @ A @ P2 ) ) ) ) ) ) ).
% Least_Min
thf(fact_3230_enumerate__0,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A] :
( ( infini527867602293511546merate @ A @ S3 @ ( zero_zero @ nat ) )
= ( ord_Least @ A
@ ^ [N6: A] : ( member @ A @ N6 @ S3 ) ) ) ) ).
% enumerate_0
thf(fact_3231_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T4: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ S3 )
= ( groups1027152243600224163dd_sum @ B @ A @ G3 @ T4 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_3232_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T4: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ T4 )
= ( groups1027152243600224163dd_sum @ B @ A @ G3 @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_3233_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T4: set @ B,H2: B > A,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( H2 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G3 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ S3 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_3234_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T4: set @ B,G3: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G3 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ T4 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_3235_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ).
% abs_add_one_gt_zero
thf(fact_3236_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( X
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_3237_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_3238_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ Y @ ( ring_1_Ints @ A ) )
=> ( ( X = Y )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_3239_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I6 )
& ( ( G3 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I6 )
& ( ( H2 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I3: B] : ( plus_plus @ A @ ( G3 @ I3 ) @ ( H2 @ I3 ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G3 @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_3240_sum_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ B @ A )
= ( ^ [P7: B > A,I8: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I8 )
& ( ( P7 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P7
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I8 )
& ( ( P7 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_3241_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_3242_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,X: A > B,A2: A > B,B2: B,Delta: B] :
( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I6 )
= ( one_one @ B ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A2 @ I2 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( times_times @ B @ ( A2 @ I3 ) @ ( X @ I3 ) )
@ I6 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_3243_enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ S3 )
=> ( ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S8: A] :
( ( member @ A @ S8 @ S3 )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ N ) @ S8 ) ) ) ) ) ) ).
% enumerate_Suc''
thf(fact_3244_enumerate__Suc,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) )
= ( infini527867602293511546merate @ A
@ ( minus_minus @ ( set @ A ) @ S3
@ ( insert @ A
@ ( ord_Least @ A
@ ^ [N6: A] : ( member @ A @ N6 @ S3 ) )
@ ( bot_bot @ ( set @ A ) ) ) )
@ N ) ) ) ).
% enumerate_Suc
thf(fact_3245_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I6: set @ A,F3: A > B,I: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ I6 )
& ( ( F3 @ I3 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I6 ) @ ( F3 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I6 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_3246_lemma__interval,axiom,
! [A2: real,X: real,B2: real] :
( ( ord_less @ real @ A2 @ X )
=> ( ( ord_less @ real @ X @ B2 )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y4 ) ) @ D3 )
=> ( ( ord_less_eq @ real @ A2 @ Y4 )
& ( ord_less_eq @ real @ Y4 @ B2 ) ) ) ) ) ) ).
% lemma_interval
thf(fact_3247_lemma__interval__lt,axiom,
! [A2: real,X: real,B2: real] :
( ( ord_less @ real @ A2 @ X )
=> ( ( ord_less @ real @ X @ B2 )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y4 ) ) @ D3 )
=> ( ( ord_less @ real @ A2 @ Y4 )
& ( ord_less @ real @ Y4 @ B2 ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_3248_set__vebt__succ_H,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ? [X5: nat] :
( ( member @ nat @ X5 @ ( vEBT_set_vebt @ T2 ) )
& ( ord_less @ nat @ X @ X5 ) )
=> ( ( vEBT_vebt_succ @ T2 @ X )
= ( some @ nat
@ ( ord_Least @ nat
@ ^ [Y5: nat] :
( ( member @ nat @ Y5 @ ( vEBT_set_vebt @ T2 ) )
& ( ord_less @ nat @ X @ Y5 ) ) ) ) ) )
& ( ~ ? [X3: nat] :
( ( member @ nat @ X3 @ ( vEBT_set_vebt @ T2 ) )
& ( ord_less @ nat @ X @ X3 ) )
=> ( ( vEBT_vebt_succ @ T2 @ X )
= ( none @ nat ) ) ) ) ) ).
% set_vebt_succ'
thf(fact_3249_arctan__add,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( plus_plus @ real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( arctan @ ( divide_divide @ real @ ( plus_plus @ real @ X @ Y ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( times_times @ real @ X @ Y ) ) ) ) ) ) ) ).
% arctan_add
thf(fact_3250_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_3251_bex__empty,axiom,
! [A: $tType,P2: A > $o] :
~ ? [X5: A] :
( ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) )
& ( P2 @ X5 ) ) ).
% bex_empty
thf(fact_3252_finite__Collect__bex,axiom,
! [B: $tType,A: $tType,A5: set @ A,Q2: B > A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
? [Y5: A] :
( ( member @ A @ Y5 @ A5 )
& ( Q2 @ X4 @ Y5 ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Y5: B] : ( Q2 @ Y5 @ X4 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_3253_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_3254_zero__less__arctan__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( arctan @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% zero_less_arctan_iff
thf(fact_3255_arctan__less__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( arctan @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% arctan_less_zero_iff
thf(fact_3256_zero__le__arctan__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arctan @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% zero_le_arctan_iff
thf(fact_3257_arctan__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( arctan @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% arctan_le_zero_iff
thf(fact_3258_sin__coeff__0,axiom,
( ( sin_coeff @ ( zero_zero @ nat ) )
= ( zero_zero @ real ) ) ).
% sin_coeff_0
thf(fact_3259_set__vebt__pred_H,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ? [X5: nat] :
( ( member @ nat @ X5 @ ( vEBT_set_vebt @ T2 ) )
& ( ord_less @ nat @ X5 @ X ) )
=> ( ( vEBT_vebt_pred @ T2 @ X )
= ( some @ nat
@ ( order_Greatest @ nat
@ ^ [Y5: nat] :
( ( member @ nat @ Y5 @ ( vEBT_set_vebt @ T2 ) )
& ( ord_less @ nat @ Y5 @ X ) ) ) ) ) )
& ( ~ ? [X3: nat] :
( ( member @ nat @ X3 @ ( vEBT_set_vebt @ T2 ) )
& ( ord_less @ nat @ X3 @ X ) )
=> ( ( vEBT_vebt_pred @ T2 @ X )
= ( none @ nat ) ) ) ) ) ).
% set_vebt_pred'
thf(fact_3260_Bex__def,axiom,
! [A: $tType] :
( ( bex @ A )
= ( ^ [A6: set @ A,P5: A > $o] :
? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P5 @ X4 ) ) ) ) ).
% Bex_def
thf(fact_3261_arctan__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% arctan_less_iff
thf(fact_3262_arctan__monotone,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% arctan_monotone
thf(fact_3263_arctan__monotone_H,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% arctan_monotone'
thf(fact_3264_arctan__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% arctan_le_iff
thf(fact_3265_image__def,axiom,
! [B: $tType,A: $tType] :
( ( image @ A @ B )
= ( ^ [F4: A > B,A6: set @ A] :
( collect @ B
@ ^ [Y5: B] :
? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( Y5
= ( F4 @ X4 ) ) ) ) ) ) ).
% image_def
thf(fact_3266_Bex__fold,axiom,
! [A: $tType,A5: set @ A,P2: A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P2 @ X4 ) ) )
= ( finite_fold @ A @ $o
@ ^ [K3: A,S8: $o] :
( S8
| ( P2 @ K3 ) )
@ $false
@ A5 ) ) ) ).
% Bex_fold
thf(fact_3267_finite__int__iff__bounded__le,axiom,
( ( finite_finite2 @ int )
= ( ^ [S6: set @ int] :
? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image @ int @ int @ ( abs_abs @ int ) @ S6 ) @ ( set_ord_atMost @ int @ K3 ) ) ) ) ).
% finite_int_iff_bounded_le
thf(fact_3268_finite__int__iff__bounded,axiom,
( ( finite_finite2 @ int )
= ( ^ [S6: set @ int] :
? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image @ int @ int @ ( abs_abs @ int ) @ S6 ) @ ( set_ord_lessThan @ int @ K3 ) ) ) ) ).
% finite_int_iff_bounded
thf(fact_3269_infinite__int__iff__unbounded__le,axiom,
! [S3: set @ int] :
( ( ~ ( finite_finite2 @ int @ S3 ) )
= ( ! [M5: int] :
? [N6: int] :
( ( ord_less_eq @ int @ M5 @ ( abs_abs @ int @ N6 ) )
& ( member @ int @ N6 @ S3 ) ) ) ) ).
% infinite_int_iff_unbounded_le
thf(fact_3270_infinite__int__iff__unbounded,axiom,
! [S3: set @ int] :
( ( ~ ( finite_finite2 @ int @ S3 ) )
= ( ! [M5: int] :
? [N6: int] :
( ( ord_less @ int @ M5 @ ( abs_abs @ int @ N6 ) )
& ( member @ int @ N6 @ S3 ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_3271_zabs__def,axiom,
( ( abs_abs @ int )
= ( ^ [I3: int] : ( if @ int @ ( ord_less @ int @ I3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ int @ I3 ) @ I3 ) ) ) ).
% zabs_def
thf(fact_3272_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_3273_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F3: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq @ nat @ M @ I2 )
& ( ord_less @ nat @ I2 @ N ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( suc @ I2 ) ) @ ( F3 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ int @ ( F3 @ M ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F3 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ M @ I2 )
& ( ord_less_eq @ nat @ I2 @ N )
& ( ( F3 @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_3274_incr__lemma,axiom,
! [D2: int,Z: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ Z @ ( plus_plus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z ) ) @ ( one_one @ int ) ) @ D2 ) ) ) ) ).
% incr_lemma
thf(fact_3275_decr__lemma,axiom,
! [D2: int,X: int,Z: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z ) ) @ ( one_one @ int ) ) @ D2 ) ) @ Z ) ) ).
% decr_lemma
thf(fact_3276_nat__ivt__aux,axiom,
! [N: nat,F3: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( suc @ I2 ) ) @ ( F3 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F3 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F3 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
& ( ( F3 @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_3277_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_3278_nat0__intermed__int__val,axiom,
! [N: nat,F3: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) @ ( F3 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F3 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F3 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
& ( ( F3 @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_3279_ln__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_3280_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_3281_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_3282_sorted__find__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P2: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
& ( P2 @ X5 ) )
=> ( ( find @ A @ P2 @ Xs )
= ( some @ A
@ ( lattic643756798350308766er_Min @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P2 @ X4 ) ) ) ) ) ) ) ) ) ).
% sorted_find_Min
thf(fact_3283_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A8: A,Xs2: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N6: nat] : ( times_times @ A @ ( F4 @ ( nth @ B @ Xs2 @ N6 ) ) @ ( power_power @ A @ A8 @ N6 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_3284_ln__inj__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( ln_ln @ real @ X )
= ( ln_ln @ real @ Y ) )
= ( X = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_3285_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_3286_ln__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_3287_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ln_ln @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( one_one @ real ) ) ) ) ).
% ln_eq_zero_iff
thf(fact_3288_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
= ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_3289_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_less_zero_iff
thf(fact_3290_exp__ln,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( exp @ real @ ( ln_ln @ real @ X ) )
= X ) ) ).
% exp_ln
thf(fact_3291_exp__ln__iff,axiom,
! [X: real] :
( ( ( exp @ real @ ( ln_ln @ real @ X ) )
= X )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% exp_ln_iff
thf(fact_3292_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_ge_zero_iff
thf(fact_3293_ln__le__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_le_zero_iff
thf(fact_3294_find__SomeD_I1_J,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P2 @ Xs )
= ( some @ A @ X ) )
=> ( P2 @ X ) ) ).
% find_SomeD(1)
thf(fact_3295_ln__less__self,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( ln_ln @ real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_3296_GreatestI__ex__nat,axiom,
! [P2: nat > $o,B2: nat] :
( ? [X_12: nat] : ( P2 @ X_12 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( P2 @ ( order_Greatest @ nat @ P2 ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_3297_Greatest__le__nat,axiom,
! [P2: nat > $o,K: nat,B2: nat] :
( ( P2 @ K )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P2 ) ) ) ) ).
% Greatest_le_nat
thf(fact_3298_GreatestI__nat,axiom,
! [P2: nat > $o,K: nat,B2: nat] :
( ( P2 @ K )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( P2 @ ( order_Greatest @ nat @ P2 ) ) ) ) ).
% GreatestI_nat
thf(fact_3299_ln__bound,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ X ) ) ).
% ln_bound
thf(fact_3300_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) ) ) ).
% ln_gt_zero
thf(fact_3301_ln__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% ln_less_zero
thf(fact_3302_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_3303_ln__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) ) ) ).
% ln_ge_zero
thf(fact_3304_size__list__estimation,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: nat,F3: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less @ nat @ Y @ ( F3 @ X ) )
=> ( ord_less @ nat @ Y @ ( size_list @ A @ F3 @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_3305_size__list__pointwise,axiom,
! [A: $tType,Xs: list @ A,F3: A > nat,G3: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ nat @ ( size_list @ A @ F3 @ Xs ) @ ( size_list @ A @ G3 @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_3306_size__list__estimation_H,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: nat,F3: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ Y @ ( F3 @ X ) )
=> ( ord_less_eq @ nat @ Y @ ( size_list @ A @ F3 @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_3307_find__SomeD_I2_J,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P2 @ Xs )
= ( some @ A @ X ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% find_SomeD(2)
thf(fact_3308_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P2: A > $o,X: A,Q2: A > $o] :
( ( P2 @ X )
=> ( ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( ! [Y4: A] :
( ( P2 @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_Greatest @ A @ P2 ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_3309_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P2: A > $o,X: A] :
( ( P2 @ X )
=> ( ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( order_Greatest @ A @ P2 )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_3310_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_3311_ln__add__one__self__le__self,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self
thf(fact_3312_ln__mult,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ln_ln @ real @ ( times_times @ real @ X @ Y ) )
= ( plus_plus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) ) ) ) ) ).
% ln_mult
thf(fact_3313_ln__eq__minus__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ln_ln @ real @ X )
= ( minus_minus @ real @ X @ ( one_one @ real ) ) )
=> ( X
= ( one_one @ real ) ) ) ) ).
% ln_eq_minus_one
thf(fact_3314_ln__div,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ln_ln @ real @ ( divide_divide @ real @ X @ Y ) )
= ( minus_minus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) ) ) ) ) ).
% ln_div
thf(fact_3315_ln__ge__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ Y @ ( ln_ln @ real @ X ) )
= ( ord_less_eq @ real @ ( exp @ real @ Y ) @ X ) ) ) ).
% ln_ge_iff
thf(fact_3316_ln__x__over__x__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( ln_ln @ real @ Y ) @ Y ) @ ( divide_divide @ real @ ( ln_ln @ real @ X ) @ X ) ) ) ) ).
% ln_x_over_x_mono
thf(fact_3317_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A8: A,Xs2: list @ B] :
( foldr @ B @ A
@ ^ [X4: B,B8: A] : ( plus_plus @ A @ ( F4 @ X4 ) @ ( times_times @ A @ A8 @ B8 ) )
@ Xs2
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_3318_ln__le__minus__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( minus_minus @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_le_minus_one
thf(fact_3319_ln__diff__le,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ X @ Y ) @ Y ) ) ) ) ).
% ln_diff_le
thf(fact_3320_ln__add__one__self__le__self2,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self2
thf(fact_3321_ln__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( power_power @ real @ X @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( ln_ln @ real @ X ) ) ) ) ).
% ln_realpow
thf(fact_3322_ln__one__minus__pos__upper__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X ) ) @ ( uminus_uminus @ real @ X ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_3323_ln__prod,axiom,
! [A: $tType,I6: set @ A,F3: A > real] :
( ( finite_finite2 @ A @ I6 )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ I2 ) ) )
=> ( ( ln_ln @ real @ ( groups7121269368397514597t_prod @ A @ real @ F3 @ I6 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X4: A] : ( ln_ln @ real @ ( F3 @ X4 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_3324_find__Some__iff2,axiom,
! [A: $tType,X: A,P2: A > $o,Xs: list @ A] :
( ( ( some @ A @ X )
= ( find @ A @ P2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P2 @ ( nth @ A @ Xs @ I3 ) )
& ( X
= ( nth @ A @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I3 )
=> ~ ( P2 @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_3325_find__Some__iff,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P2 @ Xs )
= ( some @ A @ X ) )
= ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P2 @ ( nth @ A @ Xs @ I3 ) )
& ( X
= ( nth @ A @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I3 )
=> ~ ( P2 @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_3326_list_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X21: A,X22: list @ A] :
( ( size_list @ A @ X @ ( cons @ A @ X21 @ X22 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X @ X21 ) @ ( size_list @ A @ X @ X22 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_3327_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [F3: nat > B,Ns: list @ nat] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_eq @ nat @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ nat @ B @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ nat ) @ Ns ) ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F3 @ Ns ) ) ) ) ) ) ).
% sorted_wrt_less_sum_mono_lowerbound
thf(fact_3328_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs: list @ A,X6: set @ A,F3: A > nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X6 )
=> ( ( finite_finite2 @ A @ X6 )
=> ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X4 ) @ ( F3 @ X4 ) )
@ X6 ) ) ) ) ).
% sum_list_map_eq_sum_count2
thf(fact_3329_dual__Max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Max @ A
@ ^ [X4: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X4 ) )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% dual_Max
thf(fact_3330_dual__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Min @ A
@ ^ [X4: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X4 ) )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% dual_Min
thf(fact_3331_upto_Opinduct,axiom,
! [A0: int,A12: int,P2: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A12 ) )
=> ( ! [I2: int,J2: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I2 @ J2 ) )
=> ( ( ( ord_less_eq @ int @ I2 @ J2 )
=> ( P2 @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J2 ) )
=> ( P2 @ I2 @ J2 ) ) )
=> ( P2 @ A0 @ A12 ) ) ) ).
% upto.pinduct
thf(fact_3332_sum__list__eq__0__iff,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [Ns: list @ A] :
( ( ( groups8242544230860333062m_list @ A @ Ns )
= ( zero_zero @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ns ) )
=> ( X4
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_3333_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( zero_zero @ A )
@ Xs ) )
= ( zero_zero @ A ) ) ) ).
% sum_list_0
thf(fact_3334_linorder_OMax_Ocong,axiom,
! [A: $tType] :
( ( lattices_Max @ A )
= ( lattices_Max @ A ) ) ).
% linorder.Max.cong
thf(fact_3335_linorder_OMin_Ocong,axiom,
! [A: $tType] :
( ( lattices_Min @ A )
= ( lattices_Min @ A ) ) ).
% linorder.Min.cong
thf(fact_3336_member__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% member_le_sum_list
thf(fact_3337_Groups__List_Osum__list__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_3338_sum__list__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
=> ( ( ( groups8242544230860333062m_list @ A @ Xs )
= ( zero_zero @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( X4
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_3339_sum__list__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_nonpos
thf(fact_3340_sum__list__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [Xs: list @ A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( groups8242544230860333062m_list @ A @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ A @ A @ ( abs_abs @ A ) @ Xs ) ) ) ) ).
% sum_list_abs
thf(fact_3341_sum__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( ^ [Xs2: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs2 @ ( zero_zero @ A ) ) ) ) ) ).
% sum_list.eq_foldr
thf(fact_3342_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( ordere6658533253407199908up_add @ B ) )
=> ! [Xs: list @ A,F3: A > B,G3: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F3 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G3 @ Xs ) ) ) ) ) ).
% sum_list_mono
thf(fact_3343_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_3344_sum__list__Suc,axiom,
! [A: $tType,F3: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat
@ ( map @ A @ nat
@ ^ [X4: A] : ( suc @ ( F3 @ X4 ) )
@ Xs ) )
= ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% sum_list_Suc
thf(fact_3345_card__length__sum__list__rec,axiom,
! [M: nat,N8: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L4: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L4 )
= M )
& ( ( groups8242544230860333062m_list @ nat @ L4 )
= N8 ) ) ) )
= ( plus_plus @ nat
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L4: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L4 )
= ( minus_minus @ nat @ M @ ( one_one @ nat ) ) )
& ( ( groups8242544230860333062m_list @ nat @ L4 )
= N8 ) ) ) )
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L4: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L4 )
= M )
& ( ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ L4 ) @ ( one_one @ nat ) )
= N8 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_3346_sum__list__sum__nth,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ( ( groups8242544230860333062m_list @ B )
= ( ^ [Xs2: list @ B] : ( groups7311177749621191930dd_sum @ nat @ B @ ( nth @ B @ Xs2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ).
% sum_list_sum_nth
thf(fact_3347_sum__list__update,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [K: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X ) @ ( nth @ A @ Xs @ K ) ) ) ) ) ).
% sum_list_update
thf(fact_3348_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_3349_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_3350_max__extp_Omax__extI,axiom,
! [A: $tType,X6: set @ A,Y7: set @ A,R3: A > A > $o] :
( ( finite_finite2 @ A @ X6 )
=> ( ( finite_finite2 @ A @ Y7 )
=> ( ( Y7
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ Y7 )
& ( R3 @ X3 @ Xa ) ) )
=> ( max_extp @ A @ R3 @ X6 @ Y7 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_3351_max__extp_Osimps,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R4: A > A > $o,A13: set @ A,A24: set @ A] :
( ( finite_finite2 @ A @ A13 )
& ( finite_finite2 @ A @ A24 )
& ( A24
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A13 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A24 )
& ( R4 @ X4 @ Y5 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_3352_unique__quotient,axiom,
! [A2: int,B2: int,Q3: int,R2: int,Q7: int,R6: int] :
( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q7 @ R6 ) )
=> ( Q3 = Q7 ) ) ) ).
% unique_quotient
thf(fact_3353_unique__remainder,axiom,
! [A2: int,B2: int,Q3: int,R2: int,Q7: int,R6: int] :
( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q7 @ R6 ) )
=> ( R2 = R6 ) ) ) ).
% unique_remainder
thf(fact_3354_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_3355_mod__int__unique,axiom,
! [K: int,L: int,Q3: int,R2: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( ( modulo_modulo @ int @ K @ L )
= R2 ) ) ).
% mod_int_unique
thf(fact_3356_div__int__unique,axiom,
! [K: int,L: int,Q3: int,R2: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( ( divide_divide @ int @ K @ L )
= Q3 ) ) ).
% div_int_unique
thf(fact_3357_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_3358_eucl__rel__int,axiom,
! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ ( divide_divide @ int @ K @ L ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% eucl_rel_int
thf(fact_3359_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_3360_max__extp_Ocases,axiom,
! [A: $tType,R3: A > A > $o,A12: set @ A,A23: set @ A] :
( ( max_extp @ A @ R3 @ A12 @ A23 )
=> ~ ( ( finite_finite2 @ A @ A12 )
=> ( ( finite_finite2 @ A @ A23 )
=> ( ( A23
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ A12 )
=> ? [Xa4: A] :
( ( member @ A @ Xa4 @ A23 )
& ( R3 @ X5 @ Xa4 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_3361_finite__enumerate,axiom,
! [S3: set @ nat] :
( ( finite_finite2 @ nat @ S3 )
=> ? [R: nat > nat] :
( ( strict_mono_on @ nat @ nat @ R @ ( set_ord_lessThan @ nat @ ( finite_card @ nat @ S3 ) ) )
& ! [N9: nat] :
( ( ord_less @ nat @ N9 @ ( finite_card @ nat @ S3 ) )
=> ( member @ nat @ ( R @ N9 ) @ S3 ) ) ) ) ).
% finite_enumerate
thf(fact_3362_map__distinct__upd__conv,axiom,
! [B: $tType,A: $tType,I: nat,L: list @ A,F3: A > B,X: B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( distinct @ A @ L )
=> ( ( list_update @ B @ ( map @ A @ B @ F3 @ L ) @ I @ X )
= ( map @ A @ B @ ( fun_upd @ A @ B @ F3 @ ( nth @ A @ L @ I ) @ X ) @ L ) ) ) ) ).
% map_distinct_upd_conv
thf(fact_3363_in__lex__prod,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ A3 @ B3 ) ) @ ( lex_prod @ A @ B @ R2 @ S ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A3 ) @ R2 )
| ( ( A2 = A3 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_3364_bounded__linear__axioms_Ointro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ? [K6: real] :
! [X3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X3 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K6 ) )
=> ( real_V4916620083959148203axioms @ A @ B @ F3 ) ) ) ).
% bounded_linear_axioms.intro
thf(fact_3365_bounded__linear__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( real_V4916620083959148203axioms @ A @ B )
= ( ^ [F4: A > B] :
? [K5: real] :
! [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F4 @ X4 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ K5 ) ) ) ) ) ).
% bounded_linear_axioms_def
thf(fact_3366_image__update,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,F3: A > B,N: B] :
( ~ ( member @ A @ X @ A5 )
=> ( ( image @ A @ B @ ( fun_upd @ A @ B @ F3 @ X @ N ) @ A5 )
= ( image @ A @ B @ F3 @ A5 ) ) ) ).
% image_update
thf(fact_3367_finite__update__induct,axiom,
! [B: $tType,A: $tType,F3: A > B,C3: B,P2: ( A > B ) > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A8: A] :
( ( F3 @ A8 )
!= C3 ) ) )
=> ( ( P2
@ ^ [A8: A] : C3 )
=> ( ! [A4: A,B4: B,F2: A > B] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [C7: A] :
( ( F2 @ C7 )
!= C3 ) ) )
=> ( ( ( F2 @ A4 )
= C3 )
=> ( ( B4 != C3 )
=> ( ( P2 @ F2 )
=> ( P2 @ ( fun_upd @ A @ B @ F2 @ A4 @ B4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ).
% finite_update_induct
thf(fact_3368_fun__upd__image,axiom,
! [A: $tType,B: $tType,X: B,A5: set @ B,F3: B > A,Y: A] :
( ( ( member @ B @ X @ A5 )
=> ( ( image @ B @ A @ ( fun_upd @ B @ A @ F3 @ X @ Y ) @ A5 )
= ( insert @ A @ Y @ ( image @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ X @ A5 )
=> ( ( image @ B @ A @ ( fun_upd @ B @ A @ F3 @ X @ Y ) @ A5 )
= ( image @ B @ A @ F3 @ A5 ) ) ) ) ).
% fun_upd_image
thf(fact_3369_strict__mono__onD,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [F3: A > B,A5: set @ A,R2: A,S: A] :
( ( strict_mono_on @ A @ B @ F3 @ A5 )
=> ( ( member @ A @ R2 @ A5 )
=> ( ( member @ A @ S @ A5 )
=> ( ( ord_less @ A @ R2 @ S )
=> ( ord_less @ B @ ( F3 @ R2 ) @ ( F3 @ S ) ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_3370_strict__mono__onI,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [A5: set @ A,F3: A > B] :
( ! [R: A,S4: A] :
( ( member @ A @ R @ A5 )
=> ( ( member @ A @ S4 @ A5 )
=> ( ( ord_less @ A @ R @ S4 )
=> ( ord_less @ B @ ( F3 @ R ) @ ( F3 @ S4 ) ) ) ) )
=> ( strict_mono_on @ A @ B @ F3 @ A5 ) ) ) ).
% strict_mono_onI
thf(fact_3371_strict__mono__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( strict_mono_on @ A @ B )
= ( ^ [F4: A > B,A6: set @ A] :
! [R5: A,S8: A] :
( ( ( member @ A @ R5 @ A6 )
& ( member @ A @ S8 @ A6 )
& ( ord_less @ A @ R5 @ S8 ) )
=> ( ord_less @ B @ ( F4 @ R5 ) @ ( F4 @ S8 ) ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_3372_strict__mono__on__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( preorder @ B ) )
=> ! [F3: A > B,A5: set @ A,X: A,Y: A] :
( ( strict_mono_on @ A @ B @ F3 @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y @ A5 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_3373_comp__fun__commute__on_Ofold__insert__remove,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,X: A,A5: set @ A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S3 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( insert @ A @ X @ A5 ) )
= ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert_remove
thf(fact_3374_comp__fun__commute__on_Ofold__rec,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,A5: set @ A,X: A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S3 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ A5 )
= ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_rec
thf(fact_3375_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A8: A,Xs2: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N6: nat] : ( compow @ ( A > A ) @ N6 @ ( times_times @ A @ A8 ) @ ( F4 @ ( nth @ B @ Xs2 @ N6 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_3376_semiring__char__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiri4206861660011772517g_char @ A )
= ( ^ [Uu4: itself @ A] :
( gcd_Gcd @ nat
@ ( collect @ nat
@ ^ [N6: nat] :
( ( semiring_1_of_nat @ A @ N6 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% semiring_char_def
thf(fact_3377_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat )
@ ^ [X4: nat,Y5: nat] : ( ord_less_eq @ nat @ Y5 @ X4 )
@ ^ [X4: nat,Y5: nat] : ( ord_less @ nat @ Y5 @ X4 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_3378_Suc__funpow,axiom,
! [N: nat] :
( ( compow @ ( nat > nat ) @ N @ suc )
= ( plus_plus @ nat @ N ) ) ).
% Suc_funpow
thf(fact_3379_funpow__0,axiom,
! [A: $tType,F3: A > A,X: A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F3 @ X )
= X ) ).
% funpow_0
thf(fact_3380_relpowp__Suc__I2,axiom,
! [A: $tType,P2: A > A > $o,X: A,Y: A,N: nat,Z: A] :
( ( P2 @ X @ Y )
=> ( ( compow @ ( A > A > $o ) @ N @ P2 @ Y @ Z )
=> ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P2 @ X @ Z ) ) ) ).
% relpowp_Suc_I2
thf(fact_3381_relpowp__Suc__E2,axiom,
! [A: $tType,N: nat,P2: A > A > $o,X: A,Z: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P2 @ X @ Z )
=> ~ ! [Y3: A] :
( ( P2 @ X @ Y3 )
=> ~ ( compow @ ( A > A > $o ) @ N @ P2 @ Y3 @ Z ) ) ) ).
% relpowp_Suc_E2
thf(fact_3382_relpowp__Suc__D2,axiom,
! [A: $tType,N: nat,P2: A > A > $o,X: A,Z: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P2 @ X @ Z )
=> ? [Y3: A] :
( ( P2 @ X @ Y3 )
& ( compow @ ( A > A > $o ) @ N @ P2 @ Y3 @ Z ) ) ) ).
% relpowp_Suc_D2
thf(fact_3383_relpowp__Suc__I,axiom,
! [A: $tType,N: nat,P2: A > A > $o,X: A,Y: A,Z: A] :
( ( compow @ ( A > A > $o ) @ N @ P2 @ X @ Y )
=> ( ( P2 @ Y @ Z )
=> ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P2 @ X @ Z ) ) ) ).
% relpowp_Suc_I
thf(fact_3384_relpowp__Suc__E,axiom,
! [A: $tType,N: nat,P2: A > A > $o,X: A,Z: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P2 @ X @ Z )
=> ~ ! [Y3: A] :
( ( compow @ ( A > A > $o ) @ N @ P2 @ X @ Y3 )
=> ~ ( P2 @ Y3 @ Z ) ) ) ).
% relpowp_Suc_E
thf(fact_3385_relpowp_Osimps_I1_J,axiom,
! [A: $tType,R3: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ R3 )
= ( ^ [Y6: A,Z4: A] : Y6 = Z4 ) ) ).
% relpowp.simps(1)
thf(fact_3386_relpowp__0__E,axiom,
! [A: $tType,P2: A > A > $o,X: A,Y: A] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P2 @ X @ Y )
=> ( X = Y ) ) ).
% relpowp_0_E
thf(fact_3387_relpowp__0__I,axiom,
! [A: $tType,P2: A > A > $o,X: A] : ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P2 @ X @ X ) ).
% relpowp_0_I
thf(fact_3388_comp__fun__commute__on_Ofun__left__comm,axiom,
! [A: $tType,B: $tType,S3: set @ A,F3: A > B > B,X: A,Y: A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( member @ A @ X @ S3 )
=> ( ( member @ A @ Y @ S3 )
=> ( ( F3 @ Y @ ( F3 @ X @ Z ) )
= ( F3 @ X @ ( F3 @ Y @ Z ) ) ) ) ) ) ).
% comp_fun_commute_on.fun_left_comm
thf(fact_3389_funpow__swap1,axiom,
! [A: $tType,F3: A > A,N: nat,X: A] :
( ( F3 @ ( compow @ ( A > A ) @ N @ F3 @ X ) )
= ( compow @ ( A > A ) @ N @ F3 @ ( F3 @ X ) ) ) ).
% funpow_swap1
thf(fact_3390_comp__fun__commute__on_Ocomp__fun__commute__on__funpow,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,G3: A > nat] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( finite4664212375090638736ute_on @ A @ B @ S3
@ ^ [X4: A] : ( compow @ ( B > B ) @ ( G3 @ X4 ) @ ( F3 @ X4 ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on_funpow
thf(fact_3391_funpow__mult,axiom,
! [A: $tType,N: nat,M: nat,F3: A > A] :
( ( compow @ ( A > A ) @ N @ ( compow @ ( A > A ) @ M @ F3 ) )
= ( compow @ ( A > A ) @ ( times_times @ nat @ M @ N ) @ F3 ) ) ).
% funpow_mult
thf(fact_3392_relpowp__E,axiom,
! [A: $tType,N: nat,P2: A > A > $o,X: A,Z: A] :
( ( compow @ ( A > A > $o ) @ N @ P2 @ X @ Z )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z ) )
=> ~ ! [Y3: A,M3: nat] :
( ( N
= ( suc @ M3 ) )
=> ( ( compow @ ( A > A > $o ) @ M3 @ P2 @ X @ Y3 )
=> ~ ( P2 @ Y3 @ Z ) ) ) ) ) ).
% relpowp_E
thf(fact_3393_relpowp__E2,axiom,
! [A: $tType,N: nat,P2: A > A > $o,X: A,Z: A] :
( ( compow @ ( A > A > $o ) @ N @ P2 @ X @ Z )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z ) )
=> ~ ! [Y3: A,M3: nat] :
( ( N
= ( suc @ M3 ) )
=> ( ( P2 @ X @ Y3 )
=> ~ ( compow @ ( A > A > $o ) @ M3 @ P2 @ Y3 @ Z ) ) ) ) ) ).
% relpowp_E2
thf(fact_3394_relpowp__fun__conv,axiom,
! [A: $tType] :
( ( compow @ ( A > A > $o ) )
= ( ^ [N6: nat,P5: A > A > $o,X4: A,Y5: A] :
? [F4: nat > A] :
( ( ( F4 @ ( zero_zero @ nat ) )
= X4 )
& ( ( F4 @ N6 )
= Y5 )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N6 )
=> ( P5 @ ( F4 @ I3 ) @ ( F4 @ ( suc @ I3 ) ) ) ) ) ) ) ).
% relpowp_fun_conv
thf(fact_3395_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_3396_of__nat__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N6: nat] : ( compow @ ( A > A ) @ N6 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_def
thf(fact_3397_Finite__Set_Ofold__cong,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,G3: A > B > B,A5: set @ A,S: B,T2: B,B5: set @ A] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( finite4664212375090638736ute_on @ A @ B @ S3 @ G3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S3 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( F3 @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( S = T2 )
=> ( ( A5 = B5 )
=> ( ( finite_fold @ A @ B @ F3 @ S @ A5 )
= ( finite_fold @ A @ B @ G3 @ T2 @ B5 ) ) ) ) ) ) ) ) ) ).
% Finite_Set.fold_cong
thf(fact_3398_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,X: A,A5: set @ A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S3 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z @ A5 ) )
= ( finite_fold @ A @ B @ F3 @ ( F3 @ X @ Z ) @ A5 ) ) ) ) ) ).
% comp_fun_commute_on.fold_fun_left_comm
thf(fact_3399_comp__fun__commute__on_Ofold__insert2,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,X: A,A5: set @ A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S3 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( insert @ A @ X @ A5 ) )
= ( finite_fold @ A @ B @ F3 @ ( F3 @ X @ Z ) @ A5 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert2
thf(fact_3400_comp__fun__commute__on_Ofold__insert,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,X: A,A5: set @ A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S3 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( insert @ A @ X @ A5 ) )
= ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z @ A5 ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert
thf(fact_3401_Nat_Ofunpow__code__def,axiom,
! [A: $tType] :
( ( funpow @ A )
= ( compow @ ( A > A ) ) ) ).
% Nat.funpow_code_def
thf(fact_3402_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A13: int,A24: int,A33: product_prod @ int @ int] :
( ? [K3: int] :
( ( A13 = K3 )
& ( A24
= ( zero_zero @ int ) )
& ( A33
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) )
| ? [L4: int,K3: int,Q8: int] :
( ( A13 = K3 )
& ( A24 = L4 )
& ( A33
= ( product_Pair @ int @ int @ Q8 @ ( zero_zero @ int ) ) )
& ( L4
!= ( zero_zero @ int ) )
& ( K3
= ( times_times @ int @ Q8 @ L4 ) ) )
| ? [R5: int,L4: int,K3: int,Q8: int] :
( ( A13 = K3 )
& ( A24 = L4 )
& ( A33
= ( product_Pair @ int @ int @ Q8 @ R5 ) )
& ( ( sgn_sgn @ int @ R5 )
= ( sgn_sgn @ int @ L4 ) )
& ( ord_less @ int @ ( abs_abs @ int @ R5 ) @ ( abs_abs @ int @ L4 ) )
& ( K3
= ( plus_plus @ int @ ( times_times @ int @ Q8 @ L4 ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_3403_eucl__rel__int_Ocases,axiom,
! [A12: int,A23: int,A32: product_prod @ int @ int] :
( ( eucl_rel_int @ A12 @ A23 @ A32 )
=> ( ( ( A23
= ( zero_zero @ int ) )
=> ( A32
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A12 ) ) )
=> ( ! [Q5: int] :
( ( A32
= ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) )
=> ( ( A23
!= ( zero_zero @ int ) )
=> ( A12
!= ( times_times @ int @ Q5 @ A23 ) ) ) )
=> ~ ! [R: int,Q5: int] :
( ( A32
= ( product_Pair @ int @ int @ Q5 @ R ) )
=> ( ( ( sgn_sgn @ int @ R )
= ( sgn_sgn @ int @ A23 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R ) @ ( abs_abs @ int @ A23 ) )
=> ( A12
!= ( plus_plus @ int @ ( times_times @ int @ Q5 @ A23 ) @ R ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_3404_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert @ A @ X @ A5 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_3405_sgn__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_zero
thf(fact_3406_sgn__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_0
thf(fact_3407_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_3408_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_3409_length__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= ( suc @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ).
% length_insort
thf(fact_3410_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_3411_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_3412_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert @ A @ X @ A5 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_3413_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_3414_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_3415_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_3416_sgn__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ( sgn_sgn @ A @ X )
= ( zero_zero @ A ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% sgn_zero_iff
thf(fact_3417_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_3418_div__eq__sgn__abs,axiom,
! [K: int,L: int] :
( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ K @ L )
= ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ).
% div_eq_sgn_abs
thf(fact_3419_insort__key_Osimps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Y: B,Ys: list @ B] :
( ( ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ X @ ( cons @ B @ Y @ Ys ) )
= ( cons @ B @ X @ ( cons @ B @ Y @ Ys ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ X @ ( cons @ B @ Y @ Ys ) )
= ( cons @ B @ Y @ ( linorder_insort_key @ B @ A @ F3 @ X @ Ys ) ) ) ) ) ) ).
% insort_key.simps(2)
thf(fact_3420_sorted__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted_insort
thf(fact_3421_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_3422_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_3423_insort__is__Cons,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,F3: B > A,A2: B] :
( ! [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
=> ( ord_less_eq @ A @ ( F3 @ A2 ) @ ( F3 @ X3 ) ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ A2 @ Xs )
= ( cons @ B @ A2 @ Xs ) ) ) ) ).
% insort_is_Cons
thf(fact_3424_sorted__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) ) ) ) ).
% sorted_insort_key
thf(fact_3425_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X4: A] :
( if @ A
@ ( X4
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X4 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_3426_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_3427_zsgn__def,axiom,
( ( sgn_sgn @ int )
= ( ^ [I3: int] :
( if @ int
@ ( I3
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I3 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zsgn_def
thf(fact_3428_norm__sgn,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ( X
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X ) )
= ( zero_zero @ real ) ) )
& ( ( X
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X ) )
= ( one_one @ real ) ) ) ) ) ).
% norm_sgn
thf(fact_3429_div__sgn__abs__cancel,axiom,
! [V3: int,K: int,L: int] :
( ( V3
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ V3 ) @ ( abs_abs @ int @ K ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ V3 ) @ ( abs_abs @ int @ L ) ) )
= ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ).
% div_sgn_abs_cancel
thf(fact_3430_insort__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,Xs: list @ A] :
( ( member @ A @ A2 @ ( set2 @ A @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ A2
@ ( remove1 @ A @ A2 @ Xs ) )
= Xs ) ) ) ) ).
% insort_remove1
thf(fact_3431_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ A5 )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
thf(fact_3432_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_3433_sinh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sinh @ A @ X )
= ( zero_zero @ A ) )
= ( member @ A @ ( exp @ A @ X ) @ ( insert @ A @ ( one_one @ A ) @ ( insert @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% sinh_zero_iff
thf(fact_3434_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,P3: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I6 )
& ( ( P3 @ X4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P3 @ ( insert @ B @ I @ I6 ) )
= ( groups1962203154675924110t_prod @ B @ A @ P3 @ I6 ) ) )
& ( ~ ( member @ B @ I @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P3 @ ( insert @ B @ I @ I6 ) )
= ( times_times @ A @ ( P3 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P3 @ I6 ) ) ) ) ) ) ) ).
% prod.insert'
thf(fact_3435_time__array__nth,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P3: array @ A,I: nat,H2: heap_ext @ product_unit] :
( ( ( time_fails @ A @ ( array_nth @ A @ P3 @ I ) @ H2 )
=> ( ( time_time @ A @ ( array_nth @ A @ P3 @ I ) @ H2 )
= ( zero_zero @ nat ) ) )
& ( ~ ( time_fails @ A @ ( array_nth @ A @ P3 @ I ) @ H2 )
=> ( ( time_time @ A @ ( array_nth @ A @ P3 @ I ) @ H2 )
= ( one_one @ nat ) ) ) ) ) ).
% time_array_nth
thf(fact_3436_precise__extr__pure_I2_J,axiom,
! [B: $tType,A: $tType,R3: A > B > assn,P2: $o] :
( ( precise @ A @ B
@ ^ [X4: A,Y5: B] : ( times_times @ assn @ ( R3 @ X4 @ Y5 ) @ ( pure_assn @ P2 ) ) )
= ( P2
=> ( precise @ A @ B @ R3 ) ) ) ).
% precise_extr_pure(2)
thf(fact_3437_precise__extr__pure_I1_J,axiom,
! [B: $tType,A: $tType,P2: $o,R3: A > B > assn] :
( ( precise @ A @ B
@ ^ [X4: A,Y5: B] : ( times_times @ assn @ ( pure_assn @ P2 ) @ ( R3 @ X4 @ Y5 ) ) )
= ( P2
=> ( precise @ A @ B @ R3 ) ) ) ).
% precise_extr_pure(1)
thf(fact_3438_sinh__real__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( sinh @ real @ X ) @ ( sinh @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% sinh_real_less_iff
thf(fact_3439_sinh__real__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( sinh @ real @ X ) @ ( sinh @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% sinh_real_le_iff
thf(fact_3440_sinh__real__pos__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% sinh_real_pos_iff
thf(fact_3441_sinh__real__neg__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sinh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% sinh_real_neg_iff
thf(fact_3442_sgn__le__0__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sgn_sgn @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% sgn_le_0_iff
thf(fact_3443_zero__le__sgn__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sgn_sgn @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% zero_le_sgn_iff
thf(fact_3444_sinh__real__nonpos__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sinh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% sinh_real_nonpos_iff
thf(fact_3445_sinh__real__nonneg__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% sinh_real_nonneg_iff
thf(fact_3446_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_3447_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P3: B > A] :
( ( groups1962203154675924110t_prod @ B @ A @ P3 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty'
thf(fact_3448_prod_Oeq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,P3: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P3 @ I6 )
= ( groups7121269368397514597t_prod @ B @ A @ P3 @ I6 ) ) ) ) ).
% prod.eq_sum
thf(fact_3449_snga__prec,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( precise @ ( list @ A ) @ ( array @ A )
@ ^ [X4: list @ A,P7: array @ A] : ( snga_assn @ A @ P7 @ X4 ) ) ) ).
% snga_prec
thf(fact_3450_prod_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I3: B] : ( times_times @ A @ ( G3 @ I3 ) @ ( H2 @ I3 ) )
@ I6 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G3 @ I6 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I6 ) ) ) ) ) ).
% prod.distrib_triv'
thf(fact_3451_sngr__prec,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( precise @ A @ ( ref @ A )
@ ^ [X4: A,P7: ref @ A] : ( sngr_assn @ A @ P7 @ X4 ) ) ) ).
% sngr_prec
thf(fact_3452_sgn__real__def,axiom,
( ( sgn_sgn @ real )
= ( ^ [A8: real] :
( if @ real
@ ( A8
= ( zero_zero @ real ) )
@ ( zero_zero @ real )
@ ( if @ real @ ( ord_less @ real @ ( zero_zero @ real ) @ A8 ) @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ) ) ).
% sgn_real_def
thf(fact_3453_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T4: set @ B,G3: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G3 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G3 @ T4 )
= ( groups1962203154675924110t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
thf(fact_3454_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T4: set @ B,H2: B > A,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( H2 @ I2 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G3 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G3 @ S3 )
= ( groups1962203154675924110t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
thf(fact_3455_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T4: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G3 @ T4 )
= ( groups1962203154675924110t_prod @ B @ A @ G3 @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right'
thf(fact_3456_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T4: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( G3 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G3 @ S3 )
= ( groups1962203154675924110t_prod @ B @ A @ G3 @ T4 ) ) ) ) ) ).
% prod.mono_neutral_left'
thf(fact_3457_preciseD_H,axiom,
! [B: $tType,A: $tType,R3: A > B > assn,A2: A,P3: B,F5: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),A3: A,F7: assn] :
( ( precise @ A @ B @ R3 )
=> ( ( rep_assn @ ( times_times @ assn @ ( R3 @ A2 @ P3 ) @ F5 ) @ H2 )
=> ( ( rep_assn @ ( times_times @ assn @ ( R3 @ A3 @ P3 ) @ F7 ) @ H2 )
=> ( A2 = A3 ) ) ) ) ).
% preciseD'
thf(fact_3458_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I6 )
& ( ( G3 @ X4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I6 )
& ( ( H2 @ X4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I3: B] : ( times_times @ A @ ( G3 @ I3 ) @ ( H2 @ I3 ) )
@ I6 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G3 @ I6 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% prod.distrib'
thf(fact_3459_prod_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ B @ A )
= ( ^ [P7: B > A,I8: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I8 )
& ( ( P7 @ X4 )
!= ( one_one @ A ) ) ) ) )
@ ( groups7121269368397514597t_prod @ B @ A @ P7
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I8 )
& ( ( P7 @ X4 )
!= ( one_one @ A ) ) ) ) )
@ ( one_one @ A ) ) ) ) ) ).
% prod.G_def
thf(fact_3460_time__array__upd,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,X: A,P3: array @ A,H2: heap_ext @ product_unit] :
( ( ( time_fails @ ( array @ A ) @ ( array_upd @ A @ I @ X @ P3 ) @ H2 )
=> ( ( time_time @ ( array @ A ) @ ( array_upd @ A @ I @ X @ P3 ) @ H2 )
= ( zero_zero @ nat ) ) )
& ( ~ ( time_fails @ ( array @ A ) @ ( array_upd @ A @ I @ X @ P3 ) @ H2 )
=> ( ( time_time @ ( array @ A ) @ ( array_upd @ A @ I @ X @ P3 ) @ H2 )
= ( one_one @ nat ) ) ) ) ) ).
% time_array_upd
thf(fact_3461_sgn__power__injE,axiom,
! [A2: real,N: nat,X: real,B2: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ A2 ) @ ( power_power @ real @ ( abs_abs @ real @ A2 ) @ N ) )
= X )
=> ( ( X
= ( times_times @ real @ ( sgn_sgn @ real @ B2 ) @ ( power_power @ real @ ( abs_abs @ real @ B2 ) @ N ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A2 = B2 ) ) ) ) ).
% sgn_power_injE
thf(fact_3462_time__refines,axiom,
! [A: $tType,C3: heap_Time_Heap @ A,C8: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
( ( refine_Imp_refines @ A @ C3 @ C8 )
=> ( ~ ( time_fails @ A @ C8 @ H2 )
=> ( ord_less_eq @ nat @ ( time_time @ A @ C3 @ H2 ) @ ( time_time @ A @ C8 @ H2 ) ) ) ) ).
% time_refines
thf(fact_3463_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M5: nat,N6: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M5 @ N6 ) @ ( modulo_modulo @ nat @ M5 @ N6 ) ) ) ) ).
% divmod_nat_def
thf(fact_3464_comp__fun__idem__on_Ofold__insert__idem2,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,X: A,A5: set @ A,Z: B] :
( ( finite673082921795544331dem_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S3 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( insert @ A @ X @ A5 ) )
= ( finite_fold @ A @ B @ F3 @ ( F3 @ X @ Z ) @ A5 ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem2
thf(fact_3465_comp__fun__idem__on_Ofold__insert__idem,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,X: A,A5: set @ A,Z: B] :
( ( finite673082921795544331dem_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S3 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( insert @ A @ X @ A5 ) )
= ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z @ A5 ) ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem
thf(fact_3466_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_3467_graph__map__upd,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),K: A,V3: B] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( some @ B @ V3 ) ) )
= ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V3 ) @ ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( none @ B ) ) ) ) ) ).
% graph_map_upd
thf(fact_3468_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_3469_real__root__eq__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( root @ N @ Y ) )
= ( X = Y ) ) ) ).
% real_root_eq_iff
thf(fact_3470_root__0,axiom,
! [X: real] :
( ( root @ ( zero_zero @ nat ) @ X )
= ( zero_zero @ real ) ) ).
% root_0
thf(fact_3471_real__root__eq__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% real_root_eq_0_iff
thf(fact_3472_real__root__less__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ).
% real_root_less_iff
thf(fact_3473_real__root__le__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ).
% real_root_le_iff
thf(fact_3474_real__root__eq__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( one_one @ real ) )
= ( X
= ( one_one @ real ) ) ) ) ).
% real_root_eq_1_iff
thf(fact_3475_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_3476_real__root__lt__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ) ).
% real_root_lt_0_iff
thf(fact_3477_real__root__gt__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ Y ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_3478_real__root__le__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ) ).
% real_root_le_0_iff
thf(fact_3479_real__root__ge__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ Y ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_3480_real__root__lt__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ).
% real_root_lt_1_iff
thf(fact_3481_real__root__gt__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( root @ N @ Y ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_3482_real__root__le__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ).
% real_root_le_1_iff
thf(fact_3483_real__root__ge__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( root @ N @ Y ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_3484_real__root__pow__pos2,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos2
thf(fact_3485_comp__fun__idem__on_Ofun__left__idem,axiom,
! [A: $tType,B: $tType,S3: set @ A,F3: A > B > B,X: A,Z: B] :
( ( finite673082921795544331dem_on @ A @ B @ S3 @ F3 )
=> ( ( member @ A @ X @ S3 )
=> ( ( F3 @ X @ ( F3 @ X @ Z ) )
= ( F3 @ X @ Z ) ) ) ) ).
% comp_fun_idem_on.fun_left_idem
thf(fact_3486_sgn__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( sgn_sgn @ real @ ( root @ N @ X ) )
= ( sgn_sgn @ real @ X ) ) ) ).
% sgn_root
thf(fact_3487_real__root__pos__pos__le,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ X ) ) ) ).
% real_root_pos_pos_le
thf(fact_3488_comp__fun__idem__on_Oaxioms_I1_J,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B] :
( ( finite673082921795544331dem_on @ A @ B @ S3 @ F3 )
=> ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 ) ) ).
% comp_fun_idem_on.axioms(1)
thf(fact_3489_in__graphI,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),K: B,V3: A] :
( ( ( M @ K )
= ( some @ A @ V3 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V3 ) @ ( graph @ B @ A @ M ) ) ) ).
% in_graphI
thf(fact_3490_in__graphD,axiom,
! [A: $tType,B: $tType,K: A,V3: B,M: A > ( option @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V3 ) @ ( graph @ A @ B @ M ) )
=> ( ( M @ K )
= ( some @ B @ V3 ) ) ) ).
% in_graphD
thf(fact_3491_real__root__less__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_3492_real__root__le__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_3493_real__root__power,axiom,
! [N: nat,X: real,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X @ K ) )
= ( power_power @ real @ ( root @ N @ X ) @ K ) ) ) ).
% real_root_power
thf(fact_3494_real__root__abs,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( abs_abs @ real @ X ) )
= ( abs_abs @ real @ ( root @ N @ X ) ) ) ) ).
% real_root_abs
thf(fact_3495_real__root__gt__zero,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_3496_real__root__strict__decreasing,axiom,
! [N: nat,N8: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N8 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( root @ N8 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_3497_root__abs__power,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( abs_abs @ real @ ( root @ N @ ( power_power @ real @ Y @ N ) ) )
= ( abs_abs @ real @ Y ) ) ) ).
% root_abs_power
thf(fact_3498_real__root__pos__pos,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_3499_real__root__strict__increasing,axiom,
! [N: nat,N8: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N8 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N8 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_3500_real__root__decreasing,axiom,
! [N: nat,N8: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N8 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( root @ N8 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_3501_real__root__pow__pos,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos
thf(fact_3502_real__root__pos__unique,axiom,
! [N: nat,Y: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( power_power @ real @ Y @ N )
= X )
=> ( ( root @ N @ X )
= Y ) ) ) ) ).
% real_root_pos_unique
thf(fact_3503_real__root__power__cancel,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( root @ N @ ( power_power @ real @ X @ N ) )
= X ) ) ) ).
% real_root_power_cancel
thf(fact_3504_sgn__power__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ real @ ( sgn_sgn @ real @ ( root @ N @ X ) ) @ ( power_power @ real @ ( abs_abs @ real @ ( root @ N @ X ) ) @ N ) )
= X ) ) ).
% sgn_power_root
thf(fact_3505_root__sgn__power,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N ) ) )
= Y ) ) ).
% root_sgn_power
thf(fact_3506_real__root__increasing,axiom,
! [N: nat,N8: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N8 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N8 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_3507_split__root,axiom,
! [P2: real > $o,N: nat,X: real] :
( ( P2 @ ( root @ N @ X ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P2 @ ( zero_zero @ real ) ) )
& ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ! [Y5: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ Y5 ) @ ( power_power @ real @ ( abs_abs @ real @ Y5 ) @ N ) )
= X )
=> ( P2 @ Y5 ) ) ) ) ) ).
% split_root
thf(fact_3508_root__powr__inverse,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( root @ N @ X )
= ( powr @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_3509_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),X: A,Y: B] :
( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ ( some @ B @ Y ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( restrict_map @ A @ B @ M @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% restrict_upd_same
thf(fact_3510_log__base__root,axiom,
! [N: nat,B2: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( log @ ( root @ N @ B2 ) @ X )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ X ) ) ) ) ) ).
% log_base_root
thf(fact_3511_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_3512_fun__upd__None__restrict,axiom,
! [B: $tType,A: $tType,X: A,D5: set @ A,M: A > ( option @ B )] :
( ( ( member @ A @ X @ D5 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D5 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
& ( ~ ( member @ A @ X @ D5 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D5 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ D5 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_3513_powr__eq__0__iff,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [W2: A,Z: A] :
( ( ( powr @ A @ W2 @ Z )
= ( zero_zero @ A ) )
= ( W2
= ( zero_zero @ A ) ) ) ) ).
% powr_eq_0_iff
thf(fact_3514_powr__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [Z: A] :
( ( powr @ A @ ( zero_zero @ A ) @ Z )
= ( zero_zero @ A ) ) ) ).
% powr_0
thf(fact_3515_powr__zero__eq__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [X: A] :
( ( ( X
= ( zero_zero @ A ) )
=> ( ( powr @ A @ X @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) )
& ( ( X
!= ( zero_zero @ A ) )
=> ( ( powr @ A @ X @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% powr_zero_eq_one
thf(fact_3516_powr__gt__zero,axiom,
! [X: real,A2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ A2 ) )
= ( X
!= ( zero_zero @ real ) ) ) ).
% powr_gt_zero
thf(fact_3517_powr__nonneg__iff,axiom,
! [A2: real,X: real] :
( ( ord_less_eq @ real @ ( powr @ real @ A2 @ X ) @ ( zero_zero @ real ) )
= ( A2
= ( zero_zero @ real ) ) ) ).
% powr_nonneg_iff
thf(fact_3518_powr__less__cancel__iff,axiom,
! [X: real,A2: real,B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) )
= ( ord_less @ real @ A2 @ B2 ) ) ) ).
% powr_less_cancel_iff
thf(fact_3519_restrict__map__to__empty,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M @ ( bot_bot @ ( set @ A ) ) )
= ( ^ [X4: A] : ( none @ B ) ) ) ).
% restrict_map_to_empty
thf(fact_3520_zero__less__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X ) )
= ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_3521_log__less__zero__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ A2 @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_3522_one__less__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( log @ A2 @ X ) )
= ( ord_less @ real @ A2 @ X ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_3523_log__less__one__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ A2 @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ A2 ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_3524_log__less__cancel__iff,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_3525_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_3526_powr__eq__one__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ( powr @ real @ A2 @ X )
= ( one_one @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% powr_eq_one_iff
thf(fact_3527_powr__one__gt__zero__iff,axiom,
! [X: real] :
( ( ( powr @ real @ X @ ( one_one @ real ) )
= X )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% powr_one_gt_zero_iff
thf(fact_3528_powr__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( one_one @ real ) )
= X ) ) ).
% powr_one
thf(fact_3529_powr__le__cancel__iff,axiom,
! [X: real,A2: real,B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) )
= ( ord_less_eq @ real @ A2 @ B2 ) ) ) ).
% powr_le_cancel_iff
thf(fact_3530_zero__le__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_3531_log__le__zero__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_3532_one__le__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( log @ A2 @ X ) )
= ( ord_less_eq @ real @ A2 @ X ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_3533_log__le__one__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ A2 ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_3534_log__le__cancel__iff,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_3535_log__powr__cancel,axiom,
! [A2: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ A2 @ ( powr @ real @ A2 @ Y ) )
= Y ) ) ) ).
% log_powr_cancel
thf(fact_3536_powr__log__cancel,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ A2 @ ( log @ A2 @ X ) )
= X ) ) ) ) ).
% powr_log_cancel
thf(fact_3537_fun__upd__restrict__conv,axiom,
! [A: $tType,B: $tType,X: A,D5: set @ A,M: A > ( option @ B ),Y: option @ B] :
( ( member @ A @ X @ D5 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D5 ) @ X @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) ) ).
% fun_upd_restrict_conv
thf(fact_3538_restrict__fun__upd,axiom,
! [B: $tType,A: $tType,X: A,D5: set @ A,M: A > ( option @ B ),Y: option @ B] :
( ( ( member @ A @ X @ D5 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ Y ) @ D5 )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) )
& ( ~ ( member @ A @ X @ D5 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ Y ) @ D5 )
= ( restrict_map @ A @ B @ M @ D5 ) ) ) ) ).
% restrict_fun_upd
thf(fact_3539_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_3540_restrict__map__subset__eq,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),R3: set @ A,M6: A > ( option @ B ),R7: set @ A] :
( ( ( restrict_map @ A @ B @ M @ R3 )
= M6 )
=> ( ( ord_less_eq @ ( set @ A ) @ R7 @ R3 )
=> ( ( restrict_map @ A @ B @ M @ R7 )
= ( restrict_map @ A @ B @ M6 @ R7 ) ) ) ) ).
% restrict_map_subset_eq
thf(fact_3541_le__map__restrict,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [M: A > ( option @ B ),X6: set @ A] : ( ord_less_eq @ ( A > ( option @ B ) ) @ ( restrict_map @ A @ B @ M @ X6 ) @ M ) ) ).
% le_map_restrict
thf(fact_3542_restrict__map__eq_I2_J,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A5: set @ B,K: B,V3: A] :
( ( ( restrict_map @ B @ A @ M @ A5 @ K )
= ( some @ A @ V3 ) )
= ( ( ( M @ K )
= ( some @ A @ V3 ) )
& ( member @ B @ K @ A5 ) ) ) ).
% restrict_map_eq(2)
thf(fact_3543_less__log__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ Y @ ( log @ B2 @ X ) )
= ( ord_less @ real @ ( powr @ real @ B2 @ Y ) @ X ) ) ) ) ).
% less_log_iff
thf(fact_3544_log__less__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ B2 @ X ) @ Y )
= ( ord_less @ real @ X @ ( powr @ real @ B2 @ Y ) ) ) ) ) ).
% log_less_iff
thf(fact_3545_less__powr__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( powr @ real @ B2 @ Y ) )
= ( ord_less @ real @ ( log @ B2 @ X ) @ Y ) ) ) ) ).
% less_powr_iff
thf(fact_3546_powr__less__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( powr @ real @ B2 @ Y ) @ X )
= ( ord_less @ real @ Y @ ( log @ B2 @ X ) ) ) ) ) ).
% powr_less_iff
thf(fact_3547_le__log__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ Y @ ( log @ B2 @ X ) )
= ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y ) @ X ) ) ) ) ).
% le_log_iff
thf(fact_3548_log__le__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ B2 @ X ) @ Y )
= ( ord_less_eq @ real @ X @ ( powr @ real @ B2 @ Y ) ) ) ) ) ).
% log_le_iff
thf(fact_3549_le__powr__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( powr @ real @ B2 @ Y ) )
= ( ord_less_eq @ real @ ( log @ B2 @ X ) @ Y ) ) ) ) ).
% le_powr_iff
thf(fact_3550_powr__le__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y ) @ X )
= ( ord_less_eq @ real @ Y @ ( log @ B2 @ X ) ) ) ) ) ).
% powr_le_iff
thf(fact_3551_powr__less__mono2__neg,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( powr @ real @ Y @ A2 ) @ ( powr @ real @ X @ A2 ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_3552_powr__non__neg,axiom,
! [A2: real,X: real] :
~ ( ord_less @ real @ ( powr @ real @ A2 @ X ) @ ( zero_zero @ real ) ) ).
% powr_non_neg
thf(fact_3553_powr__mono2,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ).
% powr_mono2
thf(fact_3554_powr__ge__pzero,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ Y ) ) ).
% powr_ge_pzero
thf(fact_3555_powr__less__cancel,axiom,
! [X: real,A2: real,B2: real] :
( ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ A2 @ B2 ) ) ) ).
% powr_less_cancel
thf(fact_3556_powr__less__mono,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) ) ) ) ).
% powr_less_mono
thf(fact_3557_powr__mono,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) ) ) ) ).
% powr_mono
thf(fact_3558_add__log__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( plus_plus @ real @ Y @ ( log @ B2 @ X ) )
= ( log @ B2 @ ( times_times @ real @ ( powr @ real @ B2 @ Y ) @ X ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_3559_log__add__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( plus_plus @ real @ ( log @ B2 @ X ) @ Y )
= ( log @ B2 @ ( times_times @ real @ X @ ( powr @ real @ B2 @ Y ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_3560_minus__log__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( minus_minus @ real @ Y @ ( log @ B2 @ X ) )
= ( log @ B2 @ ( divide_divide @ real @ ( powr @ real @ B2 @ Y ) @ X ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_3561_graph__restrictD_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V3: B,M: A > ( option @ B ),A5: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V3 ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M @ A5 ) ) )
=> ( member @ A @ K @ A5 ) ) ).
% graph_restrictD(1)
thf(fact_3562_log__minus__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( minus_minus @ real @ ( log @ B2 @ X ) @ Y )
= ( log @ B2 @ ( times_times @ real @ X @ ( powr @ real @ B2 @ ( uminus_uminus @ real @ Y ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_3563_powr__less__mono2,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ).
% powr_less_mono2
thf(fact_3564_powr__mono2_H,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ Y @ A2 ) @ ( powr @ real @ X @ A2 ) ) ) ) ) ).
% powr_mono2'
thf(fact_3565_gr__one__powr,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less @ real @ ( one_one @ real ) @ ( powr @ real @ X @ Y ) ) ) ) ).
% gr_one_powr
thf(fact_3566_powr__inj,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ( powr @ real @ A2 @ X )
= ( powr @ real @ A2 @ Y ) )
= ( X = Y ) ) ) ) ).
% powr_inj
thf(fact_3567_ge__one__powr__ge__zero,axiom,
! [X: real,A2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( powr @ real @ X @ A2 ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_3568_powr__mono__both,axiom,
! [A2: real,B2: real,X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ B2 ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_3569_powr__le1,axiom,
! [A2: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( one_one @ real ) ) ) ) ) ).
% powr_le1
thf(fact_3570_powr__divide,axiom,
! [X: real,Y: real,A2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( powr @ real @ ( divide_divide @ real @ X @ Y ) @ A2 )
= ( divide_divide @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ).
% powr_divide
thf(fact_3571_powr__mult,axiom,
! [X: real,Y: real,A2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( powr @ real @ ( times_times @ real @ X @ Y ) @ A2 )
= ( times_times @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ).
% powr_mult
thf(fact_3572_map__restrict__insert__none__simp,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),X: B,S: set @ B] :
( ( ( M @ X )
= ( none @ A ) )
=> ( ( restrict_map @ B @ A @ M @ ( uminus_uminus @ ( set @ B ) @ ( insert @ B @ X @ S ) ) )
= ( restrict_map @ B @ A @ M @ ( uminus_uminus @ ( set @ B ) @ S ) ) ) ) ).
% map_restrict_insert_none_simp
thf(fact_3573_restrict__map__upd,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B ),S3: set @ A,K: A,V3: B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ F3 @ S3 ) @ K @ ( some @ B @ V3 ) )
= ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ K @ ( some @ B @ V3 ) ) @ ( insert @ A @ K @ S3 ) ) ) ).
% restrict_map_upd
thf(fact_3574_graph__restrictD_I2_J,axiom,
! [A: $tType,B: $tType,K: A,V3: B,M: A > ( option @ B ),A5: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V3 ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M @ A5 ) ) )
=> ( ( M @ K )
= ( some @ B @ V3 ) ) ) ).
% graph_restrictD(2)
thf(fact_3575_log__base__change,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ B2 @ X )
= ( divide_divide @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ B2 ) ) ) ) ) ).
% log_base_change
thf(fact_3576_powr__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( semiring_1_of_nat @ real @ N ) )
= ( power_power @ real @ X @ N ) ) ) ).
% powr_realpow
thf(fact_3577_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_3578_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_3579_fun__upd__restrict,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),D5: set @ A,X: A,Y: option @ B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D5 ) @ X @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) ).
% fun_upd_restrict
thf(fact_3580_powr__neg__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ X ) ) ) ).
% powr_neg_one
thf(fact_3581_powr__mult__base,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( times_times @ real @ X @ ( powr @ real @ X @ Y ) )
= ( powr @ real @ X @ ( plus_plus @ real @ ( one_one @ real ) @ Y ) ) ) ) ).
% powr_mult_base
thf(fact_3582_log__mult,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( log @ A2 @ ( times_times @ real @ X @ Y ) )
= ( plus_plus @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) ) ) ) ) ) ) ).
% log_mult
thf(fact_3583_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_3584_log__base__pow,axiom,
! [A2: real,N: nat,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( log @ ( power_power @ real @ A2 @ N ) @ X )
= ( divide_divide @ real @ ( log @ A2 @ X ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log_base_pow
thf(fact_3585_log__nat__power,axiom,
! [X: real,B2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ B2 @ ( power_power @ real @ X @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ X ) ) ) ) ).
% log_nat_power
thf(fact_3586_log__divide,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( log @ A2 @ ( divide_divide @ real @ X @ Y ) )
= ( minus_minus @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) ) ) ) ) ) ) ).
% log_divide
thf(fact_3587_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F3: A > ( option @ B ),X: A] :
( ( restrict_map @ A @ B @ F3 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( fun_upd @ A @ ( option @ B ) @ F3 @ X @ ( none @ B ) ) ) ).
% restrict_complement_singleton_eq
thf(fact_3588_map__upd__eq__restrict,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),X: A] :
( ( fun_upd @ A @ ( option @ B ) @ M @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% map_upd_eq_restrict
thf(fact_3589_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_3590_ln__powr__bound,axiom,
! [X: real,A2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( divide_divide @ real @ ( powr @ real @ X @ A2 ) @ A2 ) ) ) ) ).
% ln_powr_bound
thf(fact_3591_ln__powr__bound2,axiom,
! [X: real,A2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( powr @ real @ ( ln_ln @ real @ X ) @ A2 ) @ ( times_times @ real @ ( powr @ real @ A2 @ A2 ) @ X ) ) ) ) ).
% ln_powr_bound2
thf(fact_3592_log__eq__div__ln__mult__log,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ A2 @ X )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( ln_ln @ real @ A2 ) ) @ ( log @ B2 @ X ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_3593_powr__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( powr @ A )
= ( ^ [X4: A,A8: A] :
( if @ A
@ ( X4
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( exp @ A @ ( times_times @ A @ A8 @ ( ln_ln @ A @ X4 ) ) ) ) ) ) ) ).
% powr_def
thf(fact_3594_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_3595_ceiling__log__eq__powr__iff,axiom,
! [X: real,B2: real,K: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archimedean_ceiling @ real @ ( log @ B2 @ X ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) ) )
= ( ( ord_less @ real @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ K ) ) @ X )
& ( ord_less_eq @ real @ X @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% ceiling_log_eq_powr_iff
thf(fact_3596_restrict__map__upds,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,D5: set @ A,M: A > ( option @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ D5 )
=> ( ( restrict_map @ A @ B @ ( map_upds @ A @ B @ M @ Xs @ Ys ) @ D5 )
= ( map_upds @ A @ B @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D5 @ ( set2 @ A @ Xs ) ) ) @ Xs @ Ys ) ) ) ) ).
% restrict_map_upds
thf(fact_3597_dual__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( min @ A
@ ^ [X4: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X4 ) )
= ( ord_max @ A ) ) ) ).
% dual_min
thf(fact_3598_Id__on__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( id_on @ A @ ( set2 @ A @ Xs ) )
= ( set2 @ ( product_prod @ A @ A )
@ ( map @ A @ ( product_prod @ A @ A )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Xs ) ) ) ).
% Id_on_set
thf(fact_3599_mod__type_Ointro,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [Rep: A > int,Abs: int > A,N: int] :
( ( type_definition @ A @ int @ Rep @ Abs @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ N ) )
=> ( ( ord_less @ int @ ( one_one @ int ) @ N )
=> ( ( ( zero_zero @ A )
= ( Abs @ ( zero_zero @ int ) ) )
=> ( ( ( one_one @ A )
= ( Abs @ ( one_one @ int ) ) )
=> ( ! [X3: A,Y3: A] :
( ( plus_plus @ A @ X3 @ Y3 )
= ( Abs @ ( modulo_modulo @ int @ ( plus_plus @ int @ ( Rep @ X3 ) @ ( Rep @ Y3 ) ) @ N ) ) )
=> ( ! [X3: A,Y3: A] :
( ( times_times @ A @ X3 @ Y3 )
= ( Abs @ ( modulo_modulo @ int @ ( times_times @ int @ ( Rep @ X3 ) @ ( Rep @ Y3 ) ) @ N ) ) )
=> ( ! [X3: A,Y3: A] :
( ( minus_minus @ A @ X3 @ Y3 )
= ( Abs @ ( modulo_modulo @ int @ ( minus_minus @ int @ ( Rep @ X3 ) @ ( Rep @ Y3 ) ) @ N ) ) )
=> ( ! [X3: A] :
( ( uminus_uminus @ A @ X3 )
= ( Abs @ ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( Rep @ X3 ) ) @ N ) ) )
=> ( numeral_mod_type @ A @ N @ Rep @ Abs ) ) ) ) ) ) ) ) ) ) ).
% mod_type.intro
thf(fact_3600_subset__mset_Omin__arg__le_I1_J,axiom,
! [A: $tType,N: multiset @ A,M: multiset @ A] :
( ( subseteq_mset @ A @ N @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M @ N ) )
= ( ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M @ N )
= N ) ) ).
% subset_mset.min_arg_le(1)
thf(fact_3601_subset__mset_Omin__arg__le_I2_J,axiom,
! [A: $tType,M: multiset @ A,N: multiset @ A] :
( ( subseteq_mset @ A @ M @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M @ N ) )
= ( ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M @ N )
= M ) ) ).
% subset_mset.min_arg_le(2)
thf(fact_3602_Id__onI,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id_on @ A @ A5 ) ) ) ).
% Id_onI
thf(fact_3603_ceiling__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% ceiling_zero
thf(fact_3604_Id__on__empty,axiom,
! [A: $tType] :
( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% Id_on_empty
thf(fact_3605_map__upds__list__update2__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I: nat,M: A > ( option @ B ),Ys: list @ B,Y: B] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( map_upds @ A @ B @ M @ Xs @ ( list_update @ B @ Ys @ I @ Y ) )
= ( map_upds @ A @ B @ M @ Xs @ Ys ) ) ) ).
% map_upds_list_update2_drop
thf(fact_3606_ceiling__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% ceiling_le_zero
thf(fact_3607_zero__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% zero_less_ceiling
thf(fact_3608_ceiling__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% ceiling_less_one
thf(fact_3609_one__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% one_le_ceiling
thf(fact_3610_ceiling__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ).
% ceiling_le_one
thf(fact_3611_one__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_less_ceiling
thf(fact_3612_ceiling__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_zero
thf(fact_3613_zero__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X ) ) ) ).
% zero_le_ceiling
thf(fact_3614_mod__type_ORep__less__n,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep: A > int,Abs: int > A,X: A] :
( ( numeral_mod_type @ A @ N @ Rep @ Abs )
=> ( ord_less @ int @ ( Rep @ X ) @ N ) ) ) ).
% mod_type.Rep_less_n
thf(fact_3615_mod__type_ORep__le__n,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep: A > int,Abs: int > A,X: A] :
( ( numeral_mod_type @ A @ N @ Rep @ Abs )
=> ( ord_less_eq @ int @ ( Rep @ X ) @ N ) ) ) ).
% mod_type.Rep_le_n
thf(fact_3616_mod__type_ORep__0,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep: A > int,Abs: int > A] :
( ( numeral_mod_type @ A @ N @ Rep @ Abs )
=> ( ( Rep @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ) ).
% mod_type.Rep_0
thf(fact_3617_mod__type_Ozero__def,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep: A > int,Abs: int > A] :
( ( numeral_mod_type @ A @ N @ Rep @ Abs )
=> ( ( zero_zero @ A )
= ( Abs @ ( zero_zero @ int ) ) ) ) ) ).
% mod_type.zero_def
thf(fact_3618_Id__on__iff,axiom,
! [A: $tType,X: A,Y: A,A5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id_on @ A @ A5 ) )
= ( ( X = Y )
& ( member @ A @ X @ A5 ) ) ) ).
% Id_on_iff
thf(fact_3619_Id__on__eqI,axiom,
! [A: $tType,A2: A,B2: A,A5: set @ A] :
( ( A2 = B2 )
=> ( ( member @ A @ A2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id_on @ A @ A5 ) ) ) ) ).
% Id_on_eqI
thf(fact_3620_Id__onE,axiom,
! [A: $tType,C3: product_prod @ A @ A,A5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ C3 @ ( id_on @ A @ A5 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( C3
!= ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_3621_mod__type_Osize0,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep: A > int,Abs: int > A] :
( ( numeral_mod_type @ A @ N @ Rep @ Abs )
=> ( ord_less @ int @ ( zero_zero @ int ) @ N ) ) ) ).
% mod_type.size0
thf(fact_3622_mod__type_Osize1,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ! [N: int,Rep: A > int,Abs: int > A] :
( ( numeral_mod_type @ A @ N @ Rep @ Abs )
=> ( ord_less @ int @ ( one_one @ int ) @ N ) ) ) ).
% mod_type.size1
thf(fact_3623_ceiling__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% ceiling_mono
thf(fact_3624_ceiling__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% ceiling_less_cancel
thf(fact_3625_ceiling__add__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y ) ) ) ) ).
% ceiling_add_le
thf(fact_3626_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_3627_Id__on__fold,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( id_on @ A @ A5 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X4: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
@ A5 ) ) ) ).
% Id_on_fold
thf(fact_3628_mod__type__def,axiom,
! [A: $tType] :
( ( ( minus @ A )
& ( one @ A )
& ( plus @ A )
& ( times @ A )
& ( uminus @ A )
& ( zero @ A ) )
=> ( ( numeral_mod_type @ A )
= ( ^ [N6: int,Rep2: A > int,Abs2: int > A] :
( ( type_definition @ A @ int @ Rep2 @ Abs2 @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ N6 ) )
& ( ord_less @ int @ ( one_one @ int ) @ N6 )
& ( ( zero_zero @ A )
= ( Abs2 @ ( zero_zero @ int ) ) )
& ( ( one_one @ A )
= ( Abs2 @ ( one_one @ int ) ) )
& ! [X4: A,Y5: A] :
( ( plus_plus @ A @ X4 @ Y5 )
= ( Abs2 @ ( modulo_modulo @ int @ ( plus_plus @ int @ ( Rep2 @ X4 ) @ ( Rep2 @ Y5 ) ) @ N6 ) ) )
& ! [X4: A,Y5: A] :
( ( times_times @ A @ X4 @ Y5 )
= ( Abs2 @ ( modulo_modulo @ int @ ( times_times @ int @ ( Rep2 @ X4 ) @ ( Rep2 @ Y5 ) ) @ N6 ) ) )
& ! [X4: A,Y5: A] :
( ( minus_minus @ A @ X4 @ Y5 )
= ( Abs2 @ ( modulo_modulo @ int @ ( minus_minus @ int @ ( Rep2 @ X4 ) @ ( Rep2 @ Y5 ) ) @ N6 ) ) )
& ! [X4: A] :
( ( uminus_uminus @ A @ X4 )
= ( Abs2 @ ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( Rep2 @ X4 ) ) @ N6 ) ) ) ) ) ) ) ).
% mod_type_def
thf(fact_3629_ceiling__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: int,X: A] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ N ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ N ) @ ( one_one @ A ) ) )
=> ( ( archimedean_ceiling @ A @ X )
= ( plus_plus @ int @ N @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_eq
thf(fact_3630_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P3: 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 @ P3 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) @ P3 ) ) ) ).
% ceiling_divide_lower
thf(fact_3631_comp__fun__idem__on_Ointro,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( finite4980608107308702382axioms @ A @ B @ S3 @ F3 )
=> ( finite673082921795544331dem_on @ A @ B @ S3 @ F3 ) ) ) ).
% comp_fun_idem_on.intro
thf(fact_3632_comp__fun__idem__on__def,axiom,
! [B: $tType,A: $tType] :
( ( finite673082921795544331dem_on @ A @ B )
= ( ^ [S6: set @ A,F4: A > B > B] :
( ( finite4664212375090638736ute_on @ A @ B @ S6 @ F4 )
& ( finite4980608107308702382axioms @ A @ B @ S6 @ F4 ) ) ) ) ).
% comp_fun_idem_on_def
thf(fact_3633_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_3634_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_3635_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_3636_of__int__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W2: int,Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ W2 @ Z ) ) ) ).
% of_int_le_iff
thf(fact_3637_of__int__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W2: int,Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ W2 @ Z ) ) ) ).
% of_int_less_iff
thf(fact_3638_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_3639_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_3640_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_3641_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_3642_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_3643_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_3644_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_3645_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_3646_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_3647_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B2: int,W2: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( ord_less_eq @ int @ X @ ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_3648_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W2: nat,X: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less_eq @ int @ ( power_power @ int @ B2 @ W2 ) @ X ) ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_3649_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B2: int,W2: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( ord_less @ int @ X @ ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_3650_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W2: nat,X: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( power_power @ int @ B2 @ W2 ) @ X ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_3651_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_le_of_int
thf(fact_3652_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_less_of_int
thf(fact_3653_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ).
% ex_of_int_less
thf(fact_3654_le__of__int__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% le_of_int_ceiling
thf(fact_3655_ceiling__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ Z )
= ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% ceiling_le_iff
thf(fact_3656_ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: int] :
( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A2 ) )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ A2 ) ) ) ).
% ceiling_le
thf(fact_3657_less__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ int @ Z @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ) ).
% less_ceiling_iff
thf(fact_3658_real__of__int__div4,axiom,
! [N: int,X: int] : ( ord_less_eq @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X ) ) ) ).
% real_of_int_div4
thf(fact_3659_comp__fun__idem__on_Oaxioms_I2_J,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B] :
( ( finite673082921795544331dem_on @ A @ B @ S3 @ F3 )
=> ( finite4980608107308702382axioms @ A @ B @ S3 @ F3 ) ) ).
% comp_fun_idem_on.axioms(2)
thf(fact_3660_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_3661_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_3662_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_leD
thf(fact_3663_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_lessD
thf(fact_3664_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_3665_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [X3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X3 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) ) )
& ! [Y4: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y4 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y4 @ ( one_one @ int ) ) ) ) )
=> ( Y4 = X3 ) ) ) ) ).
% floor_exists1
thf(fact_3666_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_3667_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_3668_of__nat__less__of__int__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,X: int] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ X ) ) ) ).
% of_nat_less_of_int_iff
thf(fact_3669_int__le__real__less,axiom,
( ( ord_less_eq @ int )
= ( ^ [N6: int,M5: int] : ( ord_less @ real @ ( ring_1_of_int @ real @ N6 ) @ ( plus_plus @ real @ ( ring_1_of_int @ real @ M5 ) @ ( one_one @ real ) ) ) ) ) ).
% int_le_real_less
thf(fact_3670_int__less__real__le,axiom,
( ( ord_less @ int )
= ( ^ [N6: int,M5: int] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N6 ) @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ M5 ) ) ) ) ).
% int_less_real_le
thf(fact_3671_ceiling__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ) ).
% ceiling_correct
thf(fact_3672_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) )
=> ( ( archimedean_ceiling @ A @ X )
= Z ) ) ) ) ).
% ceiling_unique
thf(fact_3673_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: int] :
( ( ( archimedean_ceiling @ A @ X )
= A2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A2 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_3674_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P2: int > $o,T2: A] :
( ( P2 @ ( archimedean_ceiling @ A @ T2 ) )
= ( ! [I3: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) @ T2 )
& ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I3 ) ) )
=> ( P2 @ I3 ) ) ) ) ) ).
% ceiling_split
thf(fact_3675_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ Z )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_3676_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ int @ Z @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X ) ) ) ).
% le_ceiling_iff
thf(fact_3677_real__of__int__div2,axiom,
! [N: int,X: int] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X ) ) ) ) ).
% real_of_int_div2
thf(fact_3678_real__of__int__div3,axiom,
! [N: int,X: int] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X ) ) ) @ ( one_one @ real ) ) ).
% real_of_int_div3
thf(fact_3679_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less_eq @ A @ P3 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P3 @ Q3 ) ) ) @ Q3 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_3680_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_3681_powr__int,axiom,
! [X: real,I: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ I ) )
= ( power_power @ real @ X @ ( nat2 @ I ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ I ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( nat2 @ ( uminus_uminus @ int @ I ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_3682_floor__log__eq__powr__iff,axiom,
! [X: real,B2: real,K: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ B2 @ X ) )
= K )
= ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ K ) ) @ X )
& ( ord_less @ real @ X @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ) ) ) ) ) ).
% floor_log_eq_powr_iff
thf(fact_3683_mod__ring_Ocases,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: int,Rep: A > int,Abs: int > A,X: A] :
( ( numeral_mod_ring @ A @ N @ Rep @ Abs )
=> ~ ! [Z3: int] :
( ( X
= ( ring_1_of_int @ A @ Z3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ~ ( ord_less @ int @ Z3 @ N ) ) ) ) ) ).
% mod_ring.cases
thf(fact_3684_floor__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% floor_zero
thf(fact_3685_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_3686_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_3687_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_3688_zless__nat__conj,axiom,
! [W2: int,Z: int] :
( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
& ( ord_less @ int @ W2 @ Z ) ) ) ).
% zless_nat_conj
thf(fact_3689_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ nat ) ) ).
% nat_zminus_int
thf(fact_3690_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_3691_zero__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% zero_le_floor
thf(fact_3692_floor__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% floor_less_zero
thf(fact_3693_zero__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% zero_less_floor
thf(fact_3694_floor__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_le_zero
thf(fact_3695_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_3696_one__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_le_floor
thf(fact_3697_floor__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_less_one
thf(fact_3698_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_3699_nat__ceiling__le__eq,axiom,
! [X: real,A2: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) @ A2 )
= ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ A2 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_3700_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_3701_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_3702_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_3703_nat__floor__neg,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= ( zero_zero @ nat ) ) ) ).
% nat_floor_neg
thf(fact_3704_floor__eq3,axiom,
! [N: nat,X: real] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= N ) ) ) ).
% floor_eq3
thf(fact_3705_le__nat__floor,axiom,
! [X: nat,A2: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X ) @ A2 )
=> ( ord_less_eq @ nat @ X @ ( nat2 @ ( archim6421214686448440834_floor @ real @ A2 ) ) ) ) ).
% le_nat_floor
thf(fact_3706_floor__eq4,axiom,
! [N: nat,X: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= N ) ) ) ).
% floor_eq4
thf(fact_3707_of__int__floor__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X ) ) ).
% of_int_floor_le
thf(fact_3708_floor__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ).
% floor_mono
thf(fact_3709_floor__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% floor_less_cancel
thf(fact_3710_nat__zero__as__int,axiom,
( ( zero_zero @ nat )
= ( nat2 @ ( zero_zero @ int ) ) ) ).
% nat_zero_as_int
thf(fact_3711_floor__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% floor_le_ceiling
thf(fact_3712_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ X @ Y )
=> ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_3713_eq__nat__nat__iff,axiom,
! [Z: int,Z8: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z8 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z8 ) )
= ( Z = Z8 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_3714_all__nat,axiom,
( ( ^ [P4: nat > $o] :
! [X7: nat] : ( P4 @ X7 ) )
= ( ^ [P5: nat > $o] :
! [X4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
=> ( P5 @ ( nat2 @ X4 ) ) ) ) ) ).
% all_nat
thf(fact_3715_ex__nat,axiom,
( ( ^ [P4: nat > $o] :
? [X7: nat] : ( P4 @ X7 ) )
= ( ^ [P5: nat > $o] :
? [X4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
& ( P5 @ ( nat2 @ X4 ) ) ) ) ) ).
% ex_nat
thf(fact_3716_le__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ int @ Z @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ) ).
% le_floor_iff
thf(fact_3717_floor__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z )
= ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% floor_less_iff
thf(fact_3718_le__floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% le_floor_add
thf(fact_3719_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_3720_floor__eq,axiom,
! [N: int,X: real] :
( ( ord_less @ real @ ( ring_1_of_int @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X )
= N ) ) ) ).
% floor_eq
thf(fact_3721_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_3722_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_3723_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_3724_nat__mono__iff,axiom,
! [Z: int,W2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ W2 @ Z ) ) ) ).
% nat_mono_iff
thf(fact_3725_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_3726_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% nat_le_iff
thf(fact_3727_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_3728_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_3729_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_3730_word__of__int__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ring_1_of_int @ ( word @ A ) @ X )
= ( semiring_1_of_nat @ ( word @ A ) @ ( nat2 @ X ) ) ) ) ) ).
% word_of_int_nat
thf(fact_3731_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_3732_floor__eq2,axiom,
! [N: int,X: real] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X )
= N ) ) ) ).
% floor_eq2
thf(fact_3733_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_3734_ceiling__diff__floor__le__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archim6421214686448440834_floor @ A @ X ) ) @ ( one_one @ int ) ) ) ).
% ceiling_diff_floor_le_1
thf(fact_3735_nat__less__eq__zless,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ W2 @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_3736_nat__eq__iff2,axiom,
! [M: nat,W2: int] :
( ( M
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( W2
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff2
thf(fact_3737_nat__eq__iff,axiom,
! [W2: int,M: nat] :
( ( ( nat2 @ W2 )
= M )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( W2
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff
thf(fact_3738_split__nat,axiom,
! [P2: nat > $o,I: int] :
( ( P2 @ ( nat2 @ I ) )
= ( ! [N6: nat] :
( ( I
= ( semiring_1_of_nat @ int @ N6 ) )
=> ( P2 @ N6 ) )
& ( ( ord_less @ int @ I @ ( zero_zero @ int ) )
=> ( P2 @ ( zero_zero @ nat ) ) ) ) ) ).
% split_nat
thf(fact_3739_nat__le__eq__zle,axiom,
! [W2: int,Z: int] :
( ( ( ord_less @ int @ ( zero_zero @ int ) @ W2 )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) )
=> ( ( ord_less_eq @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_eq @ int @ W2 @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_3740_nat__add__distrib,axiom,
! [Z: int,Z8: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z8 )
=> ( ( nat2 @ ( plus_plus @ int @ Z @ Z8 ) )
= ( plus_plus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z8 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_3741_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_3742_Suc__as__int,axiom,
( suc
= ( ^ [A8: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A8 ) @ ( one_one @ int ) ) ) ) ) ).
% Suc_as_int
thf(fact_3743_nat__mult__distrib,axiom,
! [Z: int,Z8: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( nat2 @ ( times_times @ int @ Z @ Z8 ) )
= ( times_times @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z8 ) ) ) ) ).
% nat_mult_distrib
thf(fact_3744_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( minus_minus @ int @ X @ Y ) )
= ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_3745_nat__diff__distrib,axiom,
! [Z8: int,Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z8 )
=> ( ( ord_less_eq @ int @ Z8 @ Z )
=> ( ( nat2 @ ( minus_minus @ int @ Z @ Z8 ) )
= ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z8 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_3746_nat__div__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( nat2 @ ( divide_divide @ int @ X @ Y ) )
= ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib
thf(fact_3747_nat__div__distrib_H,axiom,
! [Y: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( divide_divide @ int @ X @ Y ) )
= ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib'
thf(fact_3748_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_3749_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_3750_nat__mod__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( modulo_modulo @ int @ X @ Y ) )
= ( modulo_modulo @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_3751_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_3752_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_3753_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P2: int > $o,T2: A] :
( ( P2 @ ( archim6421214686448440834_floor @ A @ T2 ) )
= ( ! [I3: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I3 ) @ T2 )
& ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) ) )
=> ( P2 @ I3 ) ) ) ) ) ).
% floor_split
thf(fact_3754_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: int] :
( ( ( archim6421214686448440834_floor @ A @ X )
= A2 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_3755_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X )
=> ( ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X )
= Z ) ) ) ) ).
% floor_unique
thf(fact_3756_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ int @ Z @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X ) ) ) ).
% less_floor_iff
thf(fact_3757_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_3758_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_3759_floor__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_correct
thf(fact_3760_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_3761_nat__less__iff,axiom,
! [W2: int,M: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ M )
= ( ord_less @ int @ W2 @ ( semiring_1_of_nat @ int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_3762_nat__mult__distrib__neg,axiom,
! [Z: int,Z8: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z @ Z8 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z8 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_3763_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_3764_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_3765_diff__nat__eq__if,axiom,
! [Z8: int,Z: int] :
( ( ( ord_less @ int @ Z8 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z8 ) )
= ( nat2 @ Z ) ) )
& ( ~ ( ord_less @ int @ Z8 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z8 ) )
= ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z @ Z8 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z @ Z8 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_3766_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P3: 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 @ P3 @ Q3 ) ) ) @ Q3 ) @ P3 ) ) ) ).
% floor_divide_lower
thf(fact_3767_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_3768_floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_add
thf(fact_3769_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_3770_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less @ A @ P3 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P3 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) ) ) ) ).
% floor_divide_upper
thf(fact_3771_powr__real__of__int,axiom,
! [X: real,N: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ N ) )
= ( power_power @ real @ X @ ( nat2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ N ) )
= ( inverse_inverse @ real @ ( power_power @ real @ X @ ( nat2 @ ( uminus_uminus @ int @ N ) ) ) ) ) ) ) ) ).
% powr_real_of_int
thf(fact_3772_card__UNION,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A5 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A5 )
=> ( finite_finite2 @ A @ X3 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) )
= ( nat2
@ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int
@ ^ [I8: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I8 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I8 ) ) ) )
@ ( collect @ ( set @ ( set @ A ) )
@ ^ [I8: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I8 @ A5 )
& ( I8
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% card_UNION
thf(fact_3773_floor__real__def,axiom,
( ( archim6421214686448440834_floor @ real )
= ( ^ [X4: real] :
( the @ int
@ ^ [Z2: int] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ Z2 ) @ X4 )
& ( ord_less @ real @ X4 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_3774_comp__fun__commute__product__fold,axiom,
! [A: $tType,B: $tType,B5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
@ ^ [X4: B,Z2: set @ ( product_prod @ B @ A )] :
( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y5: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y5 ) )
@ Z2
@ B5 ) ) ) ).
% comp_fun_commute_product_fold
thf(fact_3775_inverse__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% inverse_zero
thf(fact_3776_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_3777_finite__Inter,axiom,
! [A: $tType,M7: set @ ( set @ A )] :
( ? [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ M7 )
& ( finite_finite2 @ A @ X5 ) )
=> ( finite_finite2 @ A @ ( complete_Inf_Inf @ ( set @ A ) @ M7 ) ) ) ).
% finite_Inter
thf(fact_3778_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_3779_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_3780_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_3781_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_3782_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_3783_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_3784_cSup__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% cSup_singleton
thf(fact_3785_Sup__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_atLeastAtMost
thf(fact_3786_cSup__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_atLeastAtMost
thf(fact_3787_cInf__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% cInf_singleton
thf(fact_3788_cSup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_atLeastLessThan
thf(fact_3789_Sup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_atLeastLessThan
thf(fact_3790_Inf__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_atLeastAtMost
thf(fact_3791_cInf__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_atLeastAtMost
thf(fact_3792_cInf__atLeastLessThan,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_atLeastLessThan
thf(fact_3793_Inf__atLeastLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_atLeastLessThan
thf(fact_3794_finite__UN,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: A > ( set @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B5 @ A5 ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( finite_finite2 @ B @ ( B5 @ X4 ) ) ) ) ) ) ).
% finite_UN
thf(fact_3795_Inf__atMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_atMost @ A @ X ) )
= ( bot_bot @ A ) ) ) ).
% Inf_atMost
thf(fact_3796_finite__Union,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A5 )
=> ( ! [M11: set @ A] :
( ( member @ ( set @ A ) @ M11 @ A5 )
=> ( finite_finite2 @ A @ M11 ) )
=> ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) ) ) ) ).
% finite_Union
thf(fact_3797_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,C3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X4: B] : C3
@ A5 ) )
= C3 ) ) ) ).
% cSUP_const
thf(fact_3798_cINF__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,C3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X4: B] : C3
@ A5 ) )
= C3 ) ) ) ).
% cINF_const
thf(fact_3799_finite__UN__I,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: A > ( set @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( finite_finite2 @ B @ ( B5 @ A4 ) ) )
=> ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B5 @ A5 ) ) ) ) ) ).
% finite_UN_I
thf(fact_3800_finite__INT,axiom,
! [B: $tType,A: $tType,I6: set @ A,A5: A > ( set @ B )] :
( ? [X5: A] :
( ( member @ A @ X5 @ I6 )
& ( finite_finite2 @ B @ ( A5 @ X5 ) ) )
=> ( finite_finite2 @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A5 @ I6 ) ) ) ) ).
% finite_INT
thf(fact_3801_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_3802_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_3803_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_3804_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_3805_cInf__eq__minimum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z: A,X6: set @ A] :
( ( member @ A @ Z @ X6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ Z @ X3 ) )
=> ( ( complete_Inf_Inf @ A @ X6 )
= Z ) ) ) ) ).
% cInf_eq_minimum
thf(fact_3806_cInf__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_top @ A ) )
=> ! [X6: set @ A,A2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ A2 @ X3 ) )
=> ( ! [Y3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ( ord_less_eq @ A @ Y3 @ X5 ) )
=> ( ord_less_eq @ A @ Y3 @ A2 ) )
=> ( ( complete_Inf_Inf @ A @ X6 )
= A2 ) ) ) ) ).
% cInf_eq
thf(fact_3807_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z: A,X6: set @ A] :
( ( member @ A @ Z @ X6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= Z ) ) ) ) ).
% cSup_eq_maximum
thf(fact_3808_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_bot @ A ) )
=> ! [X6: set @ A,A2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ X3 @ A2 ) )
=> ( ! [Y3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ( ord_less_eq @ A @ X5 @ Y3 ) )
=> ( ord_less_eq @ A @ A2 @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= A2 ) ) ) ) ).
% cSup_eq
thf(fact_3809_Some__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ A @ ( complete_Sup_Sup @ A @ A5 ) )
= ( complete_Sup_Sup @ ( option @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A5 ) ) ) ) ) ).
% Some_Sup
thf(fact_3810_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_3811_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_3812_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_3813_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_3814_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_3815_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_3816_comp__fun__commute__const,axiom,
! [B: $tType,A: $tType,F3: B > B] :
( finite6289374366891150609ommute @ A @ B
@ ^ [Uu3: A] : F3 ) ).
% comp_fun_commute_const
thf(fact_3817_Some__SUP,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ B )
=> ! [A5: set @ A,F3: A > B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ B @ ( complete_Sup_Sup @ B @ ( image @ A @ B @ F3 @ A5 ) ) )
= ( complete_Sup_Sup @ ( option @ B )
@ ( image @ A @ ( option @ B )
@ ^ [X4: A] : ( some @ B @ ( F3 @ X4 ) )
@ A5 ) ) ) ) ) ).
% Some_SUP
thf(fact_3818_norm__inverse__le__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [R2: real,X: A] :
( ( ord_less_eq @ real @ R2 @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ X ) ) @ ( inverse_inverse @ real @ R2 ) ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_3819_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_3820_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_3821_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_3822_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_3823_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_3824_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_3825_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_3826_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_3827_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_3828_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_3829_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_3830_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,A2: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ A2 @ X3 ) )
=> ( ! [Y3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ( ord_less_eq @ A @ Y3 @ X5 ) )
=> ( ord_less_eq @ A @ Y3 @ A2 ) )
=> ( ( complete_Inf_Inf @ A @ X6 )
= A2 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_3831_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,Z: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ Z @ X3 ) )
=> ( ord_less_eq @ A @ Z @ ( complete_Inf_Inf @ A @ X6 ) ) ) ) ) ).
% cInf_greatest
thf(fact_3832_cInf__le__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,X: A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( member @ A @ X @ X6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X6 ) @ X ) ) ) ) ).
% cInf_le_finite
thf(fact_3833_cInf__lessD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,Z: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X6 ) @ Z )
=> ? [X3: A] :
( ( member @ A @ X3 @ X6 )
& ( ord_less @ A @ X3 @ Z ) ) ) ) ) ).
% cInf_lessD
thf(fact_3834_finite__imp__less__Inf,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,X: A,A2: A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( member @ A @ X @ X6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less @ A @ A2 @ X3 ) )
=> ( ord_less @ A @ A2 @ ( complete_Inf_Inf @ A @ X6 ) ) ) ) ) ) ).
% finite_imp_less_Inf
thf(fact_3835_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,A2: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ X3 @ A2 ) )
=> ( ! [Y3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ( ord_less_eq @ A @ X5 @ Y3 ) )
=> ( ord_less_eq @ A @ A2 @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= A2 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_3836_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,Z: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X6 ) @ Z ) ) ) ) ).
% cSup_least
thf(fact_3837_le__cSup__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,X: A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( member @ A @ X @ X6 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X6 ) ) ) ) ) ).
% le_cSup_finite
thf(fact_3838_less__cSupE,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [Y: A,X6: set @ A] :
( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X6 ) )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ) ) ).
% less_cSupE
thf(fact_3839_less__cSupD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,Z: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z @ ( complete_Sup_Sup @ A @ X6 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ X6 )
& ( ord_less @ A @ Z @ X3 ) ) ) ) ) ).
% less_cSupD
thf(fact_3840_comp__fun__commute__filter__fold,axiom,
! [A: $tType,P2: A > $o] :
( finite6289374366891150609ommute @ A @ ( set @ A )
@ ^ [X4: A,A11: set @ A] : ( if @ ( set @ A ) @ ( P2 @ X4 ) @ ( insert @ A @ X4 @ A11 ) @ A11 ) ) ).
% comp_fun_commute_filter_fold
thf(fact_3841_finite__imp__Sup__less,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,X: A,A2: A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( member @ A @ X @ X6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less @ A @ X3 @ A2 ) )
=> ( ord_less @ A @ ( complete_Sup_Sup @ A @ X6 ) @ A2 ) ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_3842_card__Union__le__sum__card,axiom,
! [A: $tType,U2: set @ ( set @ A )] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U2 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U2 ) ) ).
% card_Union_le_sum_card
thf(fact_3843_finite__UNION__then__finite,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A5: set @ B,A2: B] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A5 ) ) )
=> ( ( member @ B @ A2 @ A5 )
=> ( finite_finite2 @ A @ ( B5 @ A2 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_3844_comp__fun__commute_Ocomp__fun__commute__funpow,axiom,
! [B: $tType,A: $tType,F3: A > B > B,G3: A > nat] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( finite6289374366891150609ommute @ A @ B
@ ^ [X4: A] : ( compow @ ( B > B ) @ ( G3 @ X4 ) @ ( F3 @ X4 ) ) ) ) ).
% comp_fun_commute.comp_fun_commute_funpow
thf(fact_3845_comp__fun__commute_Ofoldl__f__commute,axiom,
! [B: $tType,A: $tType,F3: A > B > B,A2: A,B2: B,Xs: list @ A] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( ( F3 @ A2
@ ( foldl @ B @ A
@ ^ [A8: B,B8: A] : ( F3 @ B8 @ A8 )
@ B2
@ Xs ) )
= ( foldl @ B @ A
@ ^ [A8: B,B8: A] : ( F3 @ B8 @ A8 )
@ ( F3 @ A2 @ B2 )
@ Xs ) ) ) ).
% comp_fun_commute.foldl_f_commute
thf(fact_3846_finite__UnionD,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) )
=> ( finite_finite2 @ ( set @ A ) @ A5 ) ) ).
% finite_UnionD
thf(fact_3847_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_3848_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_3849_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_3850_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_3851_inverse__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% inverse_le_1_iff
thf(fact_3852_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_3853_one__less__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_less_inverse_iff
thf(fact_3854_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_3855_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_3856_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_3857_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_3858_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_3859_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,M: A,F3: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ M @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ M @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A5 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_3860_finite__less__Inf__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,A2: A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ A2 @ ( complete_Inf_Inf @ A @ X6 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ X6 )
=> ( ord_less @ A @ A2 @ X4 ) ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_3861_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F3: B > A,M7: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ M7 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A5 ) ) @ M7 ) ) ) ) ).
% cSUP_least
thf(fact_3862_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,A2: A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X6 ) @ A2 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ X6 )
=> ( ord_less @ A @ X4 @ A2 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_3863_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,A2: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S3 ) ) @ A2 ) ) ) ) ).
% cInf_abs_ge
thf(fact_3864_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,A2: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S3 ) ) @ A2 ) ) ) ) ).
% cSup_abs_le
thf(fact_3865_inverse__powr,axiom,
! [Y: real,A2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( powr @ real @ ( inverse_inverse @ real @ Y ) @ A2 )
= ( inverse_inverse @ real @ ( powr @ real @ Y @ A2 ) ) ) ) ).
% inverse_powr
thf(fact_3866_Max__Sup,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% Max_Sup
thf(fact_3867_cSup__eq__Max,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= ( lattic643756798349783984er_Max @ A @ X6 ) ) ) ) ) ).
% cSup_eq_Max
thf(fact_3868_card__Union__le__sum__card__weak,axiom,
! [A: $tType,U2: set @ ( set @ A )] :
( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ U2 )
=> ( finite_finite2 @ A @ X3 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U2 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U2 ) ) ) ).
% card_Union_le_sum_card_weak
thf(fact_3869_Min__Inf,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= ( complete_Inf_Inf @ A @ A5 ) ) ) ) ) ).
% Min_Inf
thf(fact_3870_cInf__eq__Min,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X6 )
= ( lattic643756798350308766er_Min @ A @ X6 ) ) ) ) ) ).
% cInf_eq_Min
thf(fact_3871_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_3872_inverse__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% inverse_less_1_iff
thf(fact_3873_one__le__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_le_inverse_iff
thf(fact_3874_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_3875_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_3876_comp__fun__commute_Ofoldr__conv__foldl,axiom,
! [B: $tType,A: $tType,F3: A > B > B,Xs: list @ A,A2: B] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( ( foldr @ A @ B @ F3 @ Xs @ A2 )
= ( foldl @ B @ A
@ ^ [A8: B,B8: A] : ( F3 @ B8 @ A8 )
@ A2
@ Xs ) ) ) ).
% comp_fun_commute.foldr_conv_foldl
thf(fact_3877_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_3878_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) ) @ X ) ) ) ).
% reals_Archimedean
thf(fact_3879_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,L: A,E3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S3 ) @ L ) ) @ E3 ) ) ) ) ).
% cInf_asclose
thf(fact_3880_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,L: A,E3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S3 ) @ L ) ) @ E3 ) ) ) ) ).
% cSup_asclose
thf(fact_3881_forall__pos__mono__1,axiom,
! [P2: real > $o,E3: real] :
( ! [D3: real,E2: real] :
( ( ord_less @ real @ D3 @ E2 )
=> ( ( P2 @ D3 )
=> ( P2 @ E2 ) ) )
=> ( ! [N2: nat] : ( P2 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ( P2 @ E3 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_3882_forall__pos__mono,axiom,
! [P2: real > $o,E3: real] :
( ! [D3: real,E2: real] :
( ( ord_less @ real @ D3 @ E2 )
=> ( ( P2 @ D3 )
=> ( P2 @ E2 ) ) )
=> ( ! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( P2 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N2 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ( P2 @ E3 ) ) ) ) ).
% forall_pos_mono
thf(fact_3883_real__arch__inverse,axiom,
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
= ( ? [N6: nat] :
( ( N6
!= ( zero_zero @ nat ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N6 ) ) )
& ( ord_less @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N6 ) ) @ E3 ) ) ) ) ).
% real_arch_inverse
thf(fact_3884_Sup__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S3: set @ A,X: A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( ( S3
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ X @ S3 ) )
= X ) )
& ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ X @ S3 ) )
= ( ord_max @ A @ X @ ( complete_Sup_Sup @ A @ S3 ) ) ) ) ) ) ) ).
% Sup_insert_finite
thf(fact_3885_ln__inverse,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( inverse_inverse @ real @ X ) )
= ( uminus_uminus @ real @ ( ln_ln @ real @ X ) ) ) ) ).
% ln_inverse
thf(fact_3886_comp__fun__commute__fold__insert,axiom,
! [B: $tType,A: $tType,F3: A > B > B,A5: set @ A,X: A,Z: B] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( insert @ A @ X @ A5 ) )
= ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z @ A5 ) ) ) ) ) ) ).
% comp_fun_commute_fold_insert
thf(fact_3887_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X8: set @ A] :
( the @ A
@ ^ [X4: A] :
( X8
= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_3888_finite__subset__Union,axiom,
! [A: $tType,A5: set @ A,B11: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ B11 ) )
=> ~ ! [F9: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F9 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F9 @ B11 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ F9 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_3889_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ X ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_3890_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( power_power @ A @ X @ ( minus_minus @ nat @ N @ M ) )
= ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ M ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_3891_Id__on__def,axiom,
! [A: $tType] :
( ( id_on @ A )
= ( ^ [A6: set @ A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X4: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
@ A6 ) ) ) ) ).
% Id_on_def
thf(fact_3892_log__inverse,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ A2 @ ( inverse_inverse @ real @ X ) )
= ( uminus_uminus @ real @ ( log @ A2 @ X ) ) ) ) ) ) ).
% log_inverse
thf(fact_3893_card__UN__le,axiom,
! [B: $tType,A: $tType,I6: set @ A,A5: A > ( set @ B )] :
( ( finite_finite2 @ A @ I6 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A5 @ I6 ) ) )
@ ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] : ( finite_card @ B @ ( A5 @ I3 ) )
@ I6 ) ) ) ).
% card_UN_le
thf(fact_3894_UN__singleton,axiom,
! [A: $tType,A5: set @ A] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ A @ ( set @ A )
@ ^ [X4: A] : ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) )
@ A5 ) )
= A5 ) ).
% UN_singleton
thf(fact_3895_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C4: set @ B,A2: A,B5: B > ( set @ A )] :
( ( ( C4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X4: B] : ( insert @ A @ A2 @ ( B5 @ X4 ) )
@ C4 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X4: B] : ( insert @ A @ A2 @ ( B5 @ X4 ) )
@ C4 ) )
= ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ C4 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_3896_ccSUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F3: B > A] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% ccSUP_empty
thf(fact_3897_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A5: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A5 ) )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X4 )
=> ? [Y5: B] :
( ( member @ B @ Y5 @ A5 )
& ( ord_less @ A @ ( F3 @ Y5 ) @ X4 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_3898_subset__mset_OcINF__const,axiom,
! [B: $tType,A: $tType,A5: set @ B,C3: multiset @ A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A )
@ ( image @ B @ ( multiset @ A )
@ ^ [X4: B] : C3
@ A5 ) )
= C3 ) ) ).
% subset_mset.cINF_const
thf(fact_3899_subset__mset_OcSUP__const,axiom,
! [B: $tType,A: $tType,A5: set @ B,C3: multiset @ A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A )
@ ( image @ B @ ( multiset @ A )
@ ^ [X4: B] : C3
@ A5 ) )
= C3 ) ) ).
% subset_mset.cSUP_const
thf(fact_3900_Sup__bot__conv_I2_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ A5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( X4
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_3901_Sup__bot__conv_I1_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( ( complete_Sup_Sup @ A @ A5 )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( X4
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_3902_Sup__nat__empty,axiom,
( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% Sup_nat_empty
thf(fact_3903_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A] :
( ( ( complete_Inf_Inf @ A @ A5 )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X4 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A5 )
& ( ord_less @ A @ Y5 @ X4 ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_3904_Sup__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Sup_empty
thf(fact_3905_ccSup__empty,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% ccSup_empty
thf(fact_3906_ccSUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X4: B] : ( bot_bot @ A )
@ A5 ) )
= ( bot_bot @ A ) ) ) ).
% ccSUP_bot
thf(fact_3907_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X4: B] : ( bot_bot @ A )
@ A5 ) )
= ( bot_bot @ A ) ) ) ).
% SUP_bot
thf(fact_3908_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: B > A,A5: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ B5 @ A5 ) )
= ( bot_bot @ A ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( B5 @ X4 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_3909_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: B > A,A5: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image @ B @ A @ B5 @ A5 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( B5 @ X4 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_3910_ccSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ B,F3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I3: B] : F3
@ A5 ) )
= F3 ) ) ) ).
% ccSUP_const
thf(fact_3911_SUP__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I3: B] : F3
@ A5 ) )
= F3 ) ) ) ).
% SUP_const
thf(fact_3912_ccINF__const,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ B,F3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I3: B] : F3
@ A5 ) )
= F3 ) ) ) ).
% ccINF_const
thf(fact_3913_INF__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I3: B] : F3
@ A5 ) )
= F3 ) ) ) ).
% INF_const
thf(fact_3914_UN__constant,axiom,
! [B: $tType,A: $tType,A5: set @ B,C3: set @ A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y5: B] : C3
@ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y5: B] : C3
@ A5 ) )
= C3 ) ) ) ).
% UN_constant
thf(fact_3915_Inf__nat__def1,axiom,
! [K4: set @ nat] :
( ( K4
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( complete_Inf_Inf @ nat @ K4 ) @ K4 ) ) ).
% Inf_nat_def1
thf(fact_3916_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S3: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I3: set @ ( product_prod @ A @ B ),X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ I3 )
@ S3 ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_3917_INF__Int__eq2,axiom,
! [B: $tType,A: $tType,S3: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I3: set @ ( product_prod @ A @ B ),X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ I3 )
@ S3 ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) ) ) ) ).
% INF_Int_eq2
thf(fact_3918_INF__INT__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S3: set @ C] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image @ C @ ( A > B > $o )
@ ^ [I3: C,X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( R2 @ I3 ) )
@ S3 ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ) ) ) ).
% INF_INT_eq2
thf(fact_3919_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S3: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image @ C @ ( A > B > $o )
@ ^ [I3: C,X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( R2 @ I3 ) )
@ S3 ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_3920_subset__mset_OcINF__greatest,axiom,
! [A: $tType,B: $tType,A5: set @ B,M: multiset @ A,F3: B > ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( subseteq_mset @ A @ M @ ( F3 @ X3 ) ) )
=> ( subseteq_mset @ A @ M @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image @ B @ ( multiset @ A ) @ F3 @ A5 ) ) ) ) ) ).
% subset_mset.cINF_greatest
thf(fact_3921_subset__mset_OcSUP__least,axiom,
! [B: $tType,A: $tType,A5: set @ B,F3: B > ( multiset @ A ),M7: multiset @ A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( subseteq_mset @ A @ ( F3 @ X3 ) @ M7 ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image @ B @ ( multiset @ A ) @ F3 @ A5 ) ) @ M7 ) ) ) ).
% subset_mset.cSUP_least
thf(fact_3922_INF__filter__not__bot,axiom,
! [I5: $tType,A: $tType,B5: set @ I5,F5: I5 > ( filter @ A )] :
( ! [X10: set @ I5] :
( ( ord_less_eq @ ( set @ I5 ) @ X10 @ B5 )
=> ( ( finite_finite2 @ I5 @ X10 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ I5 @ ( filter @ A ) @ F5 @ X10 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ I5 @ ( filter @ A ) @ F5 @ B5 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% INF_filter_not_bot
thf(fact_3923_Sup__nat__def,axiom,
( ( complete_Sup_Sup @ nat )
= ( ^ [X8: set @ nat] :
( if @ nat
@ ( X8
= ( bot_bot @ ( set @ nat ) ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat @ X8 ) ) ) ) ).
% Sup_nat_def
thf(fact_3924_ln__neg__is__const,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ln_ln @ real @ X )
= ( the @ real
@ ^ [X4: real] : $false ) ) ) ).
% ln_neg_is_const
thf(fact_3925_Sup__upper2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A5: set @ A,V3: A] :
( ( member @ A @ U @ A5 )
=> ( ( ord_less_eq @ A @ V3 @ U )
=> ( ord_less_eq @ A @ V3 @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% Sup_upper2
thf(fact_3926_Sup__le__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B2: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ B2 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X4 @ B2 ) ) ) ) ) ).
% Sup_le_iff
thf(fact_3927_Sup__upper,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A5: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ).
% Sup_upper
thf(fact_3928_Sup__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,Z: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ Z ) ) ) ).
% Sup_least
thf(fact_3929_Sup__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ? [X5: A] :
( ( member @ A @ X5 @ B5 )
& ( ord_less_eq @ A @ A4 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ).
% Sup_mono
thf(fact_3930_Sup__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,X: A] :
( ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ! [Y3: A] :
( ! [Z5: A] :
( ( member @ A @ Z5 @ A5 )
=> ( ord_less_eq @ A @ Z5 @ Y3 ) )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ A5 )
= X ) ) ) ) ).
% Sup_eqI
thf(fact_3931_less__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A2: A,S3: set @ A] :
( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ S3 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ S3 )
& ( ord_less @ A @ A2 @ X4 ) ) ) ) ) ).
% less_Sup_iff
thf(fact_3932_Inf__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,Z: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ A @ Z @ X3 ) )
=> ( ord_less_eq @ A @ Z @ ( complete_Inf_Inf @ A @ A5 ) ) ) ) ).
% Inf_greatest
thf(fact_3933_le__Inf__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: A,A5: set @ A] :
( ( ord_less_eq @ A @ B2 @ ( complete_Inf_Inf @ A @ A5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ B2 @ X4 ) ) ) ) ) ).
% le_Inf_iff
thf(fact_3934_Inf__lower2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A5: set @ A,V3: A] :
( ( member @ A @ U @ A5 )
=> ( ( ord_less_eq @ A @ U @ V3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ V3 ) ) ) ) ).
% Inf_lower2
thf(fact_3935_Inf__lower,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A5: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ X ) ) ) ).
% Inf_lower
thf(fact_3936_Inf__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: set @ A,A5: set @ A] :
( ! [B4: A] :
( ( member @ A @ B4 @ B5 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A5 )
& ( ord_less_eq @ A @ X5 @ B4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B5 ) ) ) ) ).
% Inf_mono
thf(fact_3937_Inf__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,X: A] :
( ! [I2: A] :
( ( member @ A @ I2 @ A5 )
=> ( ord_less_eq @ A @ X @ I2 ) )
=> ( ! [Y3: A] :
( ! [I4: A] :
( ( member @ A @ I4 @ A5 )
=> ( ord_less_eq @ A @ Y3 @ I4 ) )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( complete_Inf_Inf @ A @ A5 )
= X ) ) ) ) ).
% Inf_eqI
thf(fact_3938_Inf__less__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [S3: set @ A,A2: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S3 ) @ A2 )
= ( ? [X4: A] :
( ( member @ A @ X4 @ S3 )
& ( ord_less @ A @ X4 @ A2 ) ) ) ) ) ).
% Inf_less_iff
thf(fact_3939_Union__empty__conv,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ A5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A5 )
=> ( X4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_empty_conv
thf(fact_3940_empty__Union__conv,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ A5 ) )
= ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A5 )
=> ( X4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% empty_Union_conv
thf(fact_3941_Union__subsetI,axiom,
! [A: $tType,A5: set @ ( set @ A ),B5: set @ ( set @ A )] :
( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A5 )
=> ? [Y4: set @ A] :
( ( member @ ( set @ A ) @ Y4 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ X3 @ Y4 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B5 ) ) ) ).
% Union_subsetI
thf(fact_3942_Union__upper,axiom,
! [A: $tType,B5: set @ A,A5: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B5 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) ) ) ).
% Union_upper
thf(fact_3943_Union__least,axiom,
! [A: $tType,A5: set @ ( set @ A ),C4: set @ A] :
( ! [X10: set @ A] :
( ( member @ ( set @ A ) @ X10 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ X10 @ C4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) @ C4 ) ) ).
% Union_least
thf(fact_3944_Inter__greatest,axiom,
! [A: $tType,A5: set @ ( set @ A ),C4: set @ A] :
( ! [X10: set @ A] :
( ( member @ ( set @ A ) @ X10 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ C4 @ X10 ) )
=> ( ord_less_eq @ ( set @ A ) @ C4 @ ( complete_Inf_Inf @ ( set @ A ) @ A5 ) ) ) ).
% Inter_greatest
thf(fact_3945_Inter__lower,axiom,
! [A: $tType,B5: set @ A,A5: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B5 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A5 ) @ B5 ) ) ).
% Inter_lower
thf(fact_3946_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,A5: set @ A] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A5 ) )
= ( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X )
=> ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less @ A @ Y5 @ X4 ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_3947_Inf__le__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ X )
= ( ! [Y5: A] :
( ( ord_less @ A @ X @ Y5 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less @ A @ X4 @ Y5 ) ) ) ) ) ) ).
% Inf_le_iff
thf(fact_3948_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B5: set @ C,F3: B > A,G3: C > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A5 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B5 )
& ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( G3 @ X5 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B5 )
=> ? [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( ord_less_eq @ A @ ( G3 @ J2 ) @ ( F3 @ X5 ) ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A5 ) )
= ( complete_Sup_Sup @ A @ ( image @ C @ A @ G3 @ B5 ) ) ) ) ) ) ).
% SUP_eq
thf(fact_3949_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,U: A] :
( ! [V: A] :
( ( member @ A @ V @ A5 )
=> ( ord_less_eq @ A @ U @ V ) )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_3950_INF__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B5: set @ C,G3: C > A,F3: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A5 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B5 )
& ( ord_less_eq @ A @ ( G3 @ X5 ) @ ( F3 @ I2 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B5 )
=> ? [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G3 @ J2 ) ) ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A5 ) )
= ( complete_Inf_Inf @ A @ ( image @ C @ A @ G3 @ B5 ) ) ) ) ) ) ).
% INF_eq
thf(fact_3951_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ).
% Sup_subset_mono
thf(fact_3952_Inf__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,U: A] :
( ! [V: A] :
( ( member @ A @ V @ A5 )
=> ( ord_less_eq @ A @ V @ U ) )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ U ) ) ) ) ).
% Inf_less_eq
thf(fact_3953_Inf__superset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B5 ) ) ) ) ).
% Inf_superset_mono
thf(fact_3954_SUP__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F3: B > A,X: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I6 )
=> ( ( F3 @ I2 )
= X ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ I6 ) )
= X ) ) ) ) ).
% SUP_eq_const
thf(fact_3955_INF__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F3: B > A,X: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I6 )
=> ( ( F3 @ I2 )
= X ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ I6 ) )
= X ) ) ) ) ).
% INF_eq_const
thf(fact_3956_Union__mono,axiom,
! [A: $tType,A5: set @ ( set @ A ),B5: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A5 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B5 ) ) ) ).
% Union_mono
thf(fact_3957_Union__empty,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Union_empty
thf(fact_3958_Inter__anti__mono,axiom,
! [A: $tType,B5: set @ ( set @ A ),A5: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ B5 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A5 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B5 ) ) ) ).
% Inter_anti_mono
thf(fact_3959_Inter__subset,axiom,
! [A: $tType,A5: set @ ( set @ A ),B5: set @ A] :
( ! [X10: set @ A] :
( ( member @ ( set @ A ) @ X10 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ X10 @ B5 ) )
=> ( ( A5
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A5 ) @ B5 ) ) ) ).
% Inter_subset
thf(fact_3960_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A5: set @ B,U: A,F3: B > A] :
( ( member @ B @ I @ A5 )
=> ( ( ord_less_eq @ A @ U @ ( F3 @ I ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A5 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_3961_SUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A5: set @ B,U: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A5 ) ) @ U )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ U ) ) ) ) ) ).
% SUP_le_iff
thf(fact_3962_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A5: set @ B,F3: B > A] :
( ( member @ B @ I @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A5 ) ) ) ) ) ).
% SUP_upper
thf(fact_3963_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,G3: B > A,A5: set @ B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G3 @ A5 ) ) ) ) ) ).
% SUP_mono'
thf(fact_3964_SUP__least,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: B > A,U: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A5 ) ) @ U ) ) ) ).
% SUP_least
thf(fact_3965_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B5: set @ C,F3: B > A,G3: C > A] :
( ! [N2: B] :
( ( member @ B @ N2 @ A5 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B5 )
& ( ord_less_eq @ A @ ( F3 @ N2 ) @ ( G3 @ X5 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G3 @ B5 ) ) ) ) ) ).
% SUP_mono
thf(fact_3966_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: B > A,X: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ X ) )
=> ( ! [Y3: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I4 ) @ Y3 ) )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A5 ) )
= X ) ) ) ) ).
% SUP_eqI
thf(fact_3967_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A5: set @ B,Y: A,I: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A5 ) ) @ Y )
=> ( ( member @ B @ I @ A5 )
=> ( ord_less @ A @ ( F3 @ I ) @ Y ) ) ) ) ).
% SUP_lessD
thf(fact_3968_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A2: A,F3: B > A,A5: set @ B] :
( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A5 ) ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( ord_less @ A @ A2 @ ( F3 @ X4 ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_3969_INF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,U: A,F3: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A5 )
=> ( ord_less_eq @ A @ U @ ( F3 @ I2 ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A5 ) ) ) ) ) ).
% INF_greatest
thf(fact_3970_le__INF__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,F3: B > A,A5: set @ B] :
( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A5 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ U @ ( F3 @ X4 ) ) ) ) ) ) ).
% le_INF_iff
thf(fact_3971_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A5: set @ B,F3: B > A,U: A] :
( ( member @ B @ I @ A5 )
=> ( ( ord_less_eq @ A @ ( F3 @ I ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A5 ) ) @ U ) ) ) ) ).
% INF_lower2
thf(fact_3972_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,G3: B > A,A5: set @ B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G3 @ A5 ) ) ) ) ) ).
% INF_mono'
thf(fact_3973_INF__lower,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A5: set @ B,F3: B > A] :
( ( member @ B @ I @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A5 ) ) @ ( F3 @ I ) ) ) ) ).
% INF_lower
thf(fact_3974_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: set @ B,A5: set @ C,F3: C > A,G3: B > A] :
( ! [M3: B] :
( ( member @ B @ M3 @ B5 )
=> ? [X5: C] :
( ( member @ C @ X5 @ A5 )
& ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G3 @ M3 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G3 @ B5 ) ) ) ) ) ).
% INF_mono
thf(fact_3975_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,X: A,F3: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A5 )
=> ( ord_less_eq @ A @ X @ ( F3 @ I2 ) ) )
=> ( ! [Y3: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A5 )
=> ( ord_less_eq @ A @ Y3 @ ( F3 @ I4 ) ) )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A5 ) )
= X ) ) ) ) ).
% INF_eqI
thf(fact_3976_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y: A,F3: B > A,A5: set @ B,I: B] :
( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A5 ) ) )
=> ( ( member @ B @ I @ A5 )
=> ( ord_less @ A @ Y @ ( F3 @ I ) ) ) ) ) ).
% less_INF_D
thf(fact_3977_INF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A5: set @ B,A2: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A5 ) ) @ A2 )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( ord_less @ A @ ( F3 @ X4 ) @ A2 ) ) ) ) ) ).
% INF_less_iff
thf(fact_3978_UN__empty2,axiom,
! [B: $tType,A: $tType,A5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X4: B] : ( bot_bot @ ( set @ A ) )
@ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty2
thf(fact_3979_UN__empty,axiom,
! [B: $tType,A: $tType,B5: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_3980_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A5: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A5 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( B5 @ X4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_3981_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A5: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A5 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( B5 @ X4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_3982_UN__subset__iff,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),I6: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A5 @ I6 ) ) @ B5 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ( ord_less_eq @ ( set @ A ) @ ( A5 @ X4 ) @ B5 ) ) ) ) ).
% UN_subset_iff
thf(fact_3983_UN__upper,axiom,
! [B: $tType,A: $tType,A2: A,A5: set @ A,B5: A > ( set @ B )] :
( ( member @ A @ A2 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( B5 @ A2 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B5 @ A5 ) ) ) ) ).
% UN_upper
thf(fact_3984_UN__least,axiom,
! [A: $tType,B: $tType,A5: set @ A,B5: A > ( set @ B ),C4: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( B5 @ X3 ) @ C4 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B5 @ A5 ) ) @ C4 ) ) ).
% UN_least
thf(fact_3985_UN__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ A,F3: A > ( set @ B ),G3: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ G3 @ B5 ) ) ) ) ) ).
% UN_mono
thf(fact_3986_INT__subset__iff,axiom,
! [A: $tType,B: $tType,B5: set @ A,A5: B > ( set @ A ),I6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A5 @ I6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ ( A5 @ X4 ) ) ) ) ) ).
% INT_subset_iff
thf(fact_3987_INT__anti__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ A,F3: A > ( set @ B ),G3: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F3 @ B5 ) ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ G3 @ A5 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_3988_INT__greatest,axiom,
! [B: $tType,A: $tType,A5: set @ A,C4: set @ B,B5: A > ( set @ B )] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ C4 @ ( B5 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ C4 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B5 @ A5 ) ) ) ) ).
% INT_greatest
thf(fact_3989_INT__lower,axiom,
! [B: $tType,A: $tType,A2: A,A5: set @ A,B5: A > ( set @ B )] :
( ( member @ A @ A2 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B5 @ A5 ) ) @ ( B5 @ A2 ) ) ) ).
% INT_lower
thf(fact_3990_le__SUP__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,F3: B > A,A5: set @ B] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A5 ) ) )
= ( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X )
=> ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( ord_less @ A @ Y5 @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% le_SUP_iff
thf(fact_3991_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A5: set @ B,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A5 ) ) @ X )
= ( ! [Y5: A] :
( ( ord_less @ A @ X @ Y5 )
=> ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( ord_less @ A @ ( F3 @ X4 ) @ Y5 ) ) ) ) ) ) ).
% INF_le_iff
thf(fact_3992_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,C3: A,F3: B > A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I6 )
=> ( ord_less_eq @ A @ C3 @ ( F3 @ I2 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ I6 ) )
= C3 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ( ( F3 @ X4 )
= C3 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_3993_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F3: B > A,C3: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I6 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ C3 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ I6 ) )
= C3 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ( ( F3 @ X4 )
= C3 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_3994_Inf__le__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ).
% Inf_le_Sup
thf(fact_3995_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B5: set @ B,F3: B > A,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A5 @ B5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G3 @ B5 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_3996_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: set @ B,A5: set @ B,F3: B > A,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B5 @ A5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B5 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G3 @ B5 ) ) ) ) ) ) ).
% INF_superset_mono
thf(fact_3997_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,C3: A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y5: B] : C3
@ A5 ) )
= ( bot_bot @ A ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y5: B] : C3
@ A5 ) )
= C3 ) ) ) ) ).
% SUP_constant
thf(fact_3998_SUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% SUP_empty
thf(fact_3999_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C4: set @ B,A2: A,B5: B > ( set @ A )] :
( ( ( C4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ C4 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ C4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X4: B] : ( insert @ A @ A2 @ ( B5 @ X4 ) )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_4000_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A5 ) ) ) ) ) ).
% INF_le_SUP
thf(fact_4001_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X4: B] : ( insert @ A @ ( F3 @ X4 ) @ ( bot_bot @ ( set @ A ) ) )
@ A5 ) )
= ( image @ B @ A @ F3 @ A5 ) ) ).
% UNION_singleton_eq_range
thf(fact_4002_INT__extend__simps_I4_J,axiom,
! [G: $tType,H5: $tType,C4: set @ H5,A5: set @ G,B5: H5 > ( set @ G )] :
( ( ( C4
= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( minus_minus @ ( set @ G ) @ A5 @ ( complete_Sup_Sup @ ( set @ G ) @ ( image @ H5 @ ( set @ G ) @ B5 @ C4 ) ) )
= A5 ) )
& ( ( C4
!= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( minus_minus @ ( set @ G ) @ A5 @ ( complete_Sup_Sup @ ( set @ G ) @ ( image @ H5 @ ( set @ G ) @ B5 @ C4 ) ) )
= ( complete_Inf_Inf @ ( set @ G )
@ ( image @ H5 @ ( set @ G )
@ ^ [X4: H5] : ( minus_minus @ ( set @ G ) @ A5 @ ( B5 @ X4 ) )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_4003_ccpo__Sup__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% ccpo_Sup_singleton
thf(fact_4004_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F13: A > B > T,X4: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F13 @ X4 ) ) ) ) ).
% old.rec_prod_def
thf(fact_4005_floor__rat__def,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [X4: rat] :
( the @ int
@ ^ [Z2: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z2 ) @ X4 )
& ( ord_less @ rat @ X4 @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_4006_UN__image__subset,axiom,
! [C: $tType,A: $tType,B: $tType,F3: B > ( set @ A ),G3: C > ( set @ B ),X: C,X6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F3 @ ( G3 @ X ) ) ) @ X6 )
= ( ord_less_eq @ ( set @ B ) @ ( G3 @ X )
@ ( collect @ B
@ ^ [X4: B] : ( ord_less_eq @ ( set @ A ) @ ( F3 @ X4 ) @ X6 ) ) ) ) ).
% UN_image_subset
thf(fact_4007_Inf__filter__not__bot,axiom,
! [A: $tType,B5: set @ ( filter @ A )] :
( ! [X10: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X10 @ B5 )
=> ( ( finite_finite2 @ ( filter @ A ) @ X10 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ X10 )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ B5 )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% Inf_filter_not_bot
thf(fact_4008_less__eq__rat__def,axiom,
( ( ord_less_eq @ rat )
= ( ^ [X4: rat,Y5: rat] :
( ( ord_less @ rat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% less_eq_rat_def
thf(fact_4009_sgn__rat__def,axiom,
( ( sgn_sgn @ rat )
= ( ^ [A8: rat] :
( if @ rat
@ ( A8
= ( zero_zero @ rat ) )
@ ( zero_zero @ rat )
@ ( if @ rat @ ( ord_less @ rat @ ( zero_zero @ rat ) @ A8 ) @ ( one_one @ rat ) @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ) ) ) ).
% sgn_rat_def
thf(fact_4010_abs__rat__def,axiom,
( ( abs_abs @ rat )
= ( ^ [A8: rat] : ( if @ rat @ ( ord_less @ rat @ A8 @ ( zero_zero @ rat ) ) @ ( uminus_uminus @ rat @ A8 ) @ A8 ) ) ) ).
% abs_rat_def
thf(fact_4011_obtain__pos__sum,axiom,
! [R2: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 )
=> ~ ! [S4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ S4 )
=> ! [T7: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ T7 )
=> ( R2
!= ( plus_plus @ rat @ S4 @ T7 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_4012_conj__subset__def,axiom,
! [A: $tType,A5: set @ A,P2: A > $o,Q2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A5
@ ( collect @ A
@ ^ [X4: A] :
( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( collect @ A @ P2 ) )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( collect @ A @ Q2 ) ) ) ) ).
% conj_subset_def
thf(fact_4013_Maclaurin__sin__bound,axiom,
! [X: real,N: nat] :
( ord_less_eq @ real
@ ( abs_abs @ real
@ ( minus_minus @ real @ ( sin @ real @ X )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) )
@ ( times_times @ real @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( abs_abs @ real @ X ) @ N ) ) ) ).
% Maclaurin_sin_bound
thf(fact_4014_normalize__negative,axiom,
! [Q3: int,P3: int] :
( ( ord_less @ int @ Q3 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P3 @ Q3 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P3 ) @ ( uminus_uminus @ int @ Q3 ) ) ) ) ) ).
% normalize_negative
thf(fact_4015_fold__union__pair,axiom,
! [B: $tType,A: $tType,B5: set @ A,X: B,A5: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ B5 )
=> ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) )
@ ( image @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y5: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y5 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
@ B5 ) )
@ A5 )
= ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y5: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y5 ) )
@ A5
@ B5 ) ) ) ).
% fold_union_pair
thf(fact_4016_Cauchy__iff2,axiom,
( ( topolo3814608138187158403Cauchy @ real )
= ( ^ [X8: nat > real] :
! [J3: nat] :
? [M10: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M10 @ M5 )
=> ! [N6: nat] :
( ( ord_less_eq @ nat @ M10 @ N6 )
=> ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( X8 @ M5 ) @ ( X8 @ N6 ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_4017_Un__iff,axiom,
! [A: $tType,C3: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( ( member @ A @ C3 @ A5 )
| ( member @ A @ C3 @ B5 ) ) ) ).
% Un_iff
thf(fact_4018_UnCI,axiom,
! [A: $tType,C3: A,B5: set @ A,A5: set @ A] :
( ( ~ ( member @ A @ C3 @ B5 )
=> ( member @ A @ C3 @ A5 ) )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% UnCI
thf(fact_4019_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C3 ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_4020_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( ( ord_less_eq @ A @ X @ Z )
& ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% le_sup_iff
thf(fact_4021_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_4022_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_4023_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X @ Y ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_4024_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_4025_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A2: A,B2: A] :
( ( ( sup_sup @ A @ A2 @ B2 )
= ( bot_bot @ A ) )
= ( ( A2
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_4026_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
= A2 ) ) ).
% sup_bot.left_neutral
thf(fact_4027_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A2 @ B2 ) )
= ( ( A2
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_4028_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
= A2 ) ) ).
% sup_bot.right_neutral
thf(fact_4029_Un__empty,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_4030_finite__Un,axiom,
! [A: $tType,F5: set @ A,G6: set @ A] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F5 @ G6 ) )
= ( ( finite_finite2 @ A @ F5 )
& ( finite_finite2 @ A @ G6 ) ) ) ).
% finite_Un
thf(fact_4031_Un__subset__iff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) @ C4 )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_4032_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_4033_Un__insert__left,axiom,
! [A: $tType,A2: A,B5: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B5 ) @ C4 )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Un_insert_left
thf(fact_4034_Un__insert__right,axiom,
! [A: $tType,A5: set @ A,A2: A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ B5 ) )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% Un_insert_right
thf(fact_4035_Un__Diff__cancel,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B5 @ A5 ) )
= ( sup_sup @ ( set @ A ) @ A5 @ B5 ) ) ).
% Un_Diff_cancel
thf(fact_4036_Un__Diff__cancel2,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B5 @ A5 ) @ A5 )
= ( sup_sup @ ( set @ A ) @ B5 @ A5 ) ) ).
% Un_Diff_cancel2
thf(fact_4037_Compl__Diff__eq,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ B5 ) ) ).
% Compl_Diff_eq
thf(fact_4038_UN__simps_I3_J,axiom,
! [E: $tType,F: $tType,C4: set @ F,A5: set @ E,B5: F > ( set @ E )] :
( ( ( C4
= ( bot_bot @ ( set @ F ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image @ F @ ( set @ E )
@ ^ [X4: F] : ( sup_sup @ ( set @ E ) @ A5 @ ( B5 @ X4 ) )
@ C4 ) )
= ( bot_bot @ ( set @ E ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ F ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image @ F @ ( set @ E )
@ ^ [X4: F] : ( sup_sup @ ( set @ E ) @ A5 @ ( B5 @ X4 ) )
@ C4 ) )
= ( sup_sup @ ( set @ E ) @ A5 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image @ F @ ( set @ E ) @ B5 @ C4 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_4039_UN__simps_I2_J,axiom,
! [C: $tType,D: $tType,C4: set @ C,A5: C > ( set @ D ),B5: set @ D] :
( ( ( C4
= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A5 @ X4 ) @ B5 )
@ C4 ) )
= ( bot_bot @ ( set @ D ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A5 @ X4 ) @ B5 )
@ C4 ) )
= ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ A5 @ C4 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_4040_image__Un,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,B5: set @ B] :
( ( image @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F3 @ A5 ) @ ( image @ B @ A @ F3 @ B5 ) ) ) ).
% image_Un
thf(fact_4041_Un__mono,axiom,
! [A: $tType,A5: set @ A,C4: set @ A,B5: set @ A,D5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ D5 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) @ ( sup_sup @ ( set @ A ) @ C4 @ D5 ) ) ) ) ).
% Un_mono
thf(fact_4042_Un__least,axiom,
! [A: $tType,A5: set @ A,C4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) @ C4 ) ) ) ).
% Un_least
thf(fact_4043_Un__upper1,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) ) ).
% Un_upper1
thf(fact_4044_Un__upper2,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) ) ).
% Un_upper2
thf(fact_4045_Un__absorb1,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( sup_sup @ ( set @ A ) @ A5 @ B5 )
= B5 ) ) ).
% Un_absorb1
thf(fact_4046_Un__absorb2,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( sup_sup @ ( set @ A ) @ A5 @ B5 )
= A5 ) ) ).
% Un_absorb2
thf(fact_4047_subset__UnE,axiom,
! [A: $tType,C4: set @ A,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
=> ~ ! [A14: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A14 @ A5 )
=> ! [B12: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B12 @ B5 )
=> ( C4
!= ( sup_sup @ ( set @ A ) @ A14 @ B12 ) ) ) ) ) ).
% subset_UnE
thf(fact_4048_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ B7 )
= B7 ) ) ) ).
% subset_Un_eq
thf(fact_4049_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_4050_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ A2 )
=> ( ord_less_eq @ A @ C3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_4051_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B8: A] :
( ( sup_sup @ A @ A8 @ B8 )
= B8 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_4052_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B8: A,A8: A] :
( ( sup_sup @ A @ A8 @ B8 )
= A8 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_4053_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_4054_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_4055_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B8: A,A8: A] :
( A8
= ( sup_sup @ A @ A8 @ B8 ) ) ) ) ) ).
% sup.order_iff
thf(fact_4056_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C3 ) @ A2 ) ) ) ) ).
% sup.boundedI
thf(fact_4057_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C3 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% sup.boundedE
thf(fact_4058_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( sup_sup @ A @ X @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_4059_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( sup_sup @ A @ X @ Y )
= X ) ) ) ).
% sup_absorb1
thf(fact_4060_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_4061_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_4062_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F3: A > A > A,X: A,Y: A] :
( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ X3 @ ( F3 @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F3 @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A,Z3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ Z3 @ X3 )
=> ( ord_less_eq @ A @ ( F3 @ Y3 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup @ A @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_4063_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_4064_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_4065_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y5: A] :
( ( sup_sup @ A @ X4 @ Y5 )
= Y5 ) ) ) ) ).
% le_iff_sup
thf(fact_4066_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z ) @ X ) ) ) ) ).
% sup_least
thf(fact_4067_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,C3: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ ( sup_sup @ A @ C3 @ D2 ) ) ) ) ) ).
% sup_mono
thf(fact_4068_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A2: A,D2: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ A2 )
=> ( ( ord_less_eq @ A @ D2 @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C3 @ D2 ) @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_4069_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_4070_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X @ A2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_4071_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge2
thf(fact_4072_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge1
thf(fact_4073_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X )
=> ( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X ) ) ) ) ).
% le_supI
thf(fact_4074_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A2 @ X )
=> ~ ( ord_less_eq @ A @ B2 @ X ) ) ) ) ).
% le_supE
thf(fact_4075_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_4076_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_4077_boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% boolean_algebra.disj_zero_right
thf(fact_4078_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less @ A @ X @ A2 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% less_supI1
thf(fact_4079_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A2: A] :
( ( ord_less @ A @ X @ B2 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% less_supI2
thf(fact_4080_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_4081_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_4082_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C3 ) @ A2 )
=> ~ ( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_4083_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B8: A,A8: A] :
( ( A8
= ( sup_sup @ A @ A8 @ B8 ) )
& ( A8 != B8 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_4084_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ C3 @ A2 )
=> ( ord_less @ A @ C3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_4085_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_4086_foldl__un__empty__eq,axiom,
! [A: $tType,I: set @ A,Ww: list @ ( set @ A )] :
( ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ I @ Ww )
= ( sup_sup @ ( set @ A ) @ I @ ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) @ Ww ) ) ) ).
% foldl_un_empty_eq
thf(fact_4087_Un__empty__left,axiom,
! [A: $tType,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_4088_Un__empty__right,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= A5 ) ).
% Un_empty_right
thf(fact_4089_Un__Diff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) @ C4 )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ C4 ) @ ( minus_minus @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Un_Diff
thf(fact_4090_set__diff__diff__left,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ C4 )
= ( minus_minus @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% set_diff_diff_left
thf(fact_4091_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
| ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% Un_def
thf(fact_4092_Collect__disj__eq,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( P2 @ X4 )
| ( Q2 @ X4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_4093_Un__left__commute,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% Un_left_commute
thf(fact_4094_Un__left__absorb,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ A5 @ B5 ) ) ).
% Un_left_absorb
thf(fact_4095_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] : ( sup_sup @ ( set @ A ) @ B7 @ A6 ) ) ) ).
% Un_commute
thf(fact_4096_Un__absorb,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ A5 )
= A5 ) ).
% Un_absorb
thf(fact_4097_Un__assoc,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) @ C4 )
= ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Un_assoc
thf(fact_4098_ball__Un,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,P2: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
=> ( P2 @ X4 ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( P2 @ X4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ B5 )
=> ( P2 @ X4 ) ) ) ) ).
% ball_Un
thf(fact_4099_bex__Un,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,P2: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
& ( P2 @ X4 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P2 @ X4 ) )
| ? [X4: A] :
( ( member @ A @ X4 @ B5 )
& ( P2 @ X4 ) ) ) ) ).
% bex_Un
thf(fact_4100_UnI2,axiom,
! [A: $tType,C3: A,B5: set @ A,A5: set @ A] :
( ( member @ A @ C3 @ B5 )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% UnI2
thf(fact_4101_UnI1,axiom,
! [A: $tType,C3: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ A5 )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% UnI1
thf(fact_4102_UnE,axiom,
! [A: $tType,C3: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
=> ( ~ ( member @ A @ C3 @ A5 )
=> ( member @ A @ C3 @ B5 ) ) ) ).
% UnE
thf(fact_4103_infinite__Un,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) ) )
= ( ~ ( finite_finite2 @ A @ S3 )
| ~ ( finite_finite2 @ A @ T4 ) ) ) ).
% infinite_Un
thf(fact_4104_Un__infinite,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ~ ( finite_finite2 @ A @ S3 )
=> ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) ) ) ).
% Un_infinite
thf(fact_4105_finite__UnI,axiom,
! [A: $tType,F5: set @ A,G6: set @ A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( finite_finite2 @ A @ G6 )
=> ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F5 @ G6 ) ) ) ) ).
% finite_UnI
thf(fact_4106_insert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A8: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] : X4 = A8 ) ) ) ) ).
% insert_def
thf(fact_4107_Collect__imp__eq,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( P2 @ X4 )
=> ( Q2 @ X4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P2 ) ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_imp_eq
thf(fact_4108_sin__x__le__x,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( sin @ real @ X ) @ X ) ) ).
% sin_x_le_x
thf(fact_4109_sin__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( one_one @ real ) ) ).
% sin_le_one
thf(fact_4110_abs__sin__x__le__abs__x,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X ) ) @ ( abs_abs @ real @ X ) ) ).
% abs_sin_x_le_abs_x
thf(fact_4111_singleton__Un__iff,axiom,
! [A: $tType,X: A,A5: set @ A,B5: set @ A] :
( ( ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( ( ( A5
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_4112_Un__singleton__iff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A5 @ B5 )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A5
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_4113_insert__is__Un,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A8: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A8 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_4114_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_4115_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_4116_Diff__partition,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( sup_sup @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B5 @ A5 ) )
= B5 ) ) ).
% Diff_partition
thf(fact_4117_Diff__subset__conv,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ C4 )
= ( ord_less_eq @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Diff_subset_conv
thf(fact_4118_foldl__set,axiom,
! [A: $tType,L: list @ ( set @ A )] :
( ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) @ L )
= ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [X4: set @ A] : ( member @ ( set @ A ) @ X4 @ ( set2 @ ( set @ A ) @ L ) ) ) ) ) ).
% foldl_set
thf(fact_4119_finite__Sup__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( member @ A @ ( sup_sup @ A @ X3 @ Y3 ) @ A5 ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A5 ) @ A5 ) ) ) ) ) ).
% finite_Sup_in
thf(fact_4120_sin__x__ge__neg__x,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( uminus_uminus @ real @ X ) @ ( sin @ real @ X ) ) ) ).
% sin_x_ge_neg_x
thf(fact_4121_sin__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( sin @ real @ X ) ) ).
% sin_ge_minus_one
thf(fact_4122_abs__sin__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X ) ) @ ( one_one @ real ) ) ).
% abs_sin_le_one
thf(fact_4123_sup__Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B5: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A5 ) @ B5 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ B5 @ A5 ) ) ) ) ).
% sup_Sup_fold_sup
thf(fact_4124_card__Un__le,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ).
% card_Un_le
thf(fact_4125_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_4126_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_4127_Union__image__empty,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: B > ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= A5 ) ).
% Union_image_empty
thf(fact_4128_union__fold__insert,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( sup_sup @ ( set @ A ) @ A5 @ B5 )
= ( finite_fold @ A @ ( set @ A ) @ ( insert @ A ) @ B5 @ A5 ) ) ) ).
% union_fold_insert
thf(fact_4129_UN__extend__simps_I3_J,axiom,
! [E: $tType,F: $tType,C4: set @ F,A5: set @ E,B5: F > ( set @ E )] :
( ( ( C4
= ( bot_bot @ ( set @ F ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A5 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image @ F @ ( set @ E ) @ B5 @ C4 ) ) )
= A5 ) )
& ( ( C4
!= ( bot_bot @ ( set @ F ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A5 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image @ F @ ( set @ E ) @ B5 @ C4 ) ) )
= ( complete_Sup_Sup @ ( set @ E )
@ ( image @ F @ ( set @ E )
@ ^ [X4: F] : ( sup_sup @ ( set @ E ) @ A5 @ ( B5 @ X4 ) )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_4130_UN__extend__simps_I2_J,axiom,
! [D: $tType,C: $tType,C4: set @ C,A5: C > ( set @ D ),B5: set @ D] :
( ( ( C4
= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ A5 @ C4 ) ) @ B5 )
= B5 ) )
& ( ( C4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ A5 @ C4 ) ) @ B5 )
= ( complete_Sup_Sup @ ( set @ D )
@ ( image @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A5 @ X4 ) @ B5 )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_4131_Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( complete_Sup_Sup @ A @ A5 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) @ A5 ) ) ) ) ).
% Sup_fold_sup
thf(fact_4132_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_4133_ivl__disj__un__singleton_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [U: A] :
( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ U ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_ord_atMost @ A @ U ) ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_4134_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_4135_Max_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ ( lattic643756798349783984er_Max @ A @ B5 ) ) ) ) ) ) ) ) ).
% Max.union
thf(fact_4136_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X4: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X4 )
@ ^ [X4: A,Y5: A] : ( ord_less @ A @ Y5 @ X4 ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_4137_normalize__denom__pos,axiom,
! [R2: product_prod @ int @ int,P3: int,Q3: int] :
( ( ( normalize @ R2 )
= ( product_Pair @ int @ int @ P3 @ Q3 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q3 ) ) ).
% normalize_denom_pos
thf(fact_4138_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_4139_UNION__fun__upd,axiom,
! [B: $tType,A: $tType,A5: B > ( set @ A ),I: B,B5: set @ A,J4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ ( fun_upd @ B @ ( set @ A ) @ A5 @ I @ B5 ) @ J4 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ B ) @ J4 @ ( insert @ B @ I @ ( bot_bot @ ( set @ B ) ) ) ) ) ) @ ( if @ ( set @ A ) @ ( member @ B @ I @ J4 ) @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% UNION_fun_upd
thf(fact_4140_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_4141_CauchyD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A,E3: real] :
( ( topolo3814608138187158403Cauchy @ A @ X6 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M11: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M11 @ M2 )
=> ! [N9: nat] :
( ( ord_less_eq @ nat @ M11 @ N9 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ M2 ) @ ( X6 @ N9 ) ) ) @ E3 ) ) ) ) ) ) ).
% CauchyD
thf(fact_4142_CauchyI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M12: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M12 @ M3 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M12 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X6 ) ) ) ).
% CauchyI
thf(fact_4143_Cauchy__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M10: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M10 @ M5 )
=> ! [N6: nat] :
( ( ord_less_eq @ nat @ M10 @ N6 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M5 ) @ ( X8 @ N6 ) ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_4144_SUP__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: A,B5: A] :
( ( sup_sup @ A @ A5
@ ( complete_Sup_Sup @ A
@ ( image @ nat @ A
@ ^ [X4: nat] : B5
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( sup_sup @ A @ A5 @ B5 ) ) ) ).
% SUP_nat_binary
thf(fact_4145_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F3: nat > ( set @ A ),S3: set @ A] :
( ! [I2: nat] : ( ord_less_eq @ ( set @ A ) @ ( F3 @ I2 ) @ S3 )
=> ( ( finite_finite2 @ A @ S3 )
=> ( ? [N10: nat] :
( ! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ N10 )
=> ! [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N10 )
=> ( ( ord_less @ nat @ M3 @ N2 )
=> ( ord_less @ ( set @ A ) @ ( F3 @ M3 ) @ ( F3 @ N2 ) ) ) ) )
& ! [N2: nat] :
( ( ord_less_eq @ nat @ N10 @ N2 )
=> ( ( F3 @ N10 )
= ( F3 @ N2 ) ) ) )
=> ( ( F3 @ ( finite_card @ A @ S3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ F3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_4146_card__partition,axiom,
! [A: $tType,C4: set @ ( set @ A ),K: nat] :
( ( finite_finite2 @ ( set @ A ) @ C4 )
=> ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) )
=> ( ! [C2: set @ A] :
( ( member @ ( set @ A ) @ C2 @ C4 )
=> ( ( finite_card @ A @ C2 )
= K ) )
=> ( ! [C1: set @ A,C22: set @ A] :
( ( member @ ( set @ A ) @ C1 @ C4 )
=> ( ( member @ ( set @ A ) @ C22 @ C4 )
=> ( ( C1 != C22 )
=> ( ( inf_inf @ ( set @ A ) @ C1 @ C22 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( times_times @ nat @ K @ ( finite_card @ ( set @ A ) @ C4 ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_4147_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
thf(fact_4148_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_4149_IntI,axiom,
! [A: $tType,C3: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ A5 )
=> ( ( member @ A @ C3 @ B5 )
=> ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% IntI
thf(fact_4150_Int__iff,axiom,
! [A: $tType,C3: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) )
= ( ( member @ A @ C3 @ A5 )
& ( member @ A @ C3 @ B5 ) ) ) ).
% Int_iff
thf(fact_4151_mod__or__dist,axiom,
! [P2: assn,Q2: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( sup_sup @ assn @ P2 @ Q2 ) @ H2 )
= ( ( rep_assn @ P2 @ H2 )
| ( rep_assn @ Q2 @ H2 ) ) ) ).
% mod_or_dist
thf(fact_4152_merge__pure__or,axiom,
! [A2: $o,B2: $o] :
( ( sup_sup @ assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B2 ) )
= ( pure_assn
@ ( A2
| B2 ) ) ) ).
% merge_pure_or
thf(fact_4153_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C3 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% inf.bounded_iff
thf(fact_4154_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% le_inf_iff
thf(fact_4155_finite__option__UNIV,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_option_UNIV
thf(fact_4156_finite__Plus__UNIV__iff,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
& ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_4157_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_4158_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_4159_boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_right
thf(fact_4160_boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_left
thf(fact_4161_Int__UNIV,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= ( top_top @ ( set @ A ) ) )
= ( ( A5
= ( top_top @ ( set @ A ) ) )
& ( B5
= ( top_top @ ( set @ A ) ) ) ) ) ).
% Int_UNIV
thf(fact_4162_finite__Int,axiom,
! [A: $tType,F5: set @ A,G6: set @ A] :
( ( ( finite_finite2 @ A @ F5 )
| ( finite_finite2 @ A @ G6 ) )
=> ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F5 @ G6 ) ) ) ).
% finite_Int
thf(fact_4163_card__eq__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) )
= ( S3
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV
thf(fact_4164_card__eq__UNIV2,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S3: set @ A] :
( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
= ( finite_card @ A @ S3 ) )
= ( S3
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV2
thf(fact_4165_Int__subset__iff,axiom,
! [A: $tType,C4: set @ A,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C4 @ A5 )
& ( ord_less_eq @ ( set @ A ) @ C4 @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_4166_Int__insert__right__if1,axiom,
! [A: $tType,A2: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ B5 ) )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_4167_Int__insert__right__if0,axiom,
! [A: $tType,A2: A,A5: set @ A,B5: set @ A] :
( ~ ( member @ A @ A2 @ A5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_4168_insert__inter__insert,axiom,
! [A: $tType,A2: A,A5: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A5 ) @ ( insert @ A @ A2 @ B5 ) )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% insert_inter_insert
thf(fact_4169_Int__insert__left__if1,axiom,
! [A: $tType,A2: A,C4: set @ A,B5: set @ A] :
( ( member @ A @ A2 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B5 ) @ C4 )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_4170_Int__insert__left__if0,axiom,
! [A: $tType,A2: A,C4: set @ A,B5: set @ A] :
( ~ ( member @ A @ A2 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B5 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_4171_Un__Int__eq_I1_J,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_4172_Un__Int__eq_I2_J,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) @ T4 )
= T4 ) ).
% Un_Int_eq(2)
thf(fact_4173_Un__Int__eq_I3_J,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( inf_inf @ ( set @ A ) @ S3 @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_4174_Un__Int__eq_I4_J,axiom,
! [A: $tType,T4: set @ A,S3: set @ A] :
( ( inf_inf @ ( set @ A ) @ T4 @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) )
= T4 ) ).
% Un_Int_eq(4)
thf(fact_4175_Int__Un__eq_I1_J,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S3 @ T4 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_4176_Int__Un__eq_I2_J,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S3 @ T4 ) @ T4 )
= T4 ) ).
% Int_Un_eq(2)
thf(fact_4177_Int__Un__eq_I3_J,axiom,
! [A: $tType,S3: set @ A,T4: set @ A] :
( ( sup_sup @ ( set @ A ) @ S3 @ ( inf_inf @ ( set @ A ) @ S3 @ T4 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_4178_Int__Un__eq_I4_J,axiom,
! [A: $tType,T4: set @ A,S3: set @ A] :
( ( sup_sup @ ( set @ A ) @ T4 @ ( inf_inf @ ( set @ A ) @ S3 @ T4 ) )
= T4 ) ).
% Int_Un_eq(4)
thf(fact_4179_restrict__map__UNIV,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B )] :
( ( restrict_map @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
= F3 ) ).
% restrict_map_UNIV
thf(fact_4180_Collect__const,axiom,
! [A: $tType,P2: $o] :
( ( P2
=> ( ( collect @ A
@ ^ [S8: A] : P2 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P2
=> ( ( collect @ A
@ ^ [S8: A] : P2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_4181_finite__Collect__not,axiom,
! [A: $tType,P2: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
~ ( P2 @ X4 ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_Collect_not
thf(fact_4182_boolean__algebra_Ocompl__zero,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.compl_zero
thf(fact_4183_boolean__algebra_Ocompl__one,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.compl_one
thf(fact_4184_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A] :
( ( ( complete_Sup_Sup @ A @ A5 )
= ( top_top @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ X4 @ ( top_top @ A ) )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A5 )
& ( ord_less @ A @ X4 @ Y5 ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_4185_boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( uminus_uminus @ A @ X ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_right
thf(fact_4186_boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ X )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_left
thf(fact_4187_inf__compl__bot__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ ( uminus_uminus @ A @ X ) ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_right
thf(fact_4188_inf__compl__bot__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left2
thf(fact_4189_inf__compl__bot__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( inf_inf @ A @ X @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left1
thf(fact_4190_Inf__UNIV,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Inf_UNIV
thf(fact_4191_ccInf__empty,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% ccInf_empty
thf(fact_4192_Inf__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Inf_empty
thf(fact_4193_Diff__UNIV,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_4194_insert__disjoint_I1_J,axiom,
! [A: $tType,A2: A,A5: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A5 ) @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A2 @ B5 )
& ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_disjoint(1)
thf(fact_4195_insert__disjoint_I2_J,axiom,
! [A: $tType,A2: A,A5: set @ A,B5: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A5 ) @ B5 ) )
= ( ~ ( member @ A @ A2 @ B5 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_4196_disjoint__insert_I1_J,axiom,
! [A: $tType,B5: set @ A,A2: A,A5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ B5 @ ( insert @ A @ A2 @ A5 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A2 @ B5 )
& ( ( inf_inf @ ( set @ A ) @ B5 @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjoint_insert(1)
thf(fact_4197_disjoint__insert_I2_J,axiom,
! [A: $tType,A5: set @ A,B2: A,B5: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ B5 ) ) )
= ( ~ ( member @ A @ B2 @ A5 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_4198_card__ge__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S3: set @ A] :
( ( ord_less_eq @ nat @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) @ ( finite_card @ A @ S3 ) )
= ( S3
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_ge_UNIV
thf(fact_4199_Diff__disjoint,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B5 @ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_disjoint
thf(fact_4200_surj__fn,axiom,
! [A: $tType,F3: A > A,N: nat] :
( ( ( image @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_fn
thf(fact_4201_finite__compl,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_compl
thf(fact_4202_Compl__disjoint,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint
thf(fact_4203_Compl__disjoint2,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint2
thf(fact_4204_inter__compl__diff__conv,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ B5 ) )
= ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ).
% inter_compl_diff_conv
thf(fact_4205_Diff__Compl,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ B5 ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ).
% Diff_Compl
thf(fact_4206_mod__h__bot__iff_I7_J,axiom,
! [P2: assn,Q2: assn,H2: heap_ext @ product_unit] :
( ( rep_assn @ ( sup_sup @ assn @ P2 @ Q2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
| ( rep_assn @ Q2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(7)
thf(fact_4207_if__image__distrib,axiom,
! [A: $tType,B: $tType,P2: B > $o,F3: B > A,G3: B > A,S3: set @ B] :
( ( image @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P2 @ X4 ) @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ S3 )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ S3 @ ( collect @ B @ P2 ) ) )
@ ( image @ B @ A @ G3
@ ( inf_inf @ ( set @ B ) @ S3
@ ( collect @ B
@ ^ [X4: B] :
~ ( P2 @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_4208_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A5: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A5 ) )
= ( top_top @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ X4 @ ( top_top @ A ) )
=> ? [Y5: B] :
( ( member @ B @ Y5 @ A5 )
& ( ord_less @ A @ X4 @ ( F3 @ Y5 ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_4209_range__constant,axiom,
! [B: $tType,A: $tType,X: A] :
( ( image @ B @ A
@ ^ [Uu3: B] : X
@ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_4210_ccINF__empty,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F3: B > A] :
( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% ccINF_empty
thf(fact_4211_INT__constant,axiom,
! [B: $tType,A: $tType,A5: set @ B,C3: set @ A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y5: B] : C3
@ A5 ) )
= ( top_top @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y5: B] : C3
@ A5 ) )
= C3 ) ) ) ).
% INT_constant
thf(fact_4212_Inf__atMostLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( top_top @ A ) @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X ) )
= ( bot_bot @ A ) ) ) ) ).
% Inf_atMostLessThan
thf(fact_4213_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self1_no_field
thf(fact_4214_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C4: set @ D,A5: set @ C,B5: D > ( set @ C )] :
( ( ( C4
= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A5 @ ( B5 @ X4 ) )
@ C4 ) )
= ( top_top @ ( set @ C ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A5 @ ( B5 @ X4 ) )
@ C4 ) )
= ( inf_inf @ ( set @ C ) @ A5 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image @ D @ ( set @ C ) @ B5 @ C4 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_4215_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C4: set @ A,A5: A > ( set @ B ),B5: set @ B] :
( ( ( C4
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ X4 ) @ B5 )
@ C4 ) )
= ( top_top @ ( set @ B ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ X4 ) @ B5 )
@ C4 ) )
= ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A5 @ C4 ) ) @ B5 ) ) ) ) ).
% INT_simps(1)
thf(fact_4216_INT__simps_I3_J,axiom,
! [E: $tType,F: $tType,C4: set @ E,A5: E > ( set @ F ),B5: set @ F] :
( ( ( C4
= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E @ ( set @ F )
@ ^ [X4: E] : ( minus_minus @ ( set @ F ) @ ( A5 @ X4 ) @ B5 )
@ C4 ) )
= ( top_top @ ( set @ F ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E @ ( set @ F )
@ ^ [X4: E] : ( minus_minus @ ( set @ F ) @ ( A5 @ X4 ) @ B5 )
@ C4 ) )
= ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E @ ( set @ F ) @ A5 @ C4 ) ) @ B5 ) ) ) ) ).
% INT_simps(3)
thf(fact_4217_INT__simps_I4_J,axiom,
! [G: $tType,H5: $tType,C4: set @ H5,A5: set @ G,B5: H5 > ( set @ G )] :
( ( ( C4
= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G )
@ ( image @ H5 @ ( set @ G )
@ ^ [X4: H5] : ( minus_minus @ ( set @ G ) @ A5 @ ( B5 @ X4 ) )
@ C4 ) )
= ( top_top @ ( set @ G ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G )
@ ( image @ H5 @ ( set @ G )
@ ^ [X4: H5] : ( minus_minus @ ( set @ G ) @ A5 @ ( B5 @ X4 ) )
@ C4 ) )
= ( minus_minus @ ( set @ G ) @ A5 @ ( complete_Sup_Sup @ ( set @ G ) @ ( image @ H5 @ ( set @ G ) @ B5 @ C4 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_4218_Sup__inter__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B5: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ).
% Sup_inter_less_eq
thf(fact_4219_Union__Int__subset,axiom,
! [A: $tType,A5: set @ ( set @ A ),B5: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A5 @ B5 ) ) @ ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B5 ) ) ) ).
% Union_Int_subset
thf(fact_4220_subset__UNIV,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_4221_inter__eq__subsetI,axiom,
! [A: $tType,S3: set @ A,S7: set @ A,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ S7 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ S7 )
= ( inf_inf @ ( set @ A ) @ B5 @ S7 ) )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ S3 )
= ( inf_inf @ ( set @ A ) @ B5 @ S3 ) ) ) ) ).
% inter_eq_subsetI
thf(fact_4222_Int__mono,axiom,
! [A: $tType,A5: set @ A,C4: set @ A,B5: set @ A,D5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ D5 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) @ ( inf_inf @ ( set @ A ) @ C4 @ D5 ) ) ) ) ).
% Int_mono
thf(fact_4223_Int__lower1,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) @ A5 ) ).
% Int_lower1
thf(fact_4224_Int__lower2,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) @ B5 ) ).
% Int_lower2
thf(fact_4225_Int__absorb1,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= B5 ) ) ).
% Int_absorb1
thf(fact_4226_Int__absorb2,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= A5 ) ) ).
% Int_absorb2
thf(fact_4227_Int__greatest,axiom,
! [A: $tType,C4: set @ A,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ C4 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ C4 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% Int_greatest
thf(fact_4228_Int__Collect__mono,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,P2: A > $o,Q2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( P2 @ X3 )
=> ( Q2 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ ( collect @ A @ P2 ) ) @ ( inf_inf @ ( set @ A ) @ B5 @ ( collect @ A @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_4229_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_4230_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_4231_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_4232_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% inf.coboundedI2
thf(fact_4233_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% inf.coboundedI1
thf(fact_4234_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B8: A,A8: A] :
( ( inf_inf @ A @ A8 @ B8 )
= B8 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_4235_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B8: A] :
( ( inf_inf @ A @ A8 @ B8 )
= A8 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_4236_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_4237_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_4238_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B8: A] :
( A8
= ( inf_inf @ A @ A8 @ B8 ) ) ) ) ) ).
% inf.order_iff
thf(fact_4239_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Z )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ) ).
% inf_greatest
thf(fact_4240_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C3 ) ) ) ) ) ).
% inf.boundedI
thf(fact_4241_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C3 ) )
=> ~ ( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% inf.boundedE
thf(fact_4242_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( inf_inf @ A @ X @ Y )
= Y ) ) ) ).
% inf_absorb2
thf(fact_4243_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( inf_inf @ A @ X @ Y )
= X ) ) ) ).
% inf_absorb1
thf(fact_4244_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_4245_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_4246_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y5: A] :
( ( inf_inf @ A @ X4 @ Y5 )
= X4 ) ) ) ) ).
% le_iff_inf
thf(fact_4247_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F3: A > A > A,X: A,Y: A] :
( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( F3 @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( F3 @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: A,Y3: A,Z3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Z3 )
=> ( ord_less_eq @ A @ X3 @ ( F3 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf @ A @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ) ).
% inf_unique
thf(fact_4248_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_4249_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_4250_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X ) ) ) ).
% le_infI2
thf(fact_4251_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X ) ) ) ).
% le_infI1
thf(fact_4252_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C3: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ ( inf_inf @ A @ C3 @ D2 ) ) ) ) ) ).
% inf_mono
thf(fact_4253_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X @ A2 )
=> ( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% le_infI
thf(fact_4254_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq @ A @ X @ A2 )
=> ~ ( ord_less_eq @ A @ X @ B2 ) ) ) ) ).
% le_infE
thf(fact_4255_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% inf_le2
thf(fact_4256_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% inf_le1
thf(fact_4257_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% inf_sup_ord(1)
thf(fact_4258_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% inf_sup_ord(2)
thf(fact_4259_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ X @ Y )
= ( top_top @ A ) )
=> ( ( uminus_uminus @ A @ X )
= Y ) ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_4260_boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [A2: A,X: A,Y: A] :
( ( ( inf_inf @ A @ A2 @ X )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A2 @ X )
= ( top_top @ A ) )
=> ( ( ( inf_inf @ A @ A2 @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A2 @ Y )
= ( top_top @ A ) )
=> ( X = Y ) ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_4261_Inf__fold__inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( complete_Inf_Inf @ A @ A5 )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ A5 ) ) ) ) ).
% Inf_fold_inf
thf(fact_4262_sup__nat__def,axiom,
( ( sup_sup @ nat )
= ( ord_max @ nat ) ) ).
% sup_nat_def
thf(fact_4263_infinite__UNIV__char__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% infinite_UNIV_char_0
thf(fact_4264_ex__new__if__finite,axiom,
! [A: $tType,A5: set @ A] :
( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ A5 )
=> ? [A4: A] :
~ ( member @ A @ A4 @ A5 ) ) ) ).
% ex_new_if_finite
thf(fact_4265_finite__class_Ofinite__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_class.finite_UNIV
thf(fact_4266_ent__disjE,axiom,
! [A5: assn,C4: assn,B5: assn] :
( ( entails @ A5 @ C4 )
=> ( ( entails @ B5 @ C4 )
=> ( entails @ ( sup_sup @ assn @ A5 @ B5 ) @ C4 ) ) ) ).
% ent_disjE
thf(fact_4267_ent__disjI1,axiom,
! [P2: assn,Q2: assn,R3: assn] :
( ( entails @ ( sup_sup @ assn @ P2 @ Q2 ) @ R3 )
=> ( entails @ P2 @ R3 ) ) ).
% ent_disjI1
thf(fact_4268_ent__disjI2,axiom,
! [P2: assn,Q2: assn,R3: assn] :
( ( entails @ ( sup_sup @ assn @ P2 @ Q2 ) @ R3 )
=> ( entails @ Q2 @ R3 ) ) ).
% ent_disjI2
thf(fact_4269_ent__disjI1_H,axiom,
! [A5: assn,B5: assn,C4: assn] :
( ( entails @ A5 @ B5 )
=> ( entails @ A5 @ ( sup_sup @ assn @ B5 @ C4 ) ) ) ).
% ent_disjI1'
thf(fact_4270_ent__disjI2_H,axiom,
! [A5: assn,C4: assn,B5: assn] :
( ( entails @ A5 @ C4 )
=> ( entails @ A5 @ ( sup_sup @ assn @ B5 @ C4 ) ) ) ).
% ent_disjI2'
thf(fact_4271_ent__disjI1__direct,axiom,
! [A5: assn,B5: assn] : ( entails @ A5 @ ( sup_sup @ assn @ A5 @ B5 ) ) ).
% ent_disjI1_direct
thf(fact_4272_ent__disjI2__direct,axiom,
! [B5: assn,A5: assn] : ( entails @ B5 @ ( sup_sup @ assn @ A5 @ B5 ) ) ).
% ent_disjI2_direct
thf(fact_4273_star__or__dist1,axiom,
! [A5: assn,B5: assn,C4: assn] :
( ( times_times @ assn @ ( sup_sup @ assn @ A5 @ B5 ) @ C4 )
= ( sup_sup @ assn @ ( times_times @ assn @ A5 @ C4 ) @ ( times_times @ assn @ B5 @ C4 ) ) ) ).
% star_or_dist1
thf(fact_4274_star__or__dist2,axiom,
! [C4: assn,A5: assn,B5: assn] :
( ( times_times @ assn @ C4 @ ( sup_sup @ assn @ A5 @ B5 ) )
= ( sup_sup @ assn @ ( times_times @ assn @ C4 @ A5 ) @ ( times_times @ assn @ C4 @ B5 ) ) ) ).
% star_or_dist2
thf(fact_4275_Finite__Set_Ofinite__set,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% Finite_Set.finite_set
thf(fact_4276_finite__prod,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
& ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_prod
thf(fact_4277_finite__Prod__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% finite_Prod_UNIV
thf(fact_4278_finite__fun__UNIVD2,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
=> ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ).
% finite_fun_UNIVD2
thf(fact_4279_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A6 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% sup_set_def
thf(fact_4280_Un__Int__distrib2,axiom,
! [A: $tType,B5: set @ A,C4: set @ A,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) @ A5 )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B5 @ A5 ) @ ( sup_sup @ ( set @ A ) @ C4 @ A5 ) ) ) ).
% Un_Int_distrib2
thf(fact_4281_Int__Un__distrib2,axiom,
! [A: $tType,B5: set @ A,C4: set @ A,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) @ A5 )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B5 @ A5 ) @ ( inf_inf @ ( set @ A ) @ C4 @ A5 ) ) ) ).
% Int_Un_distrib2
thf(fact_4282_Un__Int__distrib,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) @ ( sup_sup @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% Un_Int_distrib
thf(fact_4283_Int__Un__distrib,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) @ ( inf_inf @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% Int_Un_distrib
thf(fact_4284_Un__UNIV__right,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_4285_Un__UNIV__left,axiom,
! [A: $tType,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B5 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_4286_Un__Int__crazy,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) ) @ ( inf_inf @ ( set @ A ) @ C4 @ A5 ) )
= ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) ) @ ( sup_sup @ ( set @ A ) @ C4 @ A5 ) ) ) ).
% Un_Int_crazy
thf(fact_4287_infinite__UNIV__nat,axiom,
~ ( finite_finite2 @ nat @ ( top_top @ ( set @ nat ) ) ) ).
% infinite_UNIV_nat
thf(fact_4288_nat__not__finite,axiom,
~ ( finite_finite2 @ nat @ ( top_top @ ( set @ nat ) ) ) ).
% nat_not_finite
thf(fact_4289_insert__UNIV,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_4290_IntE,axiom,
! [A: $tType,C3: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) )
=> ~ ( ( member @ A @ C3 @ A5 )
=> ~ ( member @ A @ C3 @ B5 ) ) ) ).
% IntE
thf(fact_4291_IntD1,axiom,
! [A: $tType,C3: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) )
=> ( member @ A @ C3 @ A5 ) ) ).
% IntD1
thf(fact_4292_IntD2,axiom,
! [A: $tType,C3: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) )
=> ( member @ A @ C3 @ B5 ) ) ).
% IntD2
thf(fact_4293_Int__assoc,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ A5 @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Int_assoc
thf(fact_4294_UNIV__eq__I,axiom,
! [A: $tType,A5: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A5 )
=> ( ( top_top @ ( set @ A ) )
= A5 ) ) ).
% UNIV_eq_I
thf(fact_4295_Int__absorb,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ A5 )
= A5 ) ).
% Int_absorb
thf(fact_4296_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] : ( inf_inf @ ( set @ A ) @ B7 @ A6 ) ) ) ).
% Int_commute
thf(fact_4297_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_4298_Int__UNIV__left,axiom,
! [A: $tType,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B5 )
= B5 ) ).
% Int_UNIV_left
thf(fact_4299_Int__UNIV__right,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) )
= A5 ) ).
% Int_UNIV_right
thf(fact_4300_Int__left__absorb,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ).
% Int_left_absorb
thf(fact_4301_Int__left__commute,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) )
= ( inf_inf @ ( set @ A ) @ B5 @ ( inf_inf @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% Int_left_commute
thf(fact_4302_Collect__conj__eq,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( P2 @ X4 )
& ( Q2 @ X4 ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_4303_Int__Collect,axiom,
! [A: $tType,X: A,A5: set @ A,P2: A > $o] :
( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ A5 @ ( collect @ A @ P2 ) ) )
= ( ( member @ A @ X @ A5 )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_4304_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $true ) ) ).
% UNIV_def
thf(fact_4305_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% Int_def
thf(fact_4306_Int__insert__right,axiom,
! [A: $tType,A2: A,A5: set @ A,B5: set @ A] :
( ( ( member @ A @ A2 @ A5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ B5 ) )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) )
& ( ~ ( member @ A @ A2 @ A5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_4307_Int__insert__left,axiom,
! [A: $tType,A2: A,C4: set @ A,B5: set @ A] :
( ( ( member @ A @ A2 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B5 ) @ C4 )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) ) ) )
& ( ~ ( member @ A @ A2 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B5 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ B5 @ C4 ) ) ) ) ).
% Int_insert_left
thf(fact_4308_Diff__Int__distrib2,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ C4 )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ C4 ) @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Diff_Int_distrib2
thf(fact_4309_Diff__Int__distrib,axiom,
! [A: $tType,C4: set @ A,A5: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ C4 @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C4 @ A5 ) @ ( inf_inf @ ( set @ A ) @ C4 @ B5 ) ) ) ).
% Diff_Int_distrib
thf(fact_4310_Diff__Diff__Int,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ).
% Diff_Diff_Int
thf(fact_4311_Diff__Int2,axiom,
! [A: $tType,A5: set @ A,C4: set @ A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ C4 ) @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ C4 ) @ B5 ) ) ).
% Diff_Int2
thf(fact_4312_Int__Diff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Int_Diff
thf(fact_4313_atLeastAtMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( bounded_lattice @ A )
=> ! [X: A,Y: A] :
( ( ( set_or1337092689740270186AtMost @ A @ X @ Y )
= ( top_top @ ( set @ A ) ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_4314_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% inf.strict_coboundedI2
thf(fact_4315_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ A2 @ C3 )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% inf.strict_coboundedI1
thf(fact_4316_inf_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less @ A )
= ( ^ [A8: A,B8: A] :
( ( A8
= ( inf_inf @ A @ A8 @ B8 ) )
& ( A8 != B8 ) ) ) ) ) ).
% inf.strict_order_iff
thf(fact_4317_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ ( inf_inf @ A @ B2 @ C3 ) )
=> ~ ( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% inf.strict_boundedE
thf(fact_4318_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_4319_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_4320_less__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A2: A] :
( ( ord_less @ A @ B2 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X ) ) ) ).
% less_infI2
thf(fact_4321_less__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less @ A @ A2 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X ) ) ) ).
% less_infI1
thf(fact_4322_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_4323_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ).
% top.extremum_strict
thf(fact_4324_disjointI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ~ ( member @ A @ X3 @ B2 ) )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjointI
thf(fact_4325_disjoint__iff__not__equal,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ B5 )
=> ( X4 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_4326_Int__empty__right,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_4327_Int__empty__left,axiom,
! [A: $tType,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_4328_disjoint__iff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ~ ( member @ A @ X4 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_4329_Int__emptyI,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ~ ( member @ A @ X3 @ B5 ) )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_4330_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_4331_rangeI,axiom,
! [A: $tType,B: $tType,F3: B > A,X: B] : ( member @ A @ ( F3 @ X ) @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_4332_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A,X: B] :
( ( B2
= ( F3 @ X ) )
=> ( member @ A @ B2 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_4333_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_4334_rangeE,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A] :
( ( member @ A @ B2 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X3: B] :
( B2
!= ( F3 @ X3 ) ) ) ).
% rangeE
thf(fact_4335_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,G3: B > C] :
( ( image @ B @ A
@ ^ [X4: B] : ( F3 @ ( G3 @ X4 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image @ C @ A @ F3 @ ( image @ B @ C @ G3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_4336_ex__distrib__or,axiom,
! [A: $tType,P2: A > assn,Q2: assn] :
( ( ex_assn @ A
@ ^ [X4: A] : ( sup_sup @ assn @ ( P2 @ X4 ) @ Q2 ) )
= ( sup_sup @ assn @ ( ex_assn @ A @ P2 ) @ Q2 ) ) ).
% ex_distrib_or
thf(fact_4337_ex__join__or,axiom,
! [A: $tType,P2: A > assn,Q2: A > assn] :
( ( ex_assn @ A
@ ^ [X4: A] : ( sup_sup @ assn @ ( P2 @ X4 ) @ ( ex_assn @ A @ Q2 ) ) )
= ( ex_assn @ A
@ ^ [X4: A] : ( sup_sup @ assn @ ( P2 @ X4 ) @ ( Q2 @ X4 ) ) ) ) ).
% ex_join_or
thf(fact_4338_distrib__sup__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% distrib_sup_le
thf(fact_4339_distrib__inf__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) @ ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% distrib_inf_le
thf(fact_4340_inf__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A2: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ A2 ) @ ( inf_inf @ A @ X @ B2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left2
thf(fact_4341_inf__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A2: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ A2 ) @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ B2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left1
thf(fact_4342_finite__fun__UNIVD1,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
=> ( ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
!= ( suc @ ( zero_zero @ nat ) ) )
=> ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_fun_UNIVD1
thf(fact_4343_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B5: set @ A,A2: A] :
( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B5 ) @ A2 )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B5 )
=> ( ( inf_inf @ A @ X4 @ A2 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_4344_mod__star__conv,axiom,
! [A5: assn,B5: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A5 @ B5 ) @ H2 )
= ( ? [Hr: heap_ext @ product_unit,As1: set @ nat,As22: set @ nat] :
( ( H2
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( rep_assn @ A5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As1 ) )
& ( rep_assn @ B5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As22 ) ) ) ) ) ).
% mod_star_conv
thf(fact_4345_star__assnI,axiom,
! [P2: assn,H2: heap_ext @ product_unit,As3: set @ nat,Q2: assn,As4: set @ nat] :
( ( rep_assn @ P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As3 ) )
=> ( ( rep_assn @ Q2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As4 ) )
=> ( ( ( inf_inf @ ( set @ nat ) @ As3 @ As4 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( rep_assn @ ( times_times @ assn @ P2 @ Q2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( sup_sup @ ( set @ nat ) @ As3 @ As4 ) ) ) ) ) ) ).
% star_assnI
thf(fact_4346_image__Int__subset,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,B5: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) ) @ ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ F3 @ A5 ) @ ( image @ B @ A @ F3 @ B5 ) ) ) ).
% image_Int_subset
thf(fact_4347_disjoint__mono,axiom,
! [A: $tType,A2: set @ A,A3: set @ A,B2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ B3 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% disjoint_mono
thf(fact_4348_disjoint__alt__simp1,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A5 @ B5 )
= A5 )
= ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp1
thf(fact_4349_disjoint__alt__simp2,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A5 @ B5 )
!= A5 )
= ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp2
thf(fact_4350_Int__Diff__disjoint,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_Diff_disjoint
thf(fact_4351_Diff__triv,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( minus_minus @ ( set @ A ) @ A5 @ B5 )
= A5 ) ) ).
% Diff_triv
thf(fact_4352_insert__partition,axiom,
! [A: $tType,X: set @ A,F5: set @ ( set @ A )] :
( ~ ( member @ ( set @ A ) @ X @ F5 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ ( insert @ ( set @ A ) @ X @ F5 ) )
=> ! [Xa4: set @ A] :
( ( member @ ( set @ A ) @ Xa4 @ ( insert @ ( set @ A ) @ X @ F5 ) )
=> ( ( X3 != Xa4 )
=> ( ( inf_inf @ ( set @ A ) @ X3 @ Xa4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ X @ ( complete_Sup_Sup @ ( set @ A ) @ F5 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_partition
thf(fact_4353_Union__disjoint,axiom,
! [A: $tType,C4: set @ ( set @ A ),A5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) @ A5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ X4 @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_disjoint
thf(fact_4354_Un__Int__assoc__eq,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) ) )
= ( ord_less_eq @ ( set @ A ) @ C4 @ A5 ) ) ).
% Un_Int_assoc_eq
thf(fact_4355_ivl__disj__int__two_I3_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(3)
thf(fact_4356_Un__Diff__Int,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) )
= A5 ) ).
% Un_Diff_Int
thf(fact_4357_Int__Diff__Un,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
= A5 ) ).
% Int_Diff_Un
thf(fact_4358_Diff__Int,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( inf_inf @ ( set @ A ) @ B5 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ ( minus_minus @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% Diff_Int
thf(fact_4359_Diff__Un,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) )
= ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ ( minus_minus @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% Diff_Un
thf(fact_4360_range__subsetD,axiom,
! [B: $tType,A: $tType,F3: B > A,B5: set @ A,I: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) @ B5 )
=> ( member @ A @ ( F3 @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_4361_Compl__Un,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) ) ) ).
% Compl_Un
thf(fact_4362_Compl__Int,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) ) ) ).
% Compl_Int
thf(fact_4363_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( ( finite_card @ A @ A5 )
= ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( A5
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_4364_not__UNIV__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) ) ) ).
% not_UNIV_le_Icc
thf(fact_4365_not__UNIV__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atMost @ A @ H2 ) ) ) ).
% not_UNIV_le_Iic
thf(fact_4366_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R3 )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ S3 ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S3 ) ) ) ) ).
% sup_Un_eq2
thf(fact_4367_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_4368_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_4369_Compl__eq__Diff__UNIV,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_4370_Compl__partition,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition
thf(fact_4371_Compl__partition2,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ A5 )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition2
thf(fact_4372_finite__range__imageI,axiom,
! [C: $tType,A: $tType,B: $tType,G3: B > A,F3: A > C] :
( ( finite_finite2 @ A @ ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ C
@ ( image @ B @ C
@ ^ [X4: B] : ( F3 @ ( G3 @ X4 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_imageI
thf(fact_4373_comp__fun__commute__Pow__fold,axiom,
! [A: $tType] :
( finite6289374366891150609ommute @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A,A6: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A6 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X4 ) @ A6 ) ) ) ).
% comp_fun_commute_Pow_fold
thf(fact_4374_comp__fun__commute__def_H,axiom,
! [B: $tType,A: $tType] :
( ( finite6289374366891150609ommute @ A @ B )
= ( finite4664212375090638736ute_on @ A @ B @ ( top_top @ ( set @ A ) ) ) ) ).
% comp_fun_commute_def'
thf(fact_4375_inf__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% inf_shunt
thf(fact_4376_sup__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( top_top @ A ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% sup_shunt
thf(fact_4377_sup__neg__inf,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [P3: A,Q3: A,R2: A] :
( ( ord_less_eq @ A @ P3 @ ( sup_sup @ A @ Q3 @ R2 ) )
= ( ord_less_eq @ A @ ( inf_inf @ A @ P3 @ ( uminus_uminus @ A @ Q3 ) ) @ R2 ) ) ) ).
% sup_neg_inf
thf(fact_4378_shunt2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ ( uminus_uminus @ A @ Y ) ) @ Z )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% shunt2
thf(fact_4379_shunt1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ Y ) @ Z ) ) ) ) ).
% shunt1
thf(fact_4380_inter__Set__filter,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= ( filter3 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A5 )
@ B5 ) ) ) ).
% inter_Set_filter
thf(fact_4381_finite__Inf__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( member @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ A5 ) ) )
=> ( member @ A @ ( complete_Inf_Inf @ A @ A5 ) @ A5 ) ) ) ) ) ).
% finite_Inf_in
thf(fact_4382_less__eq__Inf__inter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B5: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B5 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% less_eq_Inf_inter
thf(fact_4383_inf__Inf__fold__inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B5: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A5 ) @ B5 )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ B5 @ A5 ) ) ) ) ).
% inf_Inf_fold_inf
thf(fact_4384_INF__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: A,B5: A] :
( ( inf_inf @ A @ A5
@ ( complete_Inf_Inf @ A
@ ( image @ nat @ A
@ ^ [X4: nat] : B5
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( inf_inf @ A @ A5 @ B5 ) ) ) ).
% INF_nat_binary
thf(fact_4385_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_4386_ivl__disj__int__two_I7_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(7)
thf(fact_4387_ivl__disj__int__one_I4_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(4)
thf(fact_4388_ivl__disj__int__one_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(2)
thf(fact_4389_disjoint__alt__simp3,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ A5 )
= ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp3
thf(fact_4390_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_4391_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ B5 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_4392_Inter__Un__subset,axiom,
! [A: $tType,A5: set @ ( set @ A ),B5: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A5 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B5 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A5 @ B5 ) ) ) ).
% Inter_Un_subset
thf(fact_4393_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F3: B > A,A2: A,X: B] :
( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( F3 @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_4394_INF__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A] :
( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% INF_empty
thf(fact_4395_INF__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,C3: A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [Y5: B] : C3
@ A5 ) )
= ( top_top @ A ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [Y5: B] : C3
@ A5 ) )
= C3 ) ) ) ) ).
% INF_constant
thf(fact_4396_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,X: A,F3: B > A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I3: B] : ( inf_inf @ A @ X @ ( F3 @ I3 ) )
@ I6 ) )
= ( inf_inf @ A @ X @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ I6 ) ) ) ) ) ) ).
% INF_inf_const1
thf(fact_4397_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F3: B > A,X: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I3: B] : ( inf_inf @ A @ ( F3 @ I3 ) @ X )
@ I6 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ I6 ) ) @ X ) ) ) ) ).
% INF_inf_const2
thf(fact_4398_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F3 @ A5 ) ) @ ( image @ B @ A @ F3 @ ( uminus_uminus @ ( set @ B ) @ A5 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_4399_sum_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A,B5: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( if @ A @ ( member @ B @ X4 @ B5 ) @ ( G3 @ X4 ) @ ( zero_zero @ A ) )
@ A5 ) ) ) ) ).
% sum.inter_restrict
thf(fact_4400_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G3: B > A,B5: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( if @ A @ ( member @ B @ X4 @ B5 ) @ ( G3 @ X4 ) @ ( one_one @ A ) )
@ A5 ) ) ) ) ).
% prod.inter_restrict
thf(fact_4401_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C4: set @ D,A5: set @ C,B5: D > ( set @ C )] :
( ( ( C4
= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A5 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image @ D @ ( set @ C ) @ B5 @ C4 ) ) )
= A5 ) )
& ( ( C4
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A5 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image @ D @ ( set @ C ) @ B5 @ C4 ) ) )
= ( complete_Inf_Inf @ ( set @ C )
@ ( image @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A5 @ ( B5 @ X4 ) )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_4402_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C4: set @ A,A5: A > ( set @ B ),B5: set @ B] :
( ( ( C4
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A5 @ C4 ) ) @ B5 )
= B5 ) )
& ( ( C4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A5 @ C4 ) ) @ B5 )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ X4 ) @ B5 )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_4403_finite__range__Some,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_range_Some
thf(fact_4404_UN__UN__finite__eq,axiom,
! [A: $tType,A5: nat > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ nat @ ( set @ A )
@ ^ [N6: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N6 ) ) )
@ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A5 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UN_UN_finite_eq
thf(fact_4405_INT__empty,axiom,
! [B: $tType,A: $tType,B5: B > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% INT_empty
thf(fact_4406_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( folding_insort_key @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ ( set @ A ) )
@ ^ [X4: A] : X4 ) ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_4407_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S3: set @ B,H2: B > A,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( H2 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ S3 @ T4 ) )
=> ( ( G3 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S3 @ T4 ) )
=> ( ( G3 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_4408_sum_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A,B5: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ B5 ) ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_4409_sum_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ B5 ) ) ) ) ) ) ).
% sum.union_inter
thf(fact_4410_Iio__Int__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,K: A] :
( ( ( ord_less @ A @ X @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( ord_less @ A @ X @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Iio_Int_singleton
thf(fact_4411_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S3: set @ B,H2: B > A,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) )
=> ( ( H2 @ I2 )
= ( one_one @ A ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ S3 @ T4 ) )
=> ( ( G3 @ I2 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S3 @ T4 ) )
=> ( ( G3 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_4412_prod_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G3: B > A,B5: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ B5 ) ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_4413_prod_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B5 ) ) ) ) ) ) ).
% prod.union_inter
thf(fact_4414_card__Un__Int,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_4415_card__Diff__subset__Int,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_4416_finite__UNIV__card__ge__0,axiom,
! [A: $tType] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_4417_range__enumerate,axiom,
! [S3: set @ nat] :
( ~ ( finite_finite2 @ nat @ S3 )
=> ( ( image @ nat @ nat @ ( infini527867602293511546merate @ nat @ S3 ) @ ( top_top @ ( set @ nat ) ) )
= S3 ) ) ).
% range_enumerate
thf(fact_4418_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,P2: B > $o,H2: B > A,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P2 @ X4 ) @ ( H2 @ X4 ) @ ( G3 @ X4 ) )
@ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P2 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P2 ) ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_4419_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,P2: B > $o,H2: B > A,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P2 @ X4 ) @ ( H2 @ X4 ) @ ( G3 @ X4 ) )
@ A5 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P2 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P2 ) ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_4420_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_4421_UN__finite__subset,axiom,
! [A: $tType,A5: nat > ( set @ A ),C4: set @ A] :
( ! [N2: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A5 @ ( top_top @ ( set @ nat ) ) ) ) @ C4 ) ) ).
% UN_finite_subset
thf(fact_4422_UN__finite2__eq,axiom,
! [A: $tType,A5: nat > ( set @ A ),B5: nat > ( set @ A ),K: nat] :
( ! [N2: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N2 @ K ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A5 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B5 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_4423_INT__extend__simps_I3_J,axiom,
! [F: $tType,E: $tType,C4: set @ E,A5: E > ( set @ F ),B5: set @ F] :
( ( ( C4
= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E @ ( set @ F ) @ A5 @ C4 ) ) @ B5 )
= ( minus_minus @ ( set @ F ) @ ( top_top @ ( set @ F ) ) @ B5 ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E @ ( set @ F ) @ A5 @ C4 ) ) @ B5 )
= ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E @ ( set @ F )
@ ^ [X4: E] : ( minus_minus @ ( set @ F ) @ ( A5 @ X4 ) @ B5 )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_4424_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ B5 ) ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_4425_sum__Un,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A5: set @ B,B5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) ) ) ) ) ) ) ).
% sum_Un
thf(fact_4426_sum_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ( inf_inf @ ( set @ B ) @ A5 @ B5 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ B5 ) ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_4427_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) )
=> ( ( G3 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B5 ) ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_4428_prod_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ( inf_inf @ ( set @ B ) @ A5 @ B5 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B5 ) ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_4429_sum_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ B5 @ A5 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_4430_sum__Un2,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [A5: set @ A,B5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
=> ( ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ B5 @ A5 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ) ) ) ).
% sum_Un2
thf(fact_4431_card__Un__disjoint,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_4432_prod_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) )
= ( times_times @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ B5 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ B5 @ A5 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_4433_sum__Un__nat,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_4434_finite__enumerate__initial__segment,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat,S: A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ S3 @ ( set_ord_lessThan @ A @ S ) ) ) )
=> ( ( infini527867602293511546merate @ A @ ( inf_inf @ ( set @ A ) @ S3 @ ( set_ord_lessThan @ A @ S ) ) @ N )
= ( infini527867602293511546merate @ A @ S3 @ N ) ) ) ) ) ).
% finite_enumerate_initial_segment
thf(fact_4435_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F3: B > A] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% card_range_greater_zero
thf(fact_4436_distinct__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ ( list @ A ) @ Xs )
=> ( ! [Ys5: list @ A] :
( ( member @ ( list @ A ) @ Ys5 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys5 ) )
=> ( ! [Ys5: list @ A,Zs2: list @ A] :
( ( member @ ( list @ A ) @ Ys5 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys5 != Zs2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys5 ) @ ( set2 @ A @ Zs2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_4437_sum_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,A5: B > ( set @ C ),G3: C > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I6 )
=> ( finite_finite2 @ C @ ( A5 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I6 )
=> ! [Xa4: B] :
( ( member @ B @ Xa4 @ I6 )
=> ( ( X3 != Xa4 )
=> ( ( inf_inf @ ( set @ C ) @ ( A5 @ X3 ) @ ( A5 @ Xa4 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G3 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ A5 @ I6 ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( groups7311177749621191930dd_sum @ C @ A @ G3 @ ( A5 @ X4 ) )
@ I6 ) ) ) ) ) ) ).
% sum.UNION_disjoint
thf(fact_4438_prod_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,A5: B > ( set @ C ),G3: C > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I6 )
=> ( finite_finite2 @ C @ ( A5 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I6 )
=> ! [Xa4: B] :
( ( member @ B @ Xa4 @ I6 )
=> ( ( X3 != Xa4 )
=> ( ( inf_inf @ ( set @ C ) @ ( A5 @ X3 ) @ ( A5 @ Xa4 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G3 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ A5 @ I6 ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( groups7121269368397514597t_prod @ C @ A @ G3 @ ( A5 @ X4 ) )
@ I6 ) ) ) ) ) ) ).
% prod.UNION_disjoint
thf(fact_4439_word__atLeastLessThan__Suc__atLeastAtMost__union,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: word @ A,L: word @ A,U: word @ A] :
( ( M
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ L @ M )
=> ( ( ord_less_eq @ ( word @ A ) @ M @ U )
=> ( ( sup_sup @ ( set @ ( word @ A ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ L @ M ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ M @ ( one_one @ ( word @ A ) ) ) @ U ) )
= ( set_or1337092689740270186AtMost @ ( word @ A ) @ L @ U ) ) ) ) ) ) ).
% word_atLeastLessThan_Suc_atLeastAtMost_union
thf(fact_4440_UN__finite2__subset,axiom,
! [A: $tType,A5: nat > ( set @ A ),B5: nat > ( set @ A ),K: nat] :
( ! [N2: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B5 @ ( 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 ) @ A5 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B5 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_4441_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A5: set @ B,B5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) )
=> ( ( F3 @ X3 )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ B5 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_4442_range__mod,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( image @ nat @ nat
@ ^ [M5: nat] : ( modulo_modulo @ nat @ M5 @ N )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% range_mod
thf(fact_4443_comp__fun__commute__on_Ofold__set__union__disj,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,A5: set @ A,B5: set @ A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ S3 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( finite_fold @ A @ B @ F3 @ ( finite_fold @ A @ B @ F3 @ Z @ A5 ) @ B5 ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_set_union_disj
thf(fact_4444_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I6: set @ A,A5: A > ( set @ B )] :
( ( finite_finite2 @ A @ I6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ I6 )
=> ( finite_finite2 @ B @ ( A5 @ X3 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ I6 )
=> ! [Xa4: A] :
( ( member @ A @ Xa4 @ I6 )
=> ( ( X3 != Xa4 )
=> ( ( inf_inf @ ( set @ B ) @ ( A5 @ X3 ) @ ( A5 @ Xa4 ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A5 @ I6 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] : ( finite_card @ B @ ( A5 @ I3 ) )
@ I6 ) ) ) ) ) ).
% card_UN_disjoint
thf(fact_4445_card__nat,axiom,
( ( finite_card @ nat @ ( top_top @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% card_nat
thf(fact_4446_card__option,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_card @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
= ( suc @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% card_option
thf(fact_4447_one__less__int__card,axiom,
! [A: $tType] :
( ( card2 @ A )
=> ( ord_less @ int @ ( one_one @ int ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% one_less_int_card
thf(fact_4448_card__fun,axiom,
! [A: $tType,B: $tType] :
( ( ( ( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
!= ( zero_zero @ nat ) )
& ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
!= ( zero_zero @ nat ) ) )
| ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
= ( one_one @ nat ) ) )
=> ( ( finite_card @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
= ( power_power @ nat @ ( finite_card @ B @ ( top_top @ ( set @ B ) ) ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) )
& ( ~ ( ( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
!= ( zero_zero @ nat ) )
& ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
!= ( zero_zero @ nat ) ) )
| ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
= ( one_one @ nat ) ) )
=> ( ( finite_card @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
= ( zero_zero @ nat ) ) ) ) ).
% card_fun
thf(fact_4449_card__num0,axiom,
( ( finite_card @ numeral_num0 @ ( top_top @ ( set @ numeral_num0 ) ) )
= ( zero_zero @ nat ) ) ).
% card_num0
thf(fact_4450_card__UNIV__unit,axiom,
( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
= ( one_one @ nat ) ) ).
% card_UNIV_unit
thf(fact_4451_merge__true__star,axiom,
( ( times_times @ assn @ ( top_top @ assn ) @ ( top_top @ assn ) )
= ( top_top @ assn ) ) ).
% merge_true_star
thf(fact_4452_merge__pure__and,axiom,
! [A2: $o,B2: $o] :
( ( inf_inf @ assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B2 ) )
= ( pure_assn
@ ( A2
& B2 ) ) ) ).
% merge_pure_and
thf(fact_4453_assn__basic__inequalities_I1_J,axiom,
( ( top_top @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(1)
thf(fact_4454_assn__basic__inequalities_I5_J,axiom,
( ( top_top @ assn )
!= ( bot_bot @ assn ) ) ).
% assn_basic_inequalities(5)
thf(fact_4455_mod__h__bot__iff_I6_J,axiom,
! [P2: assn,Q2: assn,H2: heap_ext @ product_unit] :
( ( rep_assn @ ( inf_inf @ assn @ P2 @ Q2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
& ( rep_assn @ Q2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(6)
thf(fact_4456_card__literal,axiom,
( ( finite_card @ literal @ ( top_top @ ( set @ literal ) ) )
= ( zero_zero @ nat ) ) ).
% card_literal
thf(fact_4457_ent__conjE2,axiom,
! [B5: assn,C4: assn,A5: assn] :
( ( entails @ B5 @ C4 )
=> ( entails @ ( inf_inf @ assn @ A5 @ B5 ) @ C4 ) ) ).
% ent_conjE2
thf(fact_4458_ent__conjE1,axiom,
! [A5: assn,C4: assn,B5: assn] :
( ( entails @ A5 @ C4 )
=> ( entails @ ( inf_inf @ assn @ A5 @ B5 ) @ C4 ) ) ).
% ent_conjE1
thf(fact_4459_ent__conjI,axiom,
! [A5: assn,B5: assn,C4: assn] :
( ( entails @ A5 @ B5 )
=> ( ( entails @ A5 @ C4 )
=> ( entails @ A5 @ ( inf_inf @ assn @ B5 @ C4 ) ) ) ) ).
% ent_conjI
thf(fact_4460_mod__and__dist,axiom,
! [P2: assn,Q2: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( inf_inf @ assn @ P2 @ Q2 ) @ H2 )
= ( ( rep_assn @ P2 @ H2 )
& ( rep_assn @ Q2 @ H2 ) ) ) ).
% mod_and_dist
thf(fact_4461_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_4462_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A6 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% inf_set_def
thf(fact_4463_ent__true,axiom,
! [P2: assn] : ( entails @ P2 @ ( top_top @ assn ) ) ).
% ent_true
thf(fact_4464_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( inf_inf @ ( A > B > $o )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R3 )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ S3 ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S3 ) ) ) ) ).
% inf_Int_eq2
thf(fact_4465_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_4466_ex__distrib__and,axiom,
! [A: $tType,P2: A > assn,Q2: assn] :
( ( ex_assn @ A
@ ^ [X4: A] : ( inf_inf @ assn @ ( P2 @ X4 ) @ Q2 ) )
= ( inf_inf @ assn @ ( ex_assn @ A @ P2 ) @ Q2 ) ) ).
% ex_distrib_and
thf(fact_4467_mod__star__trueE,axiom,
! [P2: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P2 @ ( top_top @ assn ) ) @ H2 )
=> ~ ! [H6: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ P2 @ H6 ) ) ).
% mod_star_trueE
thf(fact_4468_mod__star__trueI,axiom,
! [P2: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P2 @ H2 )
=> ( rep_assn @ ( times_times @ assn @ P2 @ ( top_top @ assn ) ) @ H2 ) ) ).
% mod_star_trueI
thf(fact_4469_Inter__UNIV,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Inter_UNIV
thf(fact_4470_prec__frame_H,axiom,
! [A: $tType,P2: A > assn,X: A,F14: assn,Y: A,F24: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),R12: assn,R23: assn] :
( ( ( rep_assn @ ( inf_inf @ assn @ ( times_times @ assn @ ( P2 @ X ) @ F14 ) @ ( times_times @ assn @ ( P2 @ Y ) @ F24 ) ) @ H2 )
=> ( X = Y ) )
=> ( ( rep_assn @ ( inf_inf @ assn @ R12 @ R23 ) @ H2 )
=> ( ( entails @ R12 @ ( times_times @ assn @ ( P2 @ X ) @ F14 ) )
=> ( ( entails @ R23 @ ( times_times @ assn @ ( P2 @ Y ) @ F24 ) )
=> ( X = Y ) ) ) ) ) ).
% prec_frame'
thf(fact_4471_prec__frame__expl,axiom,
! [A: $tType,P2: A > assn,F14: assn,F24: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),R12: assn,R23: assn,X: A,Y: A] :
( ! [X3: A,Y3: A] :
( ( rep_assn @ ( inf_inf @ assn @ ( times_times @ assn @ ( P2 @ X3 ) @ F14 ) @ ( times_times @ assn @ ( P2 @ Y3 ) @ F24 ) ) @ H2 )
=> ( X3 = Y3 ) )
=> ( ( rep_assn @ ( inf_inf @ assn @ R12 @ R23 ) @ H2 )
=> ( ( entails @ R12 @ ( times_times @ assn @ ( P2 @ X ) @ F14 ) )
=> ( ( entails @ R23 @ ( times_times @ assn @ ( P2 @ Y ) @ F24 ) )
=> ( X = Y ) ) ) ) ) ).
% prec_frame_expl
thf(fact_4472_mod__h__bot__iff_I2_J,axiom,
! [H2: heap_ext @ product_unit] : ( rep_assn @ ( top_top @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% mod_h_bot_iff(2)
thf(fact_4473_bit0_Osize0,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) ) ) ).
% bit0.size0
thf(fact_4474_bit1_Osize0,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ).
% bit1.size0
thf(fact_4475_bit1_Osize1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ord_less @ int @ ( one_one @ int ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ).
% bit1.size1
thf(fact_4476_bit0_Osize1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ord_less @ int @ ( one_one @ int ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) ) ) ).
% bit0.size1
thf(fact_4477_preciseD,axiom,
! [B: $tType,A: $tType,R3: A > B > assn,A2: A,P3: B,F5: assn,A3: A,F7: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( precise @ A @ B @ R3 )
=> ( ( rep_assn @ ( inf_inf @ assn @ ( times_times @ assn @ ( R3 @ A2 @ P3 ) @ F5 ) @ ( times_times @ assn @ ( R3 @ A3 @ P3 ) @ F7 ) ) @ H2 )
=> ( A2 = A3 ) ) ) ).
% preciseD
thf(fact_4478_preciseI,axiom,
! [B: $tType,A: $tType,R3: A > B > assn] :
( ! [A4: A,A15: A,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),P8: B,F6: assn,F10: assn] :
( ( rep_assn @ ( inf_inf @ assn @ ( times_times @ assn @ ( R3 @ A4 @ P8 ) @ F6 ) @ ( times_times @ assn @ ( R3 @ A15 @ P8 ) @ F10 ) ) @ H )
=> ( A4 = A15 ) )
=> ( precise @ A @ B @ R3 ) ) ).
% preciseI
thf(fact_4479_precise__def,axiom,
! [B: $tType,A: $tType] :
( ( precise @ A @ B )
= ( ^ [R4: A > B > assn] :
! [A8: A,A16: A,H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),P7: B,F11: assn,F15: assn] :
( ( rep_assn @ ( inf_inf @ assn @ ( times_times @ assn @ ( R4 @ A8 @ P7 ) @ F11 ) @ ( times_times @ assn @ ( R4 @ A16 @ P7 ) @ F15 ) ) @ H4 )
=> ( A8 = A16 ) ) ) ) ).
% precise_def
thf(fact_4480_INF__filter__bot__base,axiom,
! [B: $tType,A: $tType,I6: set @ A,F5: A > ( filter @ B )] :
( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ I6 )
& ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X5 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ I2 ) @ ( F5 @ J2 ) ) ) ) ) )
=> ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ F5 @ I6 ) )
= ( bot_bot @ ( filter @ B ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ I6 )
& ( ( F5 @ X4 )
= ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ).
% INF_filter_bot_base
thf(fact_4481_bit1__induct,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [P2: ( numeral_bit1 @ A ) > $o,X: numeral_bit1 @ A] :
( ! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less @ int @ Z3 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) )
=> ( P2 @ ( ring_1_of_int @ ( numeral_bit1 @ A ) @ Z3 ) ) ) )
=> ( P2 @ X ) ) ) ).
% bit1_induct
thf(fact_4482_bit0__induct,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [P2: ( numeral_bit0 @ A ) > $o,X: numeral_bit0 @ A] :
( ! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less @ int @ Z3 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) )
=> ( P2 @ ( ring_1_of_int @ ( numeral_bit0 @ A ) @ Z3 ) ) ) )
=> ( P2 @ X ) ) ) ).
% bit0_induct
thf(fact_4483_bit1__cases,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit1 @ A] :
~ ! [Z3: int] :
( ( X
= ( ring_1_of_int @ ( numeral_bit1 @ A ) @ Z3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ~ ( ord_less @ int @ Z3 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ).
% bit1_cases
thf(fact_4484_bit0__cases,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit0 @ A] :
~ ! [Z3: int] :
( ( X
= ( ring_1_of_int @ ( numeral_bit0 @ A ) @ Z3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ~ ( ord_less @ int @ Z3 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) ) ) ) ) ).
% bit0_cases
thf(fact_4485_card__map__elide2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ ( word @ A ) @ ( top_top @ ( set @ ( word @ A ) ) ) ) )
=> ( ( finite_card @ ( word @ A ) @ ( image @ nat @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
= N ) ) ) ).
% card_map_elide2
thf(fact_4486_prec__frame,axiom,
! [B: $tType,A: $tType,P2: A > B > assn,R12: assn,R23: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),X: A,P3: B,F14: assn,Y: A,F24: assn] :
( ( precise @ A @ B @ P2 )
=> ( ( rep_assn @ ( inf_inf @ assn @ R12 @ R23 ) @ H2 )
=> ( ( entails @ R12 @ ( times_times @ assn @ ( P2 @ X @ P3 ) @ F14 ) )
=> ( ( entails @ R23 @ ( times_times @ assn @ ( P2 @ Y @ P3 ) @ F24 ) )
=> ( X = Y ) ) ) ) ) ).
% prec_frame
thf(fact_4487_card__map__elide,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ ( word @ A ) @ ( top_top @ ( set @ ( word @ A ) ) ) ) )
=> ( ( finite_card @ ( word @ A ) @ ( image @ nat @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( finite_card @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% card_map_elide
thf(fact_4488_zero__less__card__finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% zero_less_card_finite
thf(fact_4489_card__UNIV__sum,axiom,
! [A: $tType,B: $tType] :
( ( ( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
!= ( zero_zero @ nat ) )
& ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
!= ( zero_zero @ nat ) ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( plus_plus @ nat @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) @ ( finite_card @ B @ ( top_top @ ( set @ B ) ) ) ) ) )
& ( ~ ( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
!= ( zero_zero @ nat ) )
& ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
!= ( zero_zero @ nat ) ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( zero_zero @ nat ) ) ) ) ).
% card_UNIV_sum
thf(fact_4490_finite__UNIV__fun,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
= ( ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
& ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) )
| ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
= ( one_one @ nat ) ) ) ) ).
% finite_UNIV_fun
thf(fact_4491_one__less__card,axiom,
! [A: $tType] :
( ( card2 @ A )
=> ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% one_less_card
thf(fact_4492_one__le__card__finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% one_le_card_finite
thf(fact_4493_card__UNIV__option,axiom,
! [A: $tType] :
( ( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
= ( zero_zero @ nat ) )
=> ( ( finite_card @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
= ( zero_zero @ nat ) ) )
& ( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
!= ( zero_zero @ nat ) )
=> ( ( finite_card @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
= ( plus_plus @ nat @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_UNIV_option
thf(fact_4494_Inf__finite__empty,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% Inf_finite_empty
thf(fact_4495_Sup__finite__empty,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% Sup_finite_empty
thf(fact_4496_cclfp__def,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( order_532582986084564980_cclfp @ A )
= ( ^ [F4: A > A] :
( complete_Sup_Sup @ A
@ ( image @ nat @ A
@ ^ [I3: nat] : ( compow @ ( A > A ) @ I3 @ F4 @ ( bot_bot @ A ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% cclfp_def
thf(fact_4497_root__def,axiom,
( root
= ( ^ [N6: nat,X4: real] :
( if @ real
@ ( N6
= ( zero_zero @ nat ) )
@ ( zero_zero @ real )
@ ( the_inv_into @ real @ real @ ( top_top @ ( set @ real ) )
@ ^ [Y5: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y5 ) @ ( power_power @ real @ ( abs_abs @ real @ Y5 ) @ N6 ) )
@ X4 ) ) ) ) ).
% root_def
thf(fact_4498_less__eq__assn__def,axiom,
( ( ord_less_eq @ assn )
= ( ^ [A8: assn,B8: assn] :
( A8
= ( inf_inf @ assn @ A8 @ B8 ) ) ) ) ).
% less_eq_assn_def
thf(fact_4499_minus__assn__def,axiom,
( ( minus_minus @ assn )
= ( ^ [A8: assn,B8: assn] : ( inf_inf @ assn @ A8 @ ( uminus_uminus @ assn @ B8 ) ) ) ) ).
% minus_assn_def
thf(fact_4500_less__filter__def,axiom,
! [A: $tType] :
( ( ord_less @ ( filter @ A ) )
= ( ^ [F11: filter @ A,F15: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F11 @ F15 )
& ~ ( ord_less_eq @ ( filter @ A ) @ F15 @ F11 ) ) ) ) ).
% less_filter_def
thf(fact_4501_bot__finite__def,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ( ( bot_bot @ A )
= ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% bot_finite_def
thf(fact_4502_perfect__space__class_OUNIV__not__singleton,axiom,
! [A: $tType] :
( ( topolo8386298272705272623_space @ A )
=> ! [X: A] :
( ( top_top @ ( set @ A ) )
!= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% perfect_space_class.UNIV_not_singleton
thf(fact_4503_Pow__fold,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( pow @ A @ A5 )
= ( finite_fold @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A,A6: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A6 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X4 ) @ A6 ) )
@ ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A5 ) ) ) ).
% Pow_fold
thf(fact_4504_distinct__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ A @ ( concat @ A @ Xs ) )
= ( ( distinct @ ( list @ A ) @ ( removeAll @ ( list @ A ) @ ( nil @ A ) @ Xs ) )
& ! [Ys3: list @ A] :
( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys3 ) )
& ! [Ys3: list @ A,Zs3: list @ A] :
( ( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
& ( member @ ( list @ A ) @ Zs3 @ ( set2 @ ( list @ A ) @ Xs ) )
& ( Ys3 != Zs3 ) )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Zs3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% distinct_concat_iff
thf(fact_4505_times__assn__raw_Oelims_I3_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( times_assn_raw @ X @ Xa2 @ Xb )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ? [As12: set @ nat,As23: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As12 @ As23 ) )
& ( ( inf_inf @ ( set @ nat ) @ As12 @ As23 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As12 ) )
& ( Xa2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As23 ) ) ) ) ) ).
% times_assn_raw.elims(3)
thf(fact_4506_PowI,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( member @ ( set @ A ) @ A5 @ ( pow @ A @ B5 ) ) ) ).
% PowI
thf(fact_4507_Pow__iff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( member @ ( set @ A ) @ A5 @ ( pow @ A @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% Pow_iff
thf(fact_4508_slice__eq__bounds__empty,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( slice @ A @ I @ I @ Xs )
= ( nil @ A ) ) ).
% slice_eq_bounds_empty
thf(fact_4509_slice__Nil,axiom,
! [A: $tType,Begin: nat,End: nat] :
( ( slice @ A @ Begin @ End @ ( nil @ A ) )
= ( nil @ A ) ) ).
% slice_Nil
thf(fact_4510_length__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( zero_zero @ nat ) )
= ( Xs
= ( nil @ A ) ) ) ).
% length_0_conv
thf(fact_4511_set__empty2,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty2
thf(fact_4512_set__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( set2 @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty
thf(fact_4513_take0,axiom,
! [A: $tType] :
( ( take @ A @ ( zero_zero @ nat ) )
= ( ^ [Xs2: list @ A] : ( nil @ A ) ) ) ).
% take0
thf(fact_4514_take__eq__Nil,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( take @ A @ N @ Xs )
= ( nil @ A ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( Xs
= ( nil @ A ) ) ) ) ).
% take_eq_Nil
thf(fact_4515_take__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( nil @ A )
= ( take @ A @ N @ Xs ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( Xs
= ( nil @ A ) ) ) ) ).
% take_eq_Nil2
thf(fact_4516_sum__list_ONil,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A @ ( nil @ A ) )
= ( zero_zero @ A ) ) ) ).
% sum_list.Nil
thf(fact_4517_empty__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( nil @ A )
= ( replicate @ A @ N @ X ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_4518_replicate__empty,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( replicate @ A @ N @ X )
= ( nil @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_4519_Misc_Olast__in__set,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ( member @ A @ ( last @ A @ L ) @ ( set2 @ A @ L ) ) ) ).
% Misc.last_in_set
thf(fact_4520_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( nil @ A ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_4521_horner__sum__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F3: B > A,A2: A] :
( ( groups4207007520872428315er_sum @ B @ A @ F3 @ A2 @ ( nil @ B ) )
= ( zero_zero @ A ) ) ) ).
% horner_sum_simps(1)
thf(fact_4522_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ A5 )
= ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_4523_finite__Pow__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ ( set @ A ) @ ( pow @ A @ A5 ) )
= ( finite_finite2 @ A @ A5 ) ) ).
% finite_Pow_iff
thf(fact_4524_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_4525_Pow__UNIV,axiom,
! [A: $tType] :
( ( pow @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% Pow_UNIV
thf(fact_4526_Pow__Int__eq,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( pow @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) )
= ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow @ A @ A5 ) @ ( pow @ A @ B5 ) ) ) ).
% Pow_Int_eq
thf(fact_4527_length__greater__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( Xs
!= ( nil @ A ) ) ) ).
% length_greater_0_conv
thf(fact_4528_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( linord4507533701916653071of_set @ A @ A5 )
= ( nil @ A ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_4529_Pow__empty,axiom,
! [A: $tType] :
( ( pow @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_empty
thf(fact_4530_Pow__singleton__iff,axiom,
! [A: $tType,X6: set @ A,Y7: set @ A] :
( ( ( pow @ A @ X6 )
= ( insert @ ( set @ A ) @ Y7 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
= ( ( X6
= ( bot_bot @ ( set @ A ) ) )
& ( Y7
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pow_singleton_iff
thf(fact_4531_length__ge__1__conv,axiom,
! [A: $tType,L: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( size_size @ ( list @ A ) @ L ) )
= ( L
!= ( nil @ A ) ) ) ).
% length_ge_1_conv
thf(fact_4532_Pow__def,axiom,
! [A: $tType] :
( ( pow @ A )
= ( ^ [A6: set @ A] :
( collect @ ( set @ A )
@ ^ [B7: set @ A] : ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% Pow_def
thf(fact_4533_PowD,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( member @ ( set @ A ) @ A5 @ ( pow @ A @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% PowD
thf(fact_4534_less__assn__def,axiom,
( ( ord_less @ assn )
= ( ^ [A8: assn,B8: assn] :
( ( ord_less_eq @ assn @ A8 @ B8 )
& ( A8 != B8 ) ) ) ) ).
% less_assn_def
thf(fact_4535_Pow__bottom,axiom,
! [A: $tType,B5: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow @ A @ B5 ) ) ).
% Pow_bottom
thf(fact_4536_Pow__not__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( pow @ A @ A5 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Pow_not_empty
thf(fact_4537_Misc_Ofoldr__Cons,axiom,
! [A: $tType,Xs: list @ A] :
( ( foldr @ A @ ( list @ A ) @ ( cons @ A ) @ Xs @ ( nil @ A ) )
= Xs ) ).
% Misc.foldr_Cons
thf(fact_4538_neq__NilE,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ~ ! [X3: A,Xs3: list @ A] :
( L
!= ( cons @ A @ X3 @ Xs3 ) ) ) ).
% neq_NilE
thf(fact_4539_list__2pre__induct,axiom,
! [A: $tType,B: $tType,P2: ( list @ A ) > ( list @ B ) > $o,W1: list @ A,W22: list @ B] :
( ( P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [E2: A,W12: list @ A,W23: list @ B] :
( ( P2 @ W12 @ W23 )
=> ( P2 @ ( cons @ A @ E2 @ W12 ) @ W23 ) )
=> ( ! [E2: B,W13: list @ A,W24: list @ B] :
( ( P2 @ W13 @ W24 )
=> ( P2 @ W13 @ ( cons @ B @ E2 @ W24 ) ) )
=> ( P2 @ W1 @ W22 ) ) ) ) ).
% list_2pre_induct
thf(fact_4540_list__induct__first2,axiom,
! [A: $tType,P2: ( list @ A ) > $o,Xs: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X3: A] : ( P2 @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X15: A,X23: A,Xs3: list @ A] :
( ( P2 @ Xs3 )
=> ( P2 @ ( cons @ A @ X15 @ ( cons @ A @ X23 @ Xs3 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_induct_first2
thf(fact_4541_mergesort__by__rel__merge__induct,axiom,
! [A: $tType,B: $tType,P2: ( list @ A ) > ( list @ B ) > $o,R3: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ! [Xs3: list @ A] : ( P2 @ Xs3 @ ( nil @ B ) )
=> ( ! [X_1: list @ B] : ( P2 @ ( nil @ A ) @ X_1 )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys5: list @ B] :
( ( R3 @ X3 @ Y3 )
=> ( ( P2 @ Xs3 @ ( cons @ B @ Y3 @ Ys5 ) )
=> ( P2 @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y3 @ Ys5 ) ) ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys5: list @ B] :
( ~ ( R3 @ X3 @ Y3 )
=> ( ( P2 @ ( cons @ A @ X3 @ Xs3 ) @ Ys5 )
=> ( P2 @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y3 @ Ys5 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_4542_Pow__top,axiom,
! [A: $tType,A5: set @ A] : ( member @ ( set @ A ) @ A5 @ ( pow @ A @ A5 ) ) ).
% Pow_top
thf(fact_4543_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% n_lists.simps(1)
thf(fact_4544_length__nth__simps_I1_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% length_nth_simps(1)
thf(fact_4545_len__greater__imp__nonempty,axiom,
! [A: $tType,X: nat,L: list @ A] :
( ( ord_less @ nat @ X @ ( size_size @ ( list @ A ) @ L ) )
=> ( L
!= ( nil @ A ) ) ) ).
% len_greater_imp_nonempty
thf(fact_4546_empty__set,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ ( nil @ A ) ) ) ).
% empty_set
thf(fact_4547_sorted0,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ).
% sorted0
thf(fact_4548_strict__sorted__simps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less @ A ) @ ( nil @ A ) ) ) ).
% strict_sorted_simps(1)
thf(fact_4549_zipf_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,X: product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F2: A > B > C] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
=> ( ! [F2: A > B > C,A4: A,As: list @ A,B4: B,Bs: list @ B] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A4 @ As ) @ ( cons @ B @ B4 @ Bs ) ) ) )
=> ( ! [A4: A > B > C,V: A,Va: list @ A] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V @ Va ) @ ( nil @ B ) ) ) )
=> ~ ! [A4: A > B > C,V: B,Va: list @ B] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V @ Va ) ) ) ) ) ) ) ).
% zipf.cases
thf(fact_4550_merge_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [L22: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ L22 ) )
=> ( ! [V: A,Va: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V @ Va ) @ ( nil @ A ) ) )
=> ~ ! [X15: A,L1: list @ A,X23: A,L22: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X15 @ L1 ) @ ( cons @ A @ X23 @ L22 ) ) ) ) ) ) ).
% merge.cases
thf(fact_4551_list__all__zip_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [P: A > B > $o] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
=> ( ! [P: A > B > $o,A4: A,As: list @ A,B4: B,Bs: list @ B] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A4 @ As ) @ ( cons @ B @ B4 @ Bs ) ) ) )
=> ( ! [P: A > B > $o,V: A,Va: list @ A] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V @ Va ) @ ( nil @ B ) ) ) )
=> ~ ! [P: A > B > $o,V: B,Va: list @ B] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V @ Va ) ) ) ) ) ) ) ).
% list_all_zip.cases
thf(fact_4552_partition__rev_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) )] :
( ! [P: A > $o,Yes: list @ A,No: list @ A] :
( X
!= ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( nil @ A ) ) ) )
=> ~ ! [P: A > $o,Yes: list @ A,No: list @ A,X3: A,Xs3: list @ A] :
( X
!= ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( cons @ A @ X3 @ Xs3 ) ) ) ) ) ).
% partition_rev.cases
thf(fact_4553_quicksort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R8: A > A > $o,Sl: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
=> ~ ! [R8: A > A > $o,Sl: list @ A,X3: A,Xs3: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X3 @ Xs3 ) ) ) ) ) ).
% quicksort_by_rel.cases
thf(fact_4554_mergesort__by__rel__merge_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R8: A > A > $o,X3: A,Xs3: list @ A,Y3: A,Ys5: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ A @ Y3 @ Ys5 ) ) ) )
=> ( ! [R8: A > A > $o,Xs3: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ ( nil @ A ) ) ) )
=> ~ ! [R8: A > A > $o,V: A,Va: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V @ Va ) ) ) ) ) ) ).
% mergesort_by_rel_merge.cases
thf(fact_4555_mergesort__by__rel__split_Ocases,axiom,
! [A: $tType,X: product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A )] :
( ! [Xs1: list @ A,Xs22: list @ A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( nil @ A ) ) )
=> ( ! [Xs1: list @ A,Xs22: list @ A,X3: A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ~ ! [Xs1: list @ A,Xs22: list @ A,X15: A,X23: A,Xs3: list @ A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X15 @ ( cons @ A @ X23 @ Xs3 ) ) ) ) ) ) ).
% mergesort_by_rel_split.cases
thf(fact_4556_Pow__set_I1_J,axiom,
! [A: $tType] :
( ( pow @ A @ ( set2 @ A @ ( nil @ A ) ) )
= ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_set(1)
thf(fact_4557_take__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( take @ A @ ( zero_zero @ nat ) @ Xs )
= ( nil @ A ) ) ).
% take_0
thf(fact_4558_replicate__0,axiom,
! [A: $tType,X: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_4559_Pow__mono,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A5 ) @ ( pow @ A @ B5 ) ) ) ).
% Pow_mono
thf(fact_4560_subset__Pow__Union,axiom,
! [A: $tType,A5: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ A5 @ ( pow @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) ) ) ).
% subset_Pow_Union
thf(fact_4561_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F3: B > A,A5: set @ B,B5: set @ A] :
( ( ( image @ B @ A @ F3 @ A5 )
= B5 )
=> ( ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F3 ) @ ( pow @ B @ A5 ) )
= ( pow @ A @ B5 ) ) ) ).
% image_Pow_surj
thf(fact_4562_list_Osize__gen_I1_J,axiom,
! [A: $tType,X: A > nat] :
( ( size_list @ A @ X @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size_gen(1)
thf(fact_4563_count__list_Osimps_I1_J,axiom,
! [A: $tType,Y: A] :
( ( count_list @ A @ ( nil @ A ) @ Y )
= ( zero_zero @ nat ) ) ).
% count_list.simps(1)
thf(fact_4564_Fpow__subset__Pow,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A5 ) @ ( pow @ A @ A5 ) ) ).
% Fpow_subset_Pow
thf(fact_4565_length__compl__induct,axiom,
! [A: $tType,P2: ( list @ A ) > $o,L: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [E2: A,L2: list @ A] :
( ! [Ll: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L2 ) )
=> ( P2 @ Ll ) )
=> ( P2 @ ( cons @ A @ E2 @ L2 ) ) )
=> ( P2 @ L ) ) ) ).
% length_compl_induct
thf(fact_4566_list__decomp__1,axiom,
! [A: $tType,L: list @ A] :
( ( ( size_size @ ( list @ A ) @ L )
= ( one_one @ nat ) )
=> ? [A4: A] :
( L
= ( cons @ A @ A4 @ ( nil @ A ) ) ) ) ).
% list_decomp_1
thf(fact_4567_sorted1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% sorted1
thf(fact_4568_butlast__subset,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( butlast @ A @ Xs ) ) @ A5 ) ) ) ).
% butlast_subset
thf(fact_4569_sorted__butlast,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( butlast @ A @ Xs ) ) ) ) ) ).
% sorted_butlast
thf(fact_4570_Un__Pow__subset,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow @ A @ A5 ) @ ( pow @ A @ B5 ) ) @ ( pow @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% Un_Pow_subset
thf(fact_4571_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( bot_bot @ ( set @ B ) ) )
= ( nil @ B ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_empty
thf(fact_4572_Pow__insert,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( pow @ A @ ( insert @ A @ A2 @ A5 ) )
= ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow @ A @ A5 ) @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ A2 ) @ ( pow @ A @ A5 ) ) ) ) ).
% Pow_insert
thf(fact_4573_UN__Pow__subset,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A5: set @ B] :
( ord_less_eq @ ( set @ ( set @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image @ B @ ( set @ ( set @ A ) )
@ ^ [X4: B] : ( pow @ A @ ( B5 @ X4 ) )
@ A5 ) )
@ ( pow @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% UN_Pow_subset
thf(fact_4574_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( strict9044650504122735259up_add @ B ) )
=> ! [Xs: list @ A,F3: A > B,G3: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ B @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F3 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G3 @ Xs ) ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_4575_last__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( last @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( product_Pair @ A @ B @ ( last @ A @ Xs ) @ ( last @ B @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_4576_Fpow__Pow__finite,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A6: set @ A] : ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow @ A @ A6 ) @ ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) ) ) ) ) ).
% Fpow_Pow_finite
thf(fact_4577_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F3: B > A,A5: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ A5 ) @ B5 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F3 ) @ ( pow @ B @ A5 ) ) @ ( pow @ A @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_4578_binomial__def,axiom,
( binomial
= ( ^ [N6: nat,K3: nat] :
( finite_card @ ( set @ nat )
@ ( collect @ ( set @ nat )
@ ^ [K5: set @ nat] :
( ( member @ ( set @ nat ) @ K5 @ ( pow @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N6 ) ) )
& ( ( finite_card @ nat @ K5 )
= K3 ) ) ) ) ) ) ).
% binomial_def
thf(fact_4579_take__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_4580_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S3 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 )
= ( nil @ B ) )
= ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
thf(fact_4581_times__assn__raw_Osimps,axiom,
! [P2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H2: heap_ext @ product_unit,As3: set @ nat] :
( ( times_assn_raw @ P2 @ Q2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As3 ) )
= ( ? [As1: set @ nat,As22: set @ nat] :
( ( As3
= ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As1 ) )
& ( Q2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As22 ) ) ) ) ) ).
% times_assn_raw.simps
thf(fact_4582_times__assn__raw_Oelims_I1_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( times_assn_raw @ X @ Xa2 @ Xb )
= Y )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( Y
= ( ~ ? [As1: set @ nat,As22: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As1 ) )
& ( Xa2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As22 ) ) ) ) ) ) ) ).
% times_assn_raw.elims(1)
thf(fact_4583_times__assn__raw_Oelims_I2_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( times_assn_raw @ X @ Xa2 @ Xb )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ~ ? [As13: set @ nat,As24: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As13 @ As24 ) )
& ( ( inf_inf @ ( set @ nat ) @ As13 @ As24 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As13 ) )
& ( Xa2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As24 ) ) ) ) ) ).
% times_assn_raw.elims(2)
thf(fact_4584_times__assn__raw_Opelims_I1_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( times_assn_raw @ X @ Xa2 @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa2 @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( ( Y
= ( ? [As1: set @ nat,As22: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As1 ) )
& ( Xa2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As22 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(1)
thf(fact_4585_times__assn__raw_Opelims_I2_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( times_assn_raw @ X @ Xa2 @ Xb )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa2 @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) )
=> ~ ? [As13: set @ nat,As24: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As13 @ As24 ) )
& ( ( inf_inf @ ( set @ nat ) @ As13 @ As24 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As13 ) )
& ( Xa2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As24 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(2)
thf(fact_4586_times__assn__raw_Opelims_I3_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( times_assn_raw @ X @ Xa2 @ Xb )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa2 @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) )
=> ? [As12: set @ nat,As23: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As12 @ As23 ) )
& ( ( inf_inf @ ( set @ nat ) @ As12 @ As23 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As12 ) )
& ( Xa2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As23 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(3)
thf(fact_4587_concat__inth,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs @ ( append @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
= X ) ).
% concat_inth
thf(fact_4588_op__conc__empty__img__id,axiom,
! [A: $tType,L5: set @ ( list @ A )] :
( ( image @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ ( nil @ A ) ) @ L5 )
= L5 ) ).
% op_conc_empty_img_id
thf(fact_4589_empty__append__eq__id,axiom,
! [A: $tType] :
( ( append @ A @ ( nil @ A ) )
= ( ^ [X4: list @ A] : X4 ) ) ).
% empty_append_eq_id
thf(fact_4590_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ ( suc @ K ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ K ) ) @ ( cons @ nat @ K @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_4591_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ ( suc @ K ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ K ) ) @ ( cons @ nat @ ( suc @ K ) @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_4592_list__e__eq__lel_I2_J,axiom,
! [A: $tType,L12: list @ A,E5: A,L23: list @ A,E3: A] :
( ( ( append @ A @ L12 @ ( cons @ A @ E5 @ L23 ) )
= ( cons @ A @ E3 @ ( nil @ A ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E5 = E3 )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(2)
thf(fact_4593_list__e__eq__lel_I1_J,axiom,
! [A: $tType,E3: A,L12: list @ A,E5: A,L23: list @ A] :
( ( ( cons @ A @ E3 @ ( nil @ A ) )
= ( append @ A @ L12 @ ( cons @ A @ E5 @ L23 ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E5 = E3 )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(1)
thf(fact_4594_list__se__match_I4_J,axiom,
! [A: $tType,L23: list @ A,A2: A,L12: list @ A] :
( ( L23
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A2 @ ( nil @ A ) )
= ( append @ A @ L12 @ L23 ) )
= ( ( L12
= ( nil @ A ) )
& ( L23
= ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(4)
thf(fact_4595_list__se__match_I3_J,axiom,
! [A: $tType,L12: list @ A,A2: A,L23: list @ A] :
( ( L12
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A2 @ ( nil @ A ) )
= ( append @ A @ L12 @ L23 ) )
= ( ( L12
= ( cons @ A @ A2 @ ( nil @ A ) ) )
& ( L23
= ( nil @ A ) ) ) ) ) ).
% list_se_match(3)
thf(fact_4596_list__se__match_I2_J,axiom,
! [A: $tType,L23: list @ A,L12: list @ A,A2: A] :
( ( L23
!= ( nil @ A ) )
=> ( ( ( append @ A @ L12 @ L23 )
= ( cons @ A @ A2 @ ( nil @ A ) ) )
= ( ( L12
= ( nil @ A ) )
& ( L23
= ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(2)
thf(fact_4597_list__se__match_I1_J,axiom,
! [A: $tType,L12: list @ A,L23: list @ A,A2: A] :
( ( L12
!= ( nil @ A ) )
=> ( ( ( append @ A @ L12 @ L23 )
= ( cons @ A @ A2 @ ( nil @ A ) ) )
= ( ( L12
= ( cons @ A @ A2 @ ( nil @ A ) ) )
& ( L23
= ( nil @ A ) ) ) ) ) ).
% list_se_match(1)
thf(fact_4598_list__ee__eq__leel_I2_J,axiom,
! [A: $tType,L12: list @ A,E12: A,E23: A,L23: list @ A,E1: A,E22: A] :
( ( ( append @ A @ L12 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L23 ) ) )
= ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_4599_list__ee__eq__leel_I1_J,axiom,
! [A: $tType,E1: A,E22: A,L12: list @ A,E12: A,E23: A,L23: list @ A] :
( ( ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) )
= ( append @ A @ L12 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L23 ) ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_4600_nth__append__first,axiom,
! [A: $tType,I: nat,L: list @ A,L3: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( nth @ A @ ( append @ A @ L @ L3 ) @ I )
= ( nth @ A @ L @ I ) ) ) ).
% nth_append_first
thf(fact_4601_distinct__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( append @ A @ Xs @ Ys ) )
= ( ( distinct @ A @ Xs )
& ( distinct @ A @ Ys )
& ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% distinct_append
thf(fact_4602_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,M: A > ( option @ B ),X: A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ Ys )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M @ Xs @ Ys ) @ X @ ( some @ B @ ( nth @ B @ Ys @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% map_upds_append1
thf(fact_4603_foldl__conc__empty__eq,axiom,
! [A: $tType,I: list @ A,Ww: list @ ( list @ A )] :
( ( foldl @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ I @ Ww )
= ( append @ A @ I @ ( foldl @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ ( nil @ A ) @ Ww ) ) ) ).
% foldl_conc_empty_eq
thf(fact_4604_merge__list_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) )] :
( ( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
=> ( ! [L2: list @ A] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L2 @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [La: list @ A,Acc22: list @ ( list @ A )] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( nil @ ( list @ A ) ) ) )
=> ( ! [La: list @ A,Acc22: list @ ( list @ A ),L2: list @ A] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L2 @ ( nil @ ( list @ A ) ) ) ) )
=> ~ ! [Acc22: list @ ( list @ A ),L1: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc22 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) ) ) ) ) ) ) ).
% merge_list.cases
thf(fact_4605_list__match__lel__lel,axiom,
! [A: $tType,C12: list @ A,Qs: A,C23: list @ A,C13: list @ A,Qs2: A,C24: list @ A] :
( ( ( append @ A @ C12 @ ( cons @ A @ Qs @ C23 ) )
= ( append @ A @ C13 @ ( cons @ A @ Qs2 @ C24 ) ) )
=> ( ! [C21: list @ A] :
( ( C12
= ( append @ A @ C13 @ ( cons @ A @ Qs2 @ C21 ) ) )
=> ( C24
!= ( append @ A @ C21 @ ( cons @ A @ Qs @ C23 ) ) ) )
=> ( ( ( C13 = C12 )
=> ( ( Qs2 = Qs )
=> ( C24 != C23 ) ) )
=> ~ ! [C212: list @ A] :
( ( C13
= ( append @ A @ C12 @ ( cons @ A @ Qs @ C212 ) ) )
=> ( C23
!= ( append @ A @ C212 @ ( cons @ A @ Qs2 @ C24 ) ) ) ) ) ) ) ).
% list_match_lel_lel
thf(fact_4606_append__eq__mapE,axiom,
! [B: $tType,A: $tType,Fl: list @ A,Fl2: list @ A,F3: B > A,Ls: list @ B] :
( ( ( append @ A @ Fl @ Fl2 )
= ( map @ B @ A @ F3 @ Ls ) )
=> ~ ! [L2: list @ B,L6: list @ B] :
( ( Ls
= ( append @ B @ L2 @ L6 ) )
=> ( ( ( map @ B @ A @ F3 @ L2 )
= Fl )
=> ( ( map @ B @ A @ F3 @ L6 )
!= Fl2 ) ) ) ) ).
% append_eq_mapE
thf(fact_4607_map__eq__appendE,axiom,
! [B: $tType,A: $tType,F3: B > A,Ls: list @ B,Fl: list @ A,Fl2: list @ A] :
( ( ( map @ B @ A @ F3 @ Ls )
= ( append @ A @ Fl @ Fl2 ) )
=> ~ ! [L2: list @ B,L6: list @ B] :
( ( Ls
= ( append @ B @ L2 @ L6 ) )
=> ( ( ( map @ B @ A @ F3 @ L2 )
= Fl )
=> ( ( map @ B @ A @ F3 @ L6 )
!= Fl2 ) ) ) ) ).
% map_eq_appendE
thf(fact_4608_Misc_Oappend__eq__map__conv,axiom,
! [A: $tType,B: $tType,Fl: list @ A,Fl2: list @ A,F3: B > A,Ls: list @ B] :
( ( ( append @ A @ Fl @ Fl2 )
= ( map @ B @ A @ F3 @ Ls ) )
= ( ? [L4: list @ B,L7: list @ B] :
( ( Ls
= ( append @ B @ L4 @ L7 ) )
& ( ( map @ B @ A @ F3 @ L4 )
= Fl )
& ( ( map @ B @ A @ F3 @ L7 )
= Fl2 ) ) ) ) ).
% Misc.append_eq_map_conv
thf(fact_4609_Misc_Omap__eq__append__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Ls: list @ B,Fl: list @ A,Fl2: list @ A] :
( ( ( map @ B @ A @ F3 @ Ls )
= ( append @ A @ Fl @ Fl2 ) )
= ( ? [L4: list @ B,L7: list @ B] :
( ( Ls
= ( append @ B @ L4 @ L7 ) )
& ( ( map @ B @ A @ F3 @ L4 )
= Fl )
& ( ( map @ B @ A @ F3 @ L7 )
= Fl2 ) ) ) ) ).
% Misc.map_eq_append_conv
thf(fact_4610_neq__Nil__revE,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ~ ! [Ll2: list @ A,E2: A] :
( L
!= ( append @ A @ Ll2 @ ( cons @ A @ E2 @ ( nil @ A ) ) ) ) ) ).
% neq_Nil_revE
thf(fact_4611_rev__induct2_H,axiom,
! [A: $tType,B: $tType,P2: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs3: list @ A] : ( P2 @ ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( nil @ B ) )
=> ( ! [Y3: B,Ys5: list @ B] : ( P2 @ ( nil @ A ) @ ( append @ B @ Ys5 @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys5: list @ B] :
( ( P2 @ Xs3 @ Ys5 )
=> ( P2 @ ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( append @ B @ Ys5 @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% rev_induct2'
thf(fact_4612_neq__Nil__rev__conv,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
= ( ? [Xs2: list @ A,X4: A] :
( L
= ( append @ A @ Xs2 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) ) ).
% neq_Nil_rev_conv
thf(fact_4613_rev__nonempty__induct2_H,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P2: ( list @ A ) > ( list @ B ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ! [X3: A,Y3: B] : ( P2 @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( cons @ B @ Y3 @ ( nil @ B ) ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B] :
( ( Xs3
!= ( nil @ A ) )
=> ( P2 @ ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) )
=> ( ! [X3: A,Y3: B,Ys5: list @ B] :
( ( Ys5
!= ( nil @ B ) )
=> ( P2 @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( append @ B @ Ys5 @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys5: list @ B] :
( ( P2 @ Xs3 @ Ys5 )
=> ( ( Xs3
!= ( nil @ A ) )
=> ( ( Ys5
!= ( nil @ B ) )
=> ( P2 @ ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( append @ B @ Ys5 @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
thf(fact_4614_list__Cons__eq__append__cases,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X @ Xs ) ) )
=> ~ ! [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys4 ) )
=> ( ( append @ A @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_Cons_eq_append_cases
thf(fact_4615_list__append__eq__Cons__cases,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X @ Xs ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X @ Xs ) ) )
=> ~ ! [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys4 ) )
=> ( ( append @ A @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_append_eq_Cons_cases
thf(fact_4616_xy__in__set__cases,axiom,
! [A: $tType,X: A,L: list @ A,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ L ) )
=> ( ( member @ A @ Y @ ( set2 @ A @ L ) )
=> ( ( ( X = Y )
=> ! [L1: list @ A,L22: list @ A] :
( L
!= ( append @ A @ L1 @ ( cons @ A @ Y @ L22 ) ) ) )
=> ( ( ( X != Y )
=> ! [L1: list @ A,L22: list @ A,L32: list @ A] :
( L
!= ( append @ A @ L1 @ ( cons @ A @ X @ ( append @ A @ L22 @ ( cons @ A @ Y @ L32 ) ) ) ) ) )
=> ~ ( ( X != Y )
=> ! [L1: list @ A,L22: list @ A,L32: list @ A] :
( L
!= ( append @ A @ L1 @ ( cons @ A @ Y @ ( append @ A @ L22 @ ( cons @ A @ X @ L32 ) ) ) ) ) ) ) ) ) ) ).
% xy_in_set_cases
thf(fact_4617_in__set__list__format,axiom,
! [A: $tType,E3: A,L: list @ A] :
( ( member @ A @ E3 @ ( set2 @ A @ L ) )
=> ~ ! [L1: list @ A,L22: list @ A] :
( L
!= ( append @ A @ L1 @ ( cons @ A @ E3 @ L22 ) ) ) ) ).
% in_set_list_format
thf(fact_4618_list__rest__coinc,axiom,
! [A: $tType,S22: list @ A,S1: list @ A,R1: list @ A,R22: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ S22 ) @ ( size_size @ ( list @ A ) @ S1 ) )
=> ( ( ( append @ A @ S1 @ R1 )
= ( append @ A @ S22 @ R22 ) )
=> ? [R1p: list @ A] :
( R22
= ( append @ A @ R1p @ R1 ) ) ) ) ).
% list_rest_coinc
thf(fact_4619_map__consI_I2_J,axiom,
! [B: $tType,A: $tType,W2: list @ A,L: list @ A,F3: B > A,Ww: list @ B,A2: B] :
( ( ( append @ A @ W2 @ L )
= ( append @ A @ ( map @ B @ A @ F3 @ Ww ) @ L ) )
=> ( ( cons @ A @ ( F3 @ A2 ) @ ( append @ A @ W2 @ L ) )
= ( append @ A @ ( map @ B @ A @ F3 @ ( cons @ B @ A2 @ Ww ) ) @ L ) ) ) ).
% map_consI(2)
thf(fact_4620_set__union__code,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( set2 @ A @ ( append @ A @ Xs @ Ys ) ) ) ).
% set_union_code
thf(fact_4621_distinct__match,axiom,
! [A: $tType,Al: list @ A,E3: A,Bl: list @ A,Al2: list @ A,Bl2: list @ A] :
( ( distinct @ A @ ( append @ A @ Al @ ( cons @ A @ E3 @ Bl ) ) )
=> ( ( ( append @ A @ Al @ ( cons @ A @ E3 @ Bl ) )
= ( append @ A @ Al2 @ ( cons @ A @ E3 @ Bl2 ) ) )
= ( ( Al = Al2 )
& ( Bl = Bl2 ) ) ) ) ).
% distinct_match
thf(fact_4622_sorted__append,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ Ys ) )
= ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X4 @ Y5 ) ) ) ) ) ) ).
% sorted_append
thf(fact_4623_list__update__append1,axiom,
! [A: $tType,I: nat,Xs: list @ A,Ys: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ I @ X )
= ( append @ A @ ( list_update @ A @ Xs @ I @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_4624_butlast__eq__consE,axiom,
! [A: $tType,L: list @ A,X: A,Xs: list @ A] :
( ( ( butlast @ A @ L )
= ( cons @ A @ X @ Xs ) )
=> ~ ! [Xl: A] :
( L
!= ( cons @ A @ X @ ( append @ A @ Xs @ ( cons @ A @ Xl @ ( nil @ A ) ) ) ) ) ) ).
% butlast_eq_consE
thf(fact_4625_butlast__eq__cons__conv,axiom,
! [A: $tType,L: list @ A,X: A,Xs: list @ A] :
( ( ( butlast @ A @ L )
= ( cons @ A @ X @ Xs ) )
= ( ? [Xl2: A] :
( L
= ( cons @ A @ X @ ( append @ A @ Xs @ ( cons @ A @ Xl2 @ ( nil @ A ) ) ) ) ) ) ) ).
% butlast_eq_cons_conv
thf(fact_4626_foldl__rule__aux__P,axiom,
! [Sigma2: $tType,A: $tType,I6: Sigma2 > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F3: Sigma2 > A > Sigma2,P2: Sigma2 > $o] :
( ( I6 @ Sigma_0 @ L0 )
=> ( ! [L1: list @ A,L22: list @ A,X3: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L1 @ ( cons @ A @ X3 @ L22 ) ) )
=> ( ( I6 @ Sigma @ ( cons @ A @ X3 @ L22 ) )
=> ( I6 @ ( F3 @ Sigma @ X3 ) @ L22 ) ) )
=> ( ! [Sigma: Sigma2] :
( ( I6 @ Sigma @ ( nil @ A ) )
=> ( P2 @ Sigma ) )
=> ( P2 @ ( foldl @ Sigma2 @ A @ F3 @ Sigma_0 @ L0 ) ) ) ) ) ).
% foldl_rule_aux_P
thf(fact_4627_foldl__rule__aux,axiom,
! [Sigma2: $tType,A: $tType,I6: Sigma2 > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F3: Sigma2 > A > Sigma2] :
( ( I6 @ Sigma_0 @ L0 )
=> ( ! [L1: list @ A,L22: list @ A,X3: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L1 @ ( cons @ A @ X3 @ L22 ) ) )
=> ( ( I6 @ Sigma @ ( cons @ A @ X3 @ L22 ) )
=> ( I6 @ ( F3 @ Sigma @ X3 ) @ L22 ) ) )
=> ( I6 @ ( foldl @ Sigma2 @ A @ F3 @ Sigma_0 @ L0 ) @ ( nil @ A ) ) ) ) ).
% foldl_rule_aux
thf(fact_4628_foldl__rule__P,axiom,
! [Sigma2: $tType,A: $tType,I6: Sigma2 > ( list @ A ) > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F3: Sigma2 > A > Sigma2,P2: Sigma2 > $o] :
( ( I6 @ Sigma_0 @ ( nil @ A ) @ L0 )
=> ( ! [L1: list @ A,L22: list @ A,X3: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L1 @ ( cons @ A @ X3 @ L22 ) ) )
=> ( ( I6 @ Sigma @ L1 @ ( cons @ A @ X3 @ L22 ) )
=> ( I6 @ ( F3 @ Sigma @ X3 ) @ ( append @ A @ L1 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ L22 ) ) )
=> ( ! [Sigma: Sigma2] :
( ( I6 @ Sigma @ L0 @ ( nil @ A ) )
=> ( P2 @ Sigma ) )
=> ( P2 @ ( foldl @ Sigma2 @ A @ F3 @ Sigma_0 @ L0 ) ) ) ) ) ).
% foldl_rule_P
thf(fact_4629_foldl__rule,axiom,
! [Sigma2: $tType,A: $tType,I6: Sigma2 > ( list @ A ) > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F3: Sigma2 > A > Sigma2] :
( ( I6 @ Sigma_0 @ ( nil @ A ) @ L0 )
=> ( ! [L1: list @ A,L22: list @ A,X3: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L1 @ ( cons @ A @ X3 @ L22 ) ) )
=> ( ( I6 @ Sigma @ L1 @ ( cons @ A @ X3 @ L22 ) )
=> ( I6 @ ( F3 @ Sigma @ X3 ) @ ( append @ A @ L1 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ L22 ) ) )
=> ( I6 @ ( foldl @ Sigma2 @ A @ F3 @ Sigma_0 @ L0 ) @ L0 @ ( nil @ A ) ) ) ) ).
% foldl_rule
thf(fact_4630_lex__append__leftD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_4631_lex__append__left__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R2 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_4632_map__by__foldl,axiom,
! [B: $tType,A: $tType,F3: A > B,L: list @ A] :
( ( foldl @ ( list @ B ) @ A
@ ^ [L4: list @ B,X4: A] : ( append @ B @ L4 @ ( cons @ B @ ( F3 @ X4 ) @ ( nil @ B ) ) )
@ ( nil @ B )
@ L )
= ( map @ A @ B @ F3 @ L ) ) ).
% map_by_foldl
thf(fact_4633_length__append__singleton,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_append_singleton
thf(fact_4634_length__Suc__conv__rev,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y5: A,Ys3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ Y5 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_4635_length__Suc__rev__conv,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Ys3: list @ A,Y5: A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ Y5 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_rev_conv
thf(fact_4636_length__compl__rev__induct,axiom,
! [A: $tType,P2: ( list @ A ) > $o,L: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [L2: list @ A,E2: A] :
( ! [Ll: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L2 ) )
=> ( P2 @ Ll ) )
=> ( P2 @ ( append @ A @ L2 @ ( cons @ A @ E2 @ ( nil @ A ) ) ) ) )
=> ( P2 @ L ) ) ) ).
% length_compl_rev_induct
thf(fact_4637_nth__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
= ( nth @ A @ Xs @ N ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
= ( nth @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_4638_not__distinct__split__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ~ ( distinct @ A @ Xs )
=> ~ ! [Y3: A,Ys5: list @ A] :
( ( distinct @ A @ Ys5 )
=> ( ( member @ A @ Y3 @ ( set2 @ A @ Ys5 ) )
=> ! [Zs2: list @ A] :
( Xs
!= ( append @ A @ Ys5 @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_split_distinct
thf(fact_4639_replicate__Suc__conv__snoc,axiom,
! [A: $tType,N: nat,X: A] :
( ( replicate @ A @ ( suc @ N ) @ X )
= ( append @ A @ ( replicate @ A @ N @ X ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% replicate_Suc_conv_snoc
thf(fact_4640_list__update__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A,X: A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N @ X )
= ( append @ A @ ( list_update @ A @ Xs @ N @ X ) @ Ys ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N @ X )
= ( append @ A @ Xs @ ( list_update @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_4641_comm__append__is__replicate,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ A ) )
=> ( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Ys @ Xs ) )
=> ? [N2: nat,Zs2: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N2 @ Zs2 ) )
= ( append @ A @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_4642_listrel1E,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ~ ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 )
=> ! [Us: list @ A,Vs: list @ A] :
( ( Xs
= ( append @ A @ Us @ ( cons @ A @ X3 @ Vs ) ) )
=> ( Ys
!= ( append @ A @ Us @ ( cons @ A @ Y3 @ Vs ) ) ) ) ) ) ).
% listrel1E
thf(fact_4643_listrel1I,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Us2: list @ A,Vs2: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( ( Xs
= ( append @ A @ Us2 @ ( cons @ A @ X @ Vs2 ) ) )
=> ( ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ Y @ Vs2 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_4644_snoc__eq__iff__butlast_H,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,X: A] :
( ( Ys
= ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= ( ( Ys
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys )
= Xs )
& ( ( last @ A @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast'
thf(fact_4645_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys: list @ B] :
( ( product @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys )
= ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys ) @ ( product @ A @ B @ Xs @ Ys ) ) ) ).
% product.simps(2)
thf(fact_4646_slice__prepend,axiom,
! [A: $tType,I: nat,K: nat,Xs: list @ A,Ys: list @ A] :
( ( ord_less_eq @ nat @ I @ K )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( slice @ A @ I @ K @ Xs )
= ( slice @ A @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( plus_plus @ nat @ K @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( append @ A @ Ys @ Xs ) ) ) ) ) ).
% slice_prepend
thf(fact_4647_sorted__append__bigger,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Y: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X3 @ Y ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_append_bigger
thf(fact_4648_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_4649_sorted__insort__is__snoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X3 @ A2 ) )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ A2
@ Xs )
= ( append @ A @ Xs @ ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_insort_is_snoc
thf(fact_4650_take__Suc__conv__app__nth,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( take @ A @ ( suc @ I ) @ Xs )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( nil @ A ) ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_4651_take__update__last,axiom,
! [A: $tType,N: nat,List: list @ A,X: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ List ) )
=> ( ( list_update @ A @ ( take @ A @ ( suc @ N ) @ List ) @ N @ X )
= ( append @ A @ ( take @ A @ N @ List ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% take_update_last
thf(fact_4652_nth__repl,axiom,
! [A: $tType,M: nat,Xs: list @ A,N: nat,X: A] :
( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( M != N )
=> ( ( nth @ A @ ( append @ A @ ( take @ A @ N @ Xs ) @ ( append @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ ( drop @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) @ M )
= ( nth @ A @ Xs @ M ) ) ) ) ) ).
% nth_repl
thf(fact_4653_pos__n__replace,axiom,
! [A: $tType,N: nat,Xs: list @ A,Y: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ ( append @ A @ ( take @ A @ N @ Xs ) @ ( append @ A @ ( cons @ A @ Y @ ( nil @ A ) ) @ ( drop @ A @ ( suc @ N ) @ Xs ) ) ) ) ) ) ).
% pos_n_replace
thf(fact_4654_upto_Opelims,axiom,
! [X: int,Xa2: int,Y: list @ int] :
( ( ( upto @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X @ Xa2 )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa2 )
=> ( Y
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa2 ) ) ) ) ) ).
% upto.pelims
thf(fact_4655_drop0,axiom,
! [A: $tType] :
( ( drop @ A @ ( zero_zero @ nat ) )
= ( ^ [X4: list @ A] : X4 ) ) ).
% drop0
thf(fact_4656_drop__Suc__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( drop @ A @ ( suc @ N ) @ ( cons @ A @ X @ Xs ) )
= ( drop @ A @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_4657_drop__update__cancel,axiom,
! [A: $tType,N: nat,M: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ N @ M )
=> ( ( drop @ A @ M @ ( list_update @ A @ Xs @ N @ X ) )
= ( drop @ A @ M @ Xs ) ) ) ).
% drop_update_cancel
thf(fact_4658_drop__upd__irrelevant,axiom,
! [A: $tType,M: nat,N: nat,L: list @ A,X: A] :
( ( ord_less @ nat @ M @ N )
=> ( ( drop @ A @ N @ ( list_update @ A @ L @ M @ X ) )
= ( drop @ A @ N @ L ) ) ) ).
% drop_upd_irrelevant
thf(fact_4659_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less @ int @ J @ I )
=> ( ( upto @ I @ J )
= ( nil @ int ) ) ) ).
% upto_empty
thf(fact_4660_upto__Nil2,axiom,
! [I: int,J: int] :
( ( ( nil @ int )
= ( upto @ I @ J ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil2
thf(fact_4661_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= ( nil @ int ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil
thf(fact_4662_drop__all,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
=> ( ( drop @ A @ N @ Xs )
= ( nil @ A ) ) ) ).
% drop_all
thf(fact_4663_drop__eq__Nil,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( drop @ A @ N @ Xs )
= ( nil @ A ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_4664_drop__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( nil @ A )
= ( drop @ A @ N @ Xs ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_4665_last__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( last @ A @ ( drop @ A @ N @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_drop
thf(fact_4666_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_4667_nth__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A,I: nat] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( drop @ A @ N @ Xs ) @ I )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_4668_drop__last__conv,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ( ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L )
= ( cons @ A @ ( last @ A @ L ) @ ( nil @ A ) ) ) ) ).
% drop_last_conv
thf(fact_4669_sorted__wrt__upto,axiom,
! [I: int,J: int] : ( sorted_wrt @ int @ ( ord_less @ int ) @ ( upto @ I @ J ) ) ).
% sorted_wrt_upto
thf(fact_4670_drop__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( drop @ A @ ( zero_zero @ nat ) @ Xs )
= Xs ) ).
% drop_0
thf(fact_4671_sorted__upto,axiom,
! [M: int,N: int] : ( sorted_wrt @ int @ ( ord_less_eq @ int ) @ ( upto @ M @ N ) ) ).
% sorted_upto
thf(fact_4672_drop__eq__ConsD,axiom,
! [A: $tType,N: nat,Xs: list @ A,X: A,Xs4: list @ A] :
( ( ( drop @ A @ N @ Xs )
= ( cons @ A @ X @ Xs4 ) )
=> ( ( drop @ A @ ( suc @ N ) @ Xs )
= Xs4 ) ) ).
% drop_eq_ConsD
thf(fact_4673_set__drop__subset,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ N @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_drop_subset
thf(fact_4674_sorted__drop,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( drop @ A @ N @ Xs ) ) ) ) ).
% sorted_drop
thf(fact_4675_set__drop__subset__set__drop,axiom,
! [A: $tType,N: nat,M: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ M @ Xs ) ) @ ( set2 @ A @ ( drop @ A @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_4676_drop__update__swap,axiom,
! [A: $tType,M: nat,N: nat,Xs: list @ A,X: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( drop @ A @ M @ ( list_update @ A @ Xs @ N @ X ) )
= ( list_update @ A @ ( drop @ A @ M @ Xs ) @ ( minus_minus @ nat @ N @ M ) @ X ) ) ) ).
% drop_update_swap
thf(fact_4677_drop__take__drop__unsplit,axiom,
! [A: $tType,I: nat,J: nat,L: list @ A] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( append @ A @ ( drop @ A @ I @ ( take @ A @ J @ L ) ) @ ( drop @ A @ J @ L ) )
= ( drop @ A @ I @ L ) ) ) ).
% drop_take_drop_unsplit
thf(fact_4678_Misc_Oslice__def,axiom,
! [A: $tType] :
( ( slice @ A )
= ( ^ [From2: nat,To2: nat,List2: list @ A] : ( take @ A @ ( minus_minus @ nat @ To2 @ From2 ) @ ( drop @ A @ From2 @ List2 ) ) ) ) ).
% Misc.slice_def
thf(fact_4679_Union__take__drop__id,axiom,
! [A: $tType,N: nat,L: list @ ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ N @ L ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( take @ ( set @ A ) @ N @ L ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ L ) ) ) ).
% Union_take_drop_id
thf(fact_4680_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_4681_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_4682_drop__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ Xs ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_4683_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_4684_Cons__nth__drop__Suc,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) )
= ( drop @ A @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_4685_in__set__drop__conv__nth,axiom,
! [A: $tType,X: A,N: nat,L: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( drop @ A @ N @ L ) ) )
= ( ? [I3: nat] :
( ( ord_less_eq @ nat @ N @ I3 )
& ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) )
& ( X
= ( nth @ A @ L @ I3 ) ) ) ) ) ).
% in_set_drop_conv_nth
thf(fact_4686_upto_Osimps,axiom,
( upto
= ( ^ [I3: int,J3: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I3 @ J3 ) @ ( cons @ int @ I3 @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J3 ) ) @ ( nil @ int ) ) ) ) ).
% upto.simps
thf(fact_4687_upto_Oelims,axiom,
! [X: int,Xa2: int,Y: list @ int] :
( ( ( upto @ X @ Xa2 )
= Y )
=> ( ( ( ord_less_eq @ int @ X @ Xa2 )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa2 )
=> ( Y
= ( nil @ int ) ) ) ) ) ).
% upto.elims
thf(fact_4688_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_4689_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_4690_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs2: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Vs2 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ I @ Vs2 ) ) @ ( set2 @ A @ ( drop @ A @ J @ Vs2 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_take_disj_set_drop_if_distinct
thf(fact_4691_foldl__list__update,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,F3: B > A > B,A2: B,X: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( foldl @ B @ A @ F3 @ A2 @ ( list_update @ A @ Xs @ N @ X ) )
= ( foldl @ B @ A @ F3 @ ( F3 @ ( foldl @ B @ A @ F3 @ A2 @ ( take @ A @ N @ Xs ) ) @ X ) @ ( drop @ A @ ( suc @ N ) @ Xs ) ) ) ) ).
% foldl_list_update
thf(fact_4692_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_4693_id__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( Xs
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_4694_upd__conv__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A,A2: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ Xs @ I @ A2 )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ A2 @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_4695_Un__set__drop__extend,axiom,
! [A: $tType,J: nat,L: list @ ( set @ A )] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ ( set @ A ) ) @ L ) )
=> ( ( sup_sup @ ( set @ A ) @ ( nth @ ( set @ A ) @ L @ ( minus_minus @ nat @ J @ ( suc @ ( zero_zero @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ J @ L ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ ( minus_minus @ nat @ J @ ( suc @ ( zero_zero @ nat ) ) ) @ L ) ) ) ) ) ) ).
% Un_set_drop_extend
thf(fact_4696_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_4697_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_4698_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_4699_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_4700_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_4701_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_4702_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_4703_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_4704_max__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(3)
thf(fact_4705_max__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( zero_zero @ A ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(4)
thf(fact_4706_max__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V3: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V3 ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V3 ) )
= ( numeral_numeral @ A @ V3 ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V3 ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V3 ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(1)
thf(fact_4707_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) )
= A2 )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_4708_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( A2
= ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) )
= B2 ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_4709_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_4710_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_4711_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A2 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_4712_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_4713_max__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V3: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(2)
thf(fact_4714_max__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V3: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V3 ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V3 ) )
= ( numeral_numeral @ A @ V3 ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V3 ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V3 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(3)
thf(fact_4715_max__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V3: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(4)
thf(fact_4716_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( zero_zero @ nat ) ) ).
% nat_neg_numeral
thf(fact_4717_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( A2
= ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= B2 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_4718_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= A2 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_4719_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_4720_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_4721_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_4722_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_4723_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_4724_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_4725_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_4726_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_4727_numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V3 ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ V3 ) @ X ) ) ) ).
% numeral_le_floor
thf(fact_4728_floor__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V3 ) )
= ( ord_less @ A @ X @ ( numeral_numeral @ A @ V3 ) ) ) ) ).
% floor_less_numeral
thf(fact_4729_ceiling__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V3 ) )
= ( ord_less_eq @ A @ X @ ( numeral_numeral @ A @ V3 ) ) ) ) ).
% ceiling_le_numeral
thf(fact_4730_numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V3 ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( numeral_numeral @ A @ V3 ) @ X ) ) ) ).
% numeral_less_ceiling
thf(fact_4731_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I: num,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_4732_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_4733_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I: num,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_4734_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_4735_numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V3 ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V3 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_less_floor
thf(fact_4736_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V3 ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( numeral_numeral @ A @ V3 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_4737_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V3 ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( numeral_numeral @ A @ V3 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_4738_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V3 ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V3 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_le_ceiling
thf(fact_4739_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_4740_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_4741_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_4742_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_4743_neg__numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V3 ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) @ X ) ) ) ).
% neg_numeral_le_floor
thf(fact_4744_floor__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V3 ) ) )
= ( ord_less @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_4745_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V3 ) ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_4746_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V3 ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) @ X ) ) ) ).
% neg_numeral_less_ceiling
thf(fact_4747_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_4748_neg__numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V3 ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_less_floor
thf(fact_4749_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V3 ) ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_4750_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V3 ) ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_4751_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V3 ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_4752_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_4753_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A2 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_4754_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_4755_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A2 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_4756_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_4757_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_4758_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_4759_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_4760_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_4761_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_4762_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_4763_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_4764_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_4765_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_4766_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_4767_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_4768_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_4769_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_4770_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_4771_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_4772_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_4773_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B2: A,C3: A] :
( ( ( numeral_numeral @ A @ W2 )
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 )
= B2 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_4774_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,W2: num] :
( ( ( divide_divide @ A @ B2 @ C3 )
= ( numeral_numeral @ A @ W2 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_4775_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_4776_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_4777_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_4778_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_4779_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_4780_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_4781_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_4782_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_4783_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_4784_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_4785_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C3: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_4786_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_4787_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,W2: num] :
( ( ( divide_divide @ A @ B2 @ C3 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_4788_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B2: A,C3: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 )
= B2 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_4789_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_4790_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_4791_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_4792_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_4793_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_4794_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_4795_distinct__concat_H,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ ( list @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( nil @ A ) )
@ Xs ) )
=> ( ! [Ys5: list @ A] :
( ( member @ ( list @ A ) @ Ys5 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys5 ) )
=> ( ! [Ys5: list @ A,Zs2: list @ A] :
( ( member @ ( list @ A ) @ Ys5 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys5 != Zs2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys5 ) @ ( set2 @ A @ Zs2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat'
thf(fact_4796_merge__list__correct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Ls: list @ ( list @ A ),As3: list @ ( list @ A )] :
( ! [L2: list @ A] :
( ( member @ ( list @ A ) @ L2 @ ( set2 @ ( list @ A ) @ Ls ) )
=> ( ( distinct @ A @ L2 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L2 ) ) )
=> ( ! [L2: list @ A] :
( ( member @ ( list @ A ) @ L2 @ ( set2 @ ( list @ A ) @ As3 ) )
=> ( ( distinct @ A @ L2 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L2 ) ) )
=> ( ( distinct @ A @ ( merge_list @ A @ As3 @ Ls ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( merge_list @ A @ As3 @ Ls ) )
& ( ( set2 @ A @ ( merge_list @ A @ As3 @ Ls ) )
= ( set2 @ A @ ( concat @ A @ ( append @ ( list @ A ) @ As3 @ Ls ) ) ) ) ) ) ) ) ).
% merge_list_correct
thf(fact_4797_card__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( binomial @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% card_disjoint_shuffles
thf(fact_4798_word__gt__0__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: num] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y ) )
= ( ( zero_zero @ ( word @ A ) )
!= ( numeral_numeral @ ( word @ A ) @ Y ) ) ) ) ).
% word_gt_0_no
thf(fact_4799_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_4800_numeral__less__real__of__nat__iff,axiom,
! [W2: num,N: nat] :
( ( ord_less @ real @ ( numeral_numeral @ real @ W2 ) @ ( semiring_1_of_nat @ real @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ W2 ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_4801_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W2: num] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( numeral_numeral @ real @ W2 ) )
= ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ W2 ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_4802_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_4803_word__le__sub1__numberof,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: num] :
( ( ( numeral_numeral @ ( word @ A ) @ W2 )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ W2 ) )
= ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W2 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_le_sub1_numberof
thf(fact_4804_word__less__sub1__numberof,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: num] :
( ( ( numeral_numeral @ ( word @ A ) @ W2 )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ W2 ) )
= ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W2 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_less_sub1_numberof
thf(fact_4805_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_4806_powr__numeral,axiom,
! [X: real,N: num] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( numeral_numeral @ real @ N ) )
= ( power_power @ real @ X @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% powr_numeral
thf(fact_4807_nat__less__numeral__power__cancel__iff,axiom,
! [A2: int,X: num,N: nat] :
( ( ord_less @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_4808_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N: nat,A2: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_4809_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N: nat,A2: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_4810_nat__le__numeral__power__cancel__iff,axiom,
! [A2: int,X: num,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_4811_filter__shuffles__disjoint2_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_4812_filter__shuffles__disjoint2_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_4813_filter__shuffles__disjoint1_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_4814_filter__shuffles__disjoint1_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_4815_length__filter__le,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_filter_le
thf(fact_4816_filter__is__subset,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( filter2 @ A @ P2 @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% filter_is_subset
thf(fact_4817_sorted__filter_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,P2: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( filter2 @ A @ P2 @ L ) ) ) ) ).
% sorted_filter'
thf(fact_4818_last__filter,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ( P2 @ ( last @ A @ Xs ) )
=> ( ( last @ A @ ( filter2 @ A @ P2 @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last_filter
thf(fact_4819_sorted__same,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [G3: ( list @ A ) > A,Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( filter2 @ A
@ ^ [X4: A] :
( X4
= ( G3 @ Xs ) )
@ Xs ) ) ) ).
% sorted_same
thf(fact_4820_concat__filter__neq__Nil,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( concat @ A
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( nil @ A ) )
@ Xs ) )
= ( concat @ A @ Xs ) ) ).
% concat_filter_neq_Nil
thf(fact_4821_Cons__shuffles__subset2,axiom,
! [A: $tType,Y: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ).
% Cons_shuffles_subset2
thf(fact_4822_Cons__shuffles__subset1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys ) ) ).
% Cons_shuffles_subset1
thf(fact_4823_length__filter__less,axiom,
! [A: $tType,X: A,Xs: list @ A,P2: A > $o] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P2 @ X )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_filter_less
thf(fact_4824_sorted__filter,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,P2: B > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P2 @ Xs ) ) ) ) ) ).
% sorted_filter
thf(fact_4825_merge__list_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ) ).
% merge_list.simps(1)
thf(fact_4826_merge__list_Osimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La2: list @ A,Acc23: list @ ( list @ A ),L: list @ A] :
( ( merge_list @ A @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) ) ) ) ).
% merge_list.simps(4)
thf(fact_4827_merge__list_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La2: list @ A,Acc23: list @ ( list @ A )] :
( ( merge_list @ A @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( nil @ ( list @ A ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) ) ) ).
% merge_list.simps(3)
thf(fact_4828_merge__list_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
= L ) ) ).
% merge_list.simps(2)
thf(fact_4829_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,G3: ( list @ B ) > A,Xs: list @ B] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( map @ B @ A @ F3
@ ( filter2 @ B
@ ^ [X4: B] :
( ( F3 @ X4 )
= ( G3 @ Xs ) )
@ Xs ) ) ) ) ).
% sorted_map_same
thf(fact_4830_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [F3: B > A,P2: B > $o,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P2 @ Xs ) ) )
= ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P2 @ X4 ) @ ( F3 @ X4 ) @ ( zero_zero @ A ) )
@ Xs ) ) ) ) ).
% sum_list_map_filter'
thf(fact_4831_sum__list__filter__le__nat,axiom,
! [A: $tType,F3: A > nat,P2: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ ( filter2 @ A @ P2 @ Xs ) ) ) @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs ) ) ) ).
% sum_list_filter_le_nat
thf(fact_4832_filter__eq__snocD,axiom,
! [A: $tType,P2: A > $o,L: list @ A,L3: list @ A,X: A] :
( ( ( filter2 @ A @ P2 @ L )
= ( append @ A @ L3 @ ( cons @ A @ X @ ( nil @ A ) ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ L ) )
& ( P2 @ X ) ) ) ).
% filter_eq_snocD
thf(fact_4833_filter__conv__foldr,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P5: A > $o,Xs2: list @ A] :
( foldr @ A @ ( list @ A )
@ ^ [X4: A,Xt: list @ A] : ( if @ ( list @ A ) @ ( P5 @ X4 ) @ ( cons @ A @ X4 @ Xt ) @ Xt )
@ Xs2
@ ( nil @ A ) ) ) ) ).
% filter_conv_foldr
thf(fact_4834_powr__neg__numeral,axiom,
! [X: real,N: num] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ N ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% powr_neg_numeral
thf(fact_4835_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B,P2: B > $o,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
=> ( ~ ( P2 @ X3 )
=> ( ( F3 @ X3 )
= ( zero_zero @ A ) ) ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P2 @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ) ) ).
% sum_list_map_filter
thf(fact_4836_filter__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,P2: B > $o,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( P2 @ X )
=> ( ( filter2 @ B @ P2 @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= ( linorder_insort_key @ B @ A @ F3 @ X @ ( filter2 @ B @ P2 @ Xs ) ) ) ) ) ) ).
% filter_insort
thf(fact_4837_set__minus__filter__out,axiom,
! [A: $tType,Xs: list @ A,Y: A] :
( ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X4: A] : X4 != Y
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_4838_length__filter__conv__card,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P3 @ Xs ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P3 @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_4839_distinct__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( distinct @ A @ Ys )
=> ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( distinct @ A @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_4840_filter__nth__ex__nth,axiom,
! [A: $tType,N: nat,P2: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P2 @ Xs ) ) )
=> ? [M3: nat] :
( ( ord_less_eq @ nat @ N @ M3 )
& ( ord_less @ nat @ M3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ ( filter2 @ A @ P2 @ Xs ) @ N )
= ( nth @ A @ Xs @ M3 ) )
& ( ( filter2 @ A @ P2 @ ( take @ A @ M3 @ Xs ) )
= ( take @ A @ N @ ( filter2 @ A @ P2 @ Xs ) ) ) ) ) ).
% filter_nth_ex_nth
thf(fact_4841_transpose__aux__max,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Xss: list @ ( list @ B )] :
( ( ord_max @ nat @ ( suc @ ( size_size @ ( list @ A ) @ Xs ) )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [Xs2: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs2 ) )
@ Xss
@ ( zero_zero @ nat ) ) )
= ( suc
@ ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [X4: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X4 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( filter2 @ ( list @ B )
@ ^ [Ys3: list @ B] :
( Ys3
!= ( nil @ B ) )
@ Xss )
@ ( zero_zero @ nat ) ) ) ) ) ).
% transpose_aux_max
thf(fact_4842_mergesort__remdups__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( mergesort_remdups @ A )
= ( ^ [Xs2: list @ A] :
( merge_list @ A @ ( nil @ ( list @ A ) )
@ ( map @ A @ ( list @ A )
@ ^ [X4: A] : ( cons @ A @ X4 @ ( nil @ A ) )
@ Xs2 ) ) ) ) ) ).
% mergesort_remdups_def
thf(fact_4843_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_4844_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_4845_i0__less,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( N
!= ( zero_zero @ extended_enat ) ) ) ).
% i0_less
thf(fact_4846_enat__less__induct,axiom,
! [P2: extended_enat > $o,N: extended_enat] :
( ! [N2: extended_enat] :
( ! [M2: extended_enat] :
( ( ord_less @ extended_enat @ M2 @ N2 )
=> ( P2 @ M2 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ N ) ) ).
% enat_less_induct
thf(fact_4847_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_less @ extended_enat @ N @ ( zero_zero @ extended_enat ) ) ).
% not_iless0
thf(fact_4848_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_4849_add__diff__assoc__enat,axiom,
! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
( ( ord_less_eq @ extended_enat @ Z @ Y )
=> ( ( plus_plus @ extended_enat @ X @ ( minus_minus @ extended_enat @ Y @ Z ) )
= ( minus_minus @ extended_enat @ ( plus_plus @ extended_enat @ X @ Y ) @ Z ) ) ) ).
% add_diff_assoc_enat
thf(fact_4850_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_4851_i0__lb,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ).
% i0_lb
thf(fact_4852_insert_H__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_set_vebt @ ( vEBT_VEBT_insert @ T2 @ X ) )
= ( inf_inf @ ( set @ nat ) @ ( sup_sup @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% insert'_correct
thf(fact_4853_setceilmax,axiom,
! [S: vEBT_VEBT,M: nat,Listy: list @ vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ S @ M )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ Listy ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( M
= ( suc @ N ) )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ Listy ) )
=> ( ( semiring_1_of_nat @ int @ ( vEBT_VEBT_height @ X3 ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) )
=> ( ( ( semiring_1_of_nat @ int @ ( vEBT_VEBT_height @ S ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) )
=> ( ( semiring_1_of_nat @ int @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ S @ ( set2 @ vEBT_VEBT @ Listy ) ) ) ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ) ) ) ) ) ).
% setceilmax
thf(fact_4854_transpose__max__length,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( foldr @ ( list @ A ) @ nat
@ ^ [Xs2: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) )
@ ( transpose @ A @ Xs )
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [X4: list @ A] :
( X4
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_max_length
thf(fact_4855_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_4856_mulcomm,axiom,
! [I: nat,Va2: nat] :
( ( times_times @ nat @ I @ ( 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 ) ) ) ) ) )
= ( times_times @ nat @ ( 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 ) ) ) ) ) @ I ) ) ).
% mulcomm
thf(fact_4857_two__realpow__ge__two,axiom,
! [N: nat] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N ) )
=> ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% two_realpow_ge_two
thf(fact_4858_member__bound,axiom,
! [Tree: vEBT_VEBT,X: nat,N: nat] :
( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_invar_vebt @ Tree @ N )
=> ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% member_bound
thf(fact_4859_valid__pres__insert,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ T2 @ X ) @ N ) ) ) ).
% valid_pres_insert
thf(fact_4860_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_4861_insert__simp__mima,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi )
| ( X = Ma ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% insert_simp_mima
thf(fact_4862_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_4863_helpyd,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X )
= ( some @ nat @ Y ) )
=> ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% helpyd
thf(fact_4864_helpypredd,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X )
= ( some @ nat @ Y ) )
=> ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% helpypredd
thf(fact_4865_valid__insert__both__member__options__add,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ X ) @ X ) ) ) ).
% valid_insert_both_member_options_add
thf(fact_4866_valid__insert__both__member__options__pres,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ Y ) @ X ) ) ) ) ) ).
% valid_insert_both_member_options_pres
thf(fact_4867_post__member__pre__member,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T2 @ X ) @ Y )
=> ( ( vEBT_vebt_member @ T2 @ Y )
| ( X = Y ) ) ) ) ) ) ).
% post_member_pre_member
thf(fact_4868_log__ceil__idem,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ ( archimedean_ceiling @ real @ X ) ) ) ) ) ) ).
% log_ceil_idem
thf(fact_4869_count__buildup,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% count_buildup
thf(fact_4870_delt__out__of__range,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X @ Mi )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% delt_out_of_range
thf(fact_4871_del__single__cont,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi )
& ( X = Ma ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% del_single_cont
thf(fact_4872_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_4873_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_4874_set__n__deg__not__0,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,M: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ord_less_eq @ nat @ ( one_one @ nat ) @ N ) ) ) ).
% set_n_deg_not_0
thf(fact_4875_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less @ num @ M @ one2 ) ).
% semiring_norm(75)
thf(fact_4876_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% semiring_norm(68)
thf(fact_4877_tdeletemimi_H,axiom,
! [Deg: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Mi ) ) @ Deg @ TreeList @ Summary ) @ X ) @ ( one_one @ nat ) ) ) ).
% tdeletemimi'
thf(fact_4878_count__buildup_H,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( semiring_1_of_nat @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% count_buildup'
thf(fact_4879_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_4880_mi__ma__2__deg,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
& ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) ) ) ).
% mi_ma_2_deg
thf(fact_4881_succ__min,axiom,
! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less @ nat @ X @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( some @ nat @ Mi ) ) ) ) ).
% succ_min
thf(fact_4882_pred__max,axiom,
! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( some @ nat @ Ma ) ) ) ) ).
% pred_max
thf(fact_4883_cnt__bound_H,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ T2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( minus_minus @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ real ) ) ) ) ) ).
% cnt_bound'
thf(fact_4884_delete__bound__size__univ_H,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X: 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_V1232361888498592333_e_t_e @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ).
% delete_bound_size_univ'
thf(fact_4885_bit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H4: nat,L4: nat,D6: nat] : ( plus_plus @ nat @ ( times_times @ nat @ H4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ D6 ) ) @ L4 ) ) ) ).
% bit_concat_def
thf(fact_4886_TBOUND__vebt__memberi,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_TBOUND @ $o @ ( vEBT_V854960066525838166emberi @ T2 @ Ti @ X ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% TBOUND_vebt_memberi
thf(fact_4887_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_4888_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_4889_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% Suc_numeral
thf(fact_4890_insert__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( sup_sup @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) )
= ( vEBT_set_vebt @ ( vEBT_vebt_insert @ T2 @ X ) ) ) ) ) ).
% insert_correct
thf(fact_4891_set__vebt__insert,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_set_vebt @ ( vEBT_vebt_insert @ T2 @ X ) )
= ( sup_sup @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% set_vebt_insert
thf(fact_4892_insert__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( sup_sup @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) )
= ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_insert @ T2 @ X ) ) ) ) ) ).
% insert_corr
thf(fact_4893_zdiv__numeral__Bit0,axiom,
! [V3: num,W2: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit0 @ V3 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ V3 ) @ ( numeral_numeral @ int @ W2 ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_4894_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_4895_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% semiring_norm(69)
thf(fact_4896_mintlistlength,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( Mi != Ma )
=> ( ( ord_less @ nat @ Mi @ Ma )
& ? [M3: nat] :
( ( ( some @ nat @ M3 )
= ( vEBT_vebt_mint @ Summary ) )
& ( ord_less @ nat @ M3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% mintlistlength
thf(fact_4897_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_4898_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_4899_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_4900_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_4901_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_4902_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_4903_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_4904_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_4905_zmod__numeral__Bit0,axiom,
! [V3: num,W2: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit0 @ V3 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V3 ) @ ( numeral_numeral @ int @ W2 ) ) ) ) ).
% zmod_numeral_Bit0
thf(fact_4906_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_4907_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_4908_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_4909_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_4910_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_4911_power2__eq__iff__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( X = Y ) ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_4912_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_4913_sum__power2__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_4914_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_4915_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_4916_card__doubleton__eq__2__iff,axiom,
! [A: $tType,A2: A,B2: A] :
( ( ( finite_card @ A @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( A2 != B2 ) ) ).
% card_doubleton_eq_2_iff
thf(fact_4917_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_4918_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_4919_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_4920_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_4921_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_4922_one__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% one_less_floor
thf(fact_4923_floor__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% floor_le_one
thf(fact_4924_card__bit1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) )
= ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% card_bit1
thf(fact_4925_butlast__upd__last__eq,axiom,
! [A: $tType,L: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( list_update @ A @ ( butlast @ A @ L ) @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ X )
= ( append @ A @ ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ L ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% butlast_upd_last_eq
thf(fact_4926_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_4927_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_4928_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_4929_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_4930_p2__eq__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N )
= ( one_one @ ( word @ A ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% p2_eq_1
thf(fact_4931_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_4932_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_4933_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_4934_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_4935_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_4936_pos2,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ).
% pos2
thf(fact_4937_pos__mod__bound2,axiom,
! [A2: int] : ( ord_less @ int @ ( modulo_modulo @ int @ A2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ).
% pos_mod_bound2
thf(fact_4938_n__less__equal__power__2,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% n_less_equal_power_2
thf(fact_4939_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq @ num @ X @ one2 )
= ( X = one2 ) ) ).
% le_num_One_iff
thf(fact_4940_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_4941_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_4942_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_4943_field__less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ X @ ( divide_divide @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% field_less_half_sum
thf(fact_4944_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_4945_power2__eq__imp__eq,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( X = Y ) ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_4946_power2__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% power2_le_imp_le
thf(fact_4947_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_4948_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_4949_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_4950_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_4951_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_4952_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_4953_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_4954_abs__le__square__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( abs_abs @ A @ Y ) )
= ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_4955_nat__induct2,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ( P2 @ ( one_one @ nat ) )
=> ( ! [N2: nat] :
( ( P2 @ N2 )
=> ( P2 @ ( plus_plus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct2
thf(fact_4956_list__decomp__2,axiom,
! [A: $tType,L: list @ A] :
( ( ( size_size @ ( list @ A ) @ L )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ? [A4: A,B4: A] :
( L
= ( cons @ A @ A4 @ ( cons @ A @ B4 @ ( nil @ A ) ) ) ) ) ).
% list_decomp_2
thf(fact_4957_two__le__card,axiom,
! [A: $tType] :
( ( card2 @ A )
=> ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% two_le_card
thf(fact_4958_card__2__iff,axiom,
! [A: $tType,S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X4: A,Y5: A] :
( ( S3
= ( insert @ A @ X4 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X4 != Y5 ) ) ) ) ).
% card_2_iff
thf(fact_4959_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_4960_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_4961_realpow__square__minus__le,axiom,
! [U: real,X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( power_power @ real @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% realpow_square_minus_le
thf(fact_4962_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_4963_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_4964_pos__mod__sign2,axiom,
! [A2: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% pos_mod_sign2
thf(fact_4965_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_4966_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_4967_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_4968_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_4969_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_4970_sin__gt__zero__02,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_gt_zero_02
thf(fact_4971_mod__exp__less__eq__exp,axiom,
! [A2: int,N: nat] : ( ord_less @ int @ ( modulo_modulo @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% mod_exp_less_eq_exp
thf(fact_4972_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_4973_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_4974_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_4975_power2__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ X @ Y ) ) ) ) ).
% power2_less_imp_less
thf(fact_4976_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_4977_sum__power2__ge__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_4978_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_4979_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_power2_lt_zero
thf(fact_4980_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_4981_square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% square_le_1
thf(fact_4982_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_4983_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_4984_abs__sqrt__wlog,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P2: A > A > $o,X: A] :
( ! [X3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( P2 @ X3 @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P2 @ ( abs_abs @ A @ X ) @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_4985_power2__le__iff__abs__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ Y ) ) ) ) ).
% power2_le_iff_abs_le
thf(fact_4986_abs__square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% abs_square_le_1
thf(fact_4987_abs__square__less__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% abs_square_less_1
thf(fact_4988_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_4989_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_4990_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_4991_nat__bit__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( P2 @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( P2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
=> ( ! [N2: nat] :
( ( P2 @ N2 )
=> ( P2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_4992_two__pow__div__gt__le,axiom,
! [V3: nat,N: nat,M: nat] :
( ( ord_less @ nat @ V3 @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% two_pow_div_gt_le
thf(fact_4993_nat__add__offset__less,axiom,
! [Y: nat,N: nat,X: nat,M: nat,Sz: nat] :
( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( Sz
= ( plus_plus @ nat @ M @ N ) )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ Y ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Sz ) ) ) ) ) ).
% nat_add_offset_less
thf(fact_4994_nat__power__less__diff,axiom,
! [N: nat,Q3: nat,M: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Q3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ord_less @ nat @ Q3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% nat_power_less_diff
thf(fact_4995_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_4996_nat__le__power__trans,axiom,
! [N: nat,M: nat,K: nat] :
( ( ord_less_eq @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ K ) ) )
=> ( ( ord_less_eq @ nat @ K @ M )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% nat_le_power_trans
thf(fact_4997_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_4998_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_4999_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_5000_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_5001_word__less__two__pow__divD,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,M: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( divide_divide @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
& ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% word_less_two_pow_divD
thf(fact_5002_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_5003_exp__plus__inverse__exp,axiom,
! [X: real] : ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ ( exp @ real @ X ) @ ( inverse_inverse @ real @ ( exp @ real @ X ) ) ) ) ).
% exp_plus_inverse_exp
thf(fact_5004_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,
! [V3: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList @ Summary ) @ X )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
thf(fact_5005_triangle__def,axiom,
( nat_triangle
= ( ^ [N6: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N6 @ ( suc @ N6 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% triangle_def
thf(fact_5006_sum__squares__bound,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_squares_bound
thf(fact_5007_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_5008_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_5009_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_5010_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_5011_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_5012_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_5013_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_5014_less__two__pow__divD,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
& ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% less_two_pow_divD
thf(fact_5015_less__two__pow__divI,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less @ nat @ X @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% less_two_pow_divI
thf(fact_5016_nat__less__power__trans,axiom,
! [N: nat,M: nat,K: nat] :
( ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ K ) ) )
=> ( ( ord_less_eq @ nat @ K @ M )
=> ( ord_less @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% nat_less_power_trans
thf(fact_5017_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_5018_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_5019_L2__set__mult__ineq__lemma,axiom,
! [A2: real,C3: real,B2: real,D2: real] : ( ord_less_eq @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( times_times @ real @ A2 @ C3 ) ) @ ( times_times @ real @ B2 @ D2 ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ real @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ C3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_5020_plus__inverse__ge__2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ X @ ( inverse_inverse @ real @ X ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_5021_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_5022_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
@ ^ [Q8: nat] : ( ord_less @ nat @ Q8 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_5023_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_5024_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_5025_Suc__nat__number__of__add,axiom,
! [V3: num,N: nat] :
( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V3 ) @ N ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V3 @ one2 ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_5026_sum_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( 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
@ ^ [I3: nat] : ( plus_plus @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.in_pairs
thf(fact_5027_sum_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.in_pairs_0
thf(fact_5028_prod_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( 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
@ ^ [I3: nat] : ( times_times @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.in_pairs
thf(fact_5029_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.in_pairs_0
thf(fact_5030_card__Pow,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( set @ A ) @ ( pow @ A @ A5 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A5 ) ) ) ) ).
% card_Pow
thf(fact_5031_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ A @ X @ M ) )
| ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M ) @ M ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_5032_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_5033_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
! [Deg: nat,X: nat,Info: option @ ( product_prod @ nat @ nat ),TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) @ X )
=> ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) )
& ( ~ ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) @ X )
=> ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% VEBT_internal.insert'.simps(2)
thf(fact_5034_nat__div__eq__Suc__0__iff,axiom,
! [N: nat,M: nat] :
( ( ( divide_divide @ nat @ N @ M )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( ord_less_eq @ nat @ M @ N )
& ( ord_less @ nat @ N @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% nat_div_eq_Suc_0_iff
thf(fact_5035_power__2__mult__step__le,axiom,
! [N3: nat,N: nat,K7: nat,K: nat] :
( ( ord_less_eq @ nat @ N3 @ N )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ K7 ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ K ) )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( plus_plus @ nat @ K7 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ) ) ).
% power_2_mult_step_le
thf(fact_5036_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_5037_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_5038_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_5039_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_5040_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_5041_exp__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% exp_bound
thf(fact_5042_sb__dec__lem_H,axiom,
! [K: nat,A2: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) @ A2 )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ A2 ) ) ) ).
% sb_dec_lem'
thf(fact_5043_real__le__x__sinh,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ X @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X ) @ ( inverse_inverse @ real @ ( exp @ real @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_x_sinh
thf(fact_5044_real__le__abs__sinh,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( abs_abs @ real @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X ) @ ( inverse_inverse @ real @ ( exp @ real @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_abs_sinh
thf(fact_5045_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
@ ^ [Q8: nat] : ( ord_less @ nat @ Q8 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_5046_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% gauss_sum_nat
thf(fact_5047_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_5048_arith__geo__mean,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,X: A,Y: A] :
( ( ( power_power @ A @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ X @ Y ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ U @ ( divide_divide @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_5049_VEBT__internal_Oinsert_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_VEBT_insert @ X @ Xa2 )
= Y )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
!= ( vEBT_vebt_insert @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) @ Xa2 )
=> ( Y
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) @ Xa2 )
=> ( Y
= ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% VEBT_internal.insert'.elims
thf(fact_5050_card__UNIV__set,axiom,
! [A: $tType] :
( ( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
= ( zero_zero @ nat ) )
=> ( ( finite_card @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( zero_zero @ nat ) ) )
& ( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
!= ( zero_zero @ nat ) )
=> ( ( finite_card @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% card_UNIV_set
thf(fact_5051_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_5052_sb__inc__lem,axiom,
! [A2: int,K: nat] :
( ( ord_less @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ K ) ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ K ) ) ) ) ) ).
% sb_inc_lem
thf(fact_5053_sb__inc__lem_H,axiom,
! [A2: int,K: nat] :
( ( ord_less @ int @ A2 @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ K ) ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ K ) ) ) ) ) ).
% sb_inc_lem'
thf(fact_5054_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_5055_sb__dec__lem,axiom,
! [K: nat,A2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ A2 ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ A2 ) ) ) ).
% sb_dec_lem
thf(fact_5056_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_5057_ln__one__plus__pos__lower__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ X @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) ) ) ) ).
% ln_one_plus_pos_lower_bound
thf(fact_5058_real__exp__bound__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) ) ) ) ) ).
% real_exp_bound_lemma
thf(fact_5059_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_5060_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,D2: A,N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) ) ) ).
% double_arith_series
thf(fact_5061_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_5062_arith__series__nat,axiom,
! [A2: nat,D2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ A2 @ ( times_times @ nat @ I3 @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ nat @ N @ D2 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_5063_sinh__ln__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( sinh @ real @ ( ln_ln @ real @ X ) )
= ( divide_divide @ real @ ( minus_minus @ real @ X @ ( inverse_inverse @ real @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% sinh_ln_real
thf(fact_5064_pred__bound__size__univ_H,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X: 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 @ X ) ) @ ( 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_5065_succ__bound__size__univ_H,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X: 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 @ X ) ) @ ( 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_5066_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_5067_Sum__Icc__int,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X4: int] : X4
@ ( 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_5068_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_5069_exp__lower__Taylor__quadratic,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( divide_divide @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ X ) ) ) ).
% exp_lower_Taylor_quadratic
thf(fact_5070_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_5071_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_5072_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,D2: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_5073_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_5074_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_5075_invar__vebt_Ointros_I2_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_5076_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_5077_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_5078_binomial__code,axiom,
( binomial
= ( ^ [N6: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N6 @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N6 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N6 @ ( minus_minus @ nat @ N6 @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N6 @ K3 ) @ ( one_one @ nat ) ) @ N6 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_5079_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_5080_invar__vebt_Ointros_I3_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_5081_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_5082_arctan__double,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ X ) )
= ( arctan @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arctan_double
thf(fact_5083_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_5084_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_5085_ln__one__minus__pos__lower__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( uminus_uminus @ real @ X ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X ) ) ) ) ) ).
% ln_one_minus_pos_lower_bound
thf(fact_5086_abs__ln__one__plus__x__minus__x__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound
thf(fact_5087_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_5088_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_5089_lemma__termdiff2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [H2: A,Z: A,N: nat] :
( ( H2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ N ) @ ( power_power @ A @ Z @ N ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
= ( times_times @ A @ H2
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P7: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [Q8: nat] : ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ Q8 ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q8 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ P7 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_5090_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_5091_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_5092_length__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( foldr @ ( list @ A ) @ nat
@ ^ [Xs2: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) )
@ Xs
@ ( zero_zero @ nat ) ) ) ).
% length_transpose
thf(fact_5093_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_5094_lemma__termdiff3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [H2: A,Z: A,K4: real,N: nat] :
( ( H2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K4 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ Z @ H2 ) ) @ K4 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ N ) @ ( power_power @ A @ Z @ N ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) @ ( times_times @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( power_power @ real @ K4 @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_5095_nth__transpose,axiom,
! [A: $tType,I: nat,Xs: list @ ( list @ A )] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I )
= ( map @ ( list @ A ) @ A
@ ^ [Xs2: list @ A] : ( nth @ A @ Xs2 @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_5096_Tb__T__vebt__buildupi,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ ( vEBT_V441764108873111860ildupi @ N ) ) @ ( minus_minus @ int @ ( vEBT_VEBT_Tb @ N ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% Tb_T_vebt_buildupi
thf(fact_5097_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_5098_Tb__T__vebt__buildupi_H,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( vEBT_V9176841429113362141ildupi @ N ) @ ( minus_minus @ int @ ( vEBT_VEBT_Tb @ N ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% Tb_T_vebt_buildupi'
thf(fact_5099_htt__vebt__memberi__invar__vebt,axiom,
! [T2: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( time_htt @ $o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X )
@ ^ [R5: $o] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_member @ T2 @ X ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ) ) ).
% htt_vebt_memberi_invar_vebt
thf(fact_5100_Tb__T__vebt__buildupi_H_H,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( vEBT_V441764108873111860ildupi @ N ) @ ( minus_minus @ nat @ ( vEBT_VEBT_Tb2 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Tb_T_vebt_buildupi''
thf(fact_5101_htt__vebt__predi,axiom,
! [T2: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( time_htt @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_predi @ Ti @ X )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_pred @ T2 @ X ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ) ) ).
% htt_vebt_predi
thf(fact_5102_Tb__Tb_H,axiom,
( vEBT_VEBT_Tb
= ( ^ [T3: nat] : ( semiring_1_of_nat @ int @ ( vEBT_VEBT_Tb2 @ T3 ) ) ) ) ).
% Tb_Tb'
thf(fact_5103_T__vebt__buildupi__univ,axiom,
! [U: nat,N: nat] :
( ( U
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ nat @ ( vEBT_V441764108873111860ildupi @ N ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ U ) ) ) ).
% T_vebt_buildupi_univ
thf(fact_5104_tdeletemimi,axiom,
! [Deg: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Mi ) ) @ Deg @ TreeList @ Summary ) @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).
% tdeletemimi
thf(fact_5105_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_5106_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_5107_minNull__delete__time__bound,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T2 @ X ) )
=> ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% minNull_delete_time_bound
thf(fact_5108_Tb_H__cnt,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( vEBT_VEBT_Tb2 @ N ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_cnt2 @ ( vEBT_vebt_buildup @ N ) ) ) ) ).
% Tb'_cnt
thf(fact_5109_TBOUND__vebt__predi,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_TBOUND @ ( option @ nat ) @ ( vEBT_VEBT_vebt_predi @ T2 @ Ti @ X ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% TBOUND_vebt_predi
thf(fact_5110_TBOUND__vebt__succi,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_TBOUND @ ( option @ nat ) @ ( vEBT_VEBT_vebt_succi @ T2 @ Ti @ X ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% TBOUND_vebt_succi
thf(fact_5111_T__vebt__buildupi__cnt_H,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_V441764108873111860ildupi @ N ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) ) ) ).
% T_vebt_buildupi_cnt'
thf(fact_5112_TBOUND__buildupi,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( time_TBOUND @ vEBT_VEBTi @ ( vEBT_vebt_buildupi @ N ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% TBOUND_buildupi
thf(fact_5113_delete__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% delete_bound_height
thf(fact_5114_cnt__bound,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ T2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( minus_minus @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) @ ( divide_divide @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% cnt_bound
thf(fact_5115_TBOUND__vebt__inserti,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_TBOUND @ vEBT_VEBTi @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ T2 ) @ ( one_one @ nat ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ) ).
% TBOUND_vebt_inserti
thf(fact_5116_htt__vebt__buildupi__univ,axiom,
! [U: nat,N: nat] :
( ( U
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( time_htt @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ U ) ) ) ).
% htt_vebt_buildupi_univ
thf(fact_5117_htt__vebt__buildupi_H__univ,axiom,
! [U: nat,N: nat] :
( ( U
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( time_htt @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_V739175172307565963ildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ U ) ) ) ).
% htt_vebt_buildupi'_univ
thf(fact_5118_delete__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X: 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_d_e_l_e_t_e @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).
% delete_bound_size_univ
thf(fact_5119_time__vebt__delete,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% time_vebt_delete
thf(fact_5120_time__vebt__pred,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_p_r_e_d @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% time_vebt_pred
thf(fact_5121_time__vebt__succ,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_s_u_c_c @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% time_vebt_succ
thf(fact_5122_htt__vebt__inserti,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_htt @ vEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T2 @ X ) ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% htt_vebt_inserti
thf(fact_5123_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_5124_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_5125_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_5126_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit1 @ M ) @ one2 ) ).
% semiring_norm(70)
thf(fact_5127_TBOUND__vebt__deletei,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_TBOUND @ vEBT_VEBTi @ ( vEBT_V1365221501068881998eletei @ T2 @ Ti @ X ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T2 @ X ) ) @ ( one_one @ nat ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ) ).
% TBOUND_vebt_deletei
thf(fact_5128_htt__vebt__memberi,axiom,
! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] :
( time_htt @ $o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X )
@ ^ [R5: $o] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_member @ T2 @ X ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% htt_vebt_memberi
thf(fact_5129_zdiv__numeral__Bit1,axiom,
! [V3: num,W2: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit1 @ V3 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ V3 ) @ ( numeral_numeral @ int @ W2 ) ) ) ).
% zdiv_numeral_Bit1
thf(fact_5130_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_5131_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_5132_htt__vebt__inserti__invar__vebt,axiom,
! [T2: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( time_htt @ vEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T2 @ X ) ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ) ) ).
% htt_vebt_inserti_invar_vebt
thf(fact_5133_htt__vebt__deletei,axiom,
! [T2: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( time_htt @ vEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_deletei @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ T2 @ X ) ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ) ) ).
% htt_vebt_deletei
thf(fact_5134_htt__vebt__succi,axiom,
! [T2: vEBT_VEBT,N: nat,Ti: vEBT_VEBTi,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( time_htt @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_succi @ Ti @ X )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
@ ( pure_assn
@ ( R5
= ( vEBT_vebt_succ @ T2 @ X ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ) ) ).
% htt_vebt_succi
thf(fact_5135_Suc__div__eq__add3__div__numeral,axiom,
! [M: nat,V3: num] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V3 ) )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V3 ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_5136_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_5137_Suc__mod__eq__add3__mod__numeral,axiom,
! [M: nat,V3: num] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V3 ) )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V3 ) ) ) ).
% Suc_mod_eq_add3_mod_numeral
thf(fact_5138_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_5139_zmod__numeral__Bit1,axiom,
! [V3: num,W2: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit1 @ V3 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V3 ) @ ( numeral_numeral @ int @ W2 ) ) ) @ ( one_one @ int ) ) ) ).
% zmod_numeral_Bit1
thf(fact_5140_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
( ( vEBT_VEBT_Tb2 @ ( zero_zero @ nat ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).
% VEBT_internal.Tb'.simps(1)
thf(fact_5141_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
( ( vEBT_VEBT_Tb2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).
% VEBT_internal.Tb'.simps(2)
thf(fact_5142_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_5143_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_5144_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_5145_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_5146_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_5147_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_5148_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_5149_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_5150_VEBT__internal_OTb_Osimps_I1_J,axiom,
( ( vEBT_VEBT_Tb @ ( zero_zero @ nat ) )
= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) ) ).
% VEBT_internal.Tb.simps(1)
thf(fact_5151_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_5152_num_Osize_I6_J,axiom,
! [X33: num] :
( ( size_size @ num @ ( bit1 @ X33 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X33 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(6)
thf(fact_5153_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_5154_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_5155_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_5156_card__3__iff,axiom,
! [A: $tType,S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X4: A,Y5: A,Z2: A] :
( ( S3
= ( insert @ A @ X4 @ ( insert @ A @ Y5 @ ( insert @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X4 != Y5 )
& ( Y5 != Z2 )
& ( X4 != Z2 ) ) ) ) ).
% card_3_iff
thf(fact_5157_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_5158_exp__le,axiom,
ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ).
% exp_le
thf(fact_5159_VEBT__internal_OTb_Osimps_I2_J,axiom,
( ( vEBT_VEBT_Tb @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) ) ).
% VEBT_internal.Tb.simps(2)
thf(fact_5160_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_5161_insersimp,axiom,
! [T2: vEBT_VEBT,N: nat,Y: 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 @ Y ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ).
% insersimp
thf(fact_5162_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_5163_small__powers__of__2,axiom,
! [X: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ X )
=> ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ X @ ( one_one @ nat ) ) ) ) ) ).
% small_powers_of_2
thf(fact_5164_log__base__10__eq2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X )
= ( times_times @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ X ) ) ) ) ).
% log_base_10_eq2
thf(fact_5165_pred__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_p_r_e_d @ T2 @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% pred_bound_height
thf(fact_5166_insert__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% insert_bound_height
thf(fact_5167_succ__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_s_u_c_c @ T2 @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% succ_bound_height
thf(fact_5168_log__base__10__eq1,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( ln_ln @ real @ X ) ) ) ) ).
% log_base_10_eq1
thf(fact_5169_pred__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X: 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 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).
% pred_bound_size_univ
thf(fact_5170_succ__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X: 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 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).
% succ_bound_size_univ
thf(fact_5171_insert__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X: 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 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).
% insert_bound_size_univ
thf(fact_5172_VEBT__internal_Ovebt__buildupi__rule,axiom,
! [N: nat] : ( time_htt @ vEBT_VEBTi @ ( pure_assn @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% VEBT_internal.vebt_buildupi_rule
thf(fact_5173_vebt__buildupi__rule,axiom,
! [N: nat] : ( time_htt @ vEBT_VEBTi @ ( pure_assn @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_Intf_vebt_assn @ N @ ( bot_bot @ ( set @ nat ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% vebt_buildupi_rule
thf(fact_5174_vebt__memberi__rule,axiom,
! [N: nat,S: set @ nat,Ti: vEBT_VEBTi,X: nat] :
( time_htt @ $o @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X )
@ ^ [R5: $o] :
( times_times @ assn @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti )
@ ( pure_assn
@ ( R5
= ( member @ nat @ X @ S ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ) ).
% vebt_memberi_rule
thf(fact_5175_vebt__predi__rule,axiom,
! [N: nat,S: set @ nat,Ti: vEBT_VEBTi,X: nat,Y: nat] :
( time_htt @ ( option @ nat ) @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti ) @ ( vEBT_vebt_predi @ Ti @ X )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti )
@ ( pure_assn
@ ( ( R5
= ( some @ nat @ Y ) )
= ( vEBT_is_pred_in_set @ S @ X @ Y ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ) ).
% vebt_predi_rule
thf(fact_5176_vebt__succi__rule,axiom,
! [N: nat,S: set @ nat,Ti: vEBT_VEBTi,X: nat,Y: nat] :
( time_htt @ ( option @ nat ) @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti ) @ ( vEBT_vebt_succi @ Ti @ X )
@ ^ [R5: option @ nat] :
( times_times @ assn @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti )
@ ( pure_assn
@ ( ( R5
= ( some @ nat @ Y ) )
= ( vEBT_is_succ_in_set @ S @ X @ Y ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ) ).
% vebt_succi_rule
thf(fact_5177_vebt__inserti__rule,axiom,
! [X: nat,N: nat,S: set @ nat,Ti: vEBT_VEBTi] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( time_htt @ vEBT_VEBTi @ ( vEBT_Intf_vebt_assn @ N @ S @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_Intf_vebt_assn @ N @ ( sup_sup @ ( set @ nat ) @ S @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ) ) ).
% vebt_inserti_rule
thf(fact_5178_space__bound,axiom,
! [T2: vEBT_VEBT,N: nat,U: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_space @ T2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ U ) ) ) ) ).
% space_bound
thf(fact_5179_vebt__space__linear__bound,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_space @ T2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% vebt_space_linear_bound
thf(fact_5180_space__2__pow__bound,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_VEBT_space2 @ T2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( minus_minus @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ real ) ) ) ) ) ).
% space_2_pow_bound
thf(fact_5181_space__space_H,axiom,
! [T2: vEBT_VEBT] : ( ord_less @ nat @ ( vEBT_VEBT_space @ T2 ) @ ( vEBT_VEBT_space2 @ T2 ) ) ).
% space_space'
thf(fact_5182_space_H__bound,axiom,
! [T2: vEBT_VEBT,N: nat,U: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_space2 @ T2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ U ) ) ) ) ).
% space'_bound
thf(fact_5183_space__cnt,axiom,
! [T2: vEBT_VEBT] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_VEBT_space2 @ T2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_VEBT_cnt @ T2 ) ) ) ).
% space_cnt
thf(fact_5184_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_VEBT_space2 @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ).
% VEBT_internal.space'.simps(1)
thf(fact_5185_VEBT__internal_Ospace_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_VEBT_space @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).
% VEBT_internal.space.simps(1)
thf(fact_5186_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_VEBT_space2 @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_VEBT_space2 @ Summary ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_space2 @ TreeList ) @ ( zero_zero @ nat ) ) ) ) ).
% VEBT_internal.space'.simps(2)
thf(fact_5187_VEBT__internal_Ospace_H_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_space2 @ X )
= Y )
=> ( ( ? [A4: $o,B4: $o] :
( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_VEBT_space2 @ Summary2 ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_space2 @ TreeList2 ) @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% VEBT_internal.space'.elims
thf(fact_5188_VEBT__internal_Ospace_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_VEBT_space @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_space @ Summary ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_space @ TreeList ) @ ( zero_zero @ nat ) ) ) ) ).
% VEBT_internal.space.simps(2)
thf(fact_5189_VEBT__internal_Ospace_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_space @ X )
= Y )
=> ( ( ? [A4: $o,B4: $o] :
( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_space @ TreeList2 ) @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% VEBT_internal.space.elims
thf(fact_5190_t__build__cnt,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_V8646137997579335489_i_l_d @ N ) ) @ ( times_times @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ).
% t_build_cnt
thf(fact_5191_t__buildup__cnt,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_V8346862874174094_d_u_p @ N ) ) @ ( times_times @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ).
% t_buildup_cnt
thf(fact_5192_vebt__buildup__bound,axiom,
! [U: nat,N: nat] :
( ( U
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ nat @ ( vEBT_V8346862874174094_d_u_p @ N ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) @ U ) ) ) ).
% vebt_buildup_bound
thf(fact_5193_buildup__build__time,axiom,
! [N: nat] : ( ord_less @ nat @ ( vEBT_V8346862874174094_d_u_p @ N ) @ ( vEBT_V8646137997579335489_i_l_d @ N ) ) ).
% buildup_build_time
thf(fact_5194_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
( ( vEBT_V8346862874174094_d_u_p @ ( zero_zero @ nat ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
thf(fact_5195_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
( ( vEBT_V8646137997579335489_i_l_d @ ( zero_zero @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
thf(fact_5196_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
thf(fact_5197_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
thf(fact_5198_div__half__nat,axiom,
! [Y: nat,X: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ( ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ X @ Y ) @ ( modulo_modulo @ nat @ X @ Y ) )
= ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ Y @ ( minus_minus @ nat @ X @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) @ Y ) ) ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ X @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) @ Y ) ) @ Y ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) @ ( minus_minus @ nat @ X @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) @ Y ) ) ) ) ) ) ).
% div_half_nat
thf(fact_5199_VEBT__internal_Ospace_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_space @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel2 @ X )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_space @ TreeList2 ) @ ( zero_zero @ nat ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% VEBT_internal.space.pelims
thf(fact_5200_VEBT__internal_Ospace_H_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_space2 @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel @ X )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_VEBT_space2 @ Summary2 ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_space2 @ TreeList2 ) @ ( zero_zero @ nat ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% VEBT_internal.space'.pelims
thf(fact_5201_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_5202_VEBT__internal_Oinsert_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_VEBT_insert @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( vEBT_vebt_insert @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) @ Xa2 )
=> ( Y
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) @ Xa2 )
=> ( Y
= ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% VEBT_internal.insert'.pelims
thf(fact_5203_pred__list__to__short,axiom,
! [Deg: nat,X: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X @ Ma )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( none @ nat ) ) ) ) ) ).
% pred_list_to_short
thf(fact_5204_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X4: nat,N6: nat] : ( divide_divide @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) ) ) ) ).
% high_def
thf(fact_5205_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_5206_high__inv,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ X ) @ N )
= Y ) ) ).
% high_inv
thf(fact_5207_succ__list__to__short,axiom,
! [Deg: nat,Mi: nat,X: nat,TreeList: list @ vEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( none @ nat ) ) ) ) ) ).
% succ_list_to_short
thf(fact_5208_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or7035219750837199246ssThan @ code_integer @ L @ ( plus_plus @ code_integer @ U @ ( one_one @ code_integer ) ) )
= ( set_or1337092689740270186AtMost @ code_integer @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_5209_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_5210_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_5211_image__add__integer__atLeastLessThan,axiom,
! [L: code_integer,U: code_integer] :
( ( image @ code_integer @ code_integer
@ ^ [X4: code_integer] : ( plus_plus @ code_integer @ X4 @ L )
@ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ ( minus_minus @ code_integer @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ code_integer @ L @ U ) ) ).
% image_add_integer_atLeastLessThan
thf(fact_5212_uminus__integer__code_I1_J,axiom,
( ( uminus_uminus @ code_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% uminus_integer_code(1)
thf(fact_5213_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_5214_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_5215_finite__atLeastZeroLessThan__integer,axiom,
! [U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ U ) ) ).
% finite_atLeastZeroLessThan_integer
thf(fact_5216_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_5217_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_5218_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_5219_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_5220_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_5221_less__integer__code_I1_J,axiom,
~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(1)
thf(fact_5222_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_5223_nested__mint,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va2: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( N
= ( suc @ ( suc @ Va2 ) ) )
=> ( ~ ( ord_less @ nat @ Ma @ Mi )
=> ( ( Ma != Mi )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Va2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide @ nat @ Va2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ).
% nested_mint
thf(fact_5224_insert__simp__norm,axiom,
! [X: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less @ nat @ Mi @ X )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( ord_max @ nat @ X @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_norm
thf(fact_5225_insert__simp__excp,axiom,
! [Mi: nat,Deg: nat,TreeList: list @ vEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less @ nat @ X @ Mi )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ ( ord_max @ nat @ Mi @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_excp
thf(fact_5226_bit__split__inv,axiom,
! [X: nat,D2: nat] :
( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D2 ) @ ( vEBT_VEBT_low @ X @ D2 ) @ D2 )
= X ) ).
% bit_split_inv
thf(fact_5227_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X4: nat,N6: nat] : ( modulo_modulo @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) ) ) ) ).
% low_def
thf(fact_5228_low__inv,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ X ) @ N )
= X ) ) ).
% low_inv
thf(fact_5229_summaxma,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
=> ( ( Mi != Ma )
=> ( ( the2 @ nat @ ( vEBT_vebt_maxt @ Summary ) )
= ( vEBT_VEBT_high @ Ma @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% summaxma
thf(fact_5230_both__member__options__ding,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% both_member_options_ding
thf(fact_5231_finite__atLeastLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or7035219750837199246ssThan @ code_integer @ L @ U ) ) ).
% finite_atLeastLessThan_integer
thf(fact_5232_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
| ( X = Mi )
| ( X = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_5233_finite__atLeastAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or1337092689740270186AtMost @ code_integer @ L @ U ) ) ).
% finite_atLeastAtMost_integer
thf(fact_5234_member__inv,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
& ( ( X = Mi )
| ( X = Ma )
| ( ( ord_less @ nat @ X @ Ma )
& ( ord_less @ nat @ Mi @ X )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% member_inv
thf(fact_5235_both__member__options__from__chilf__to__complete__tree,axiom,
! [X: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_5236_del__x__not__mi__newnode__not__nil,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Newlist: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_not_mi_newnode_not_nil
thf(fact_5237_del__x__mi__lets__in__not__minNull,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( X = Mi )
& ( ord_less @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi_lets_in_not_minNull
thf(fact_5238_del__x__not__mi,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Newlist: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
= ( none @ nat ) )
@ Mi
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
& ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_not_mi
thf(fact_5239_del__x__not__mi__new__node__nil,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Sn
= ( vEBT_vebt_delete @ Summary @ H2 ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ Sn )
= ( none @ nat ) )
@ Mi
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_not_mi_new_node_nil
thf(fact_5240_del__x__not__mia,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
= ( none @ nat ) )
@ Mi
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
@ ( vEBT_vebt_delete @ Summary @ H2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).
% del_x_not_mia
thf(fact_5241_del__x__mia,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi )
& ( ord_less @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
@ ( if @ nat
@ ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
@ ( if @ nat
@ ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).
% del_x_mia
thf(fact_5242_del__x__mi__lets__in__minNull,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT,Sn: vEBT_VEBT] :
( ( ( X = Mi )
& ( ord_less @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Sn
= ( vEBT_vebt_delete @ Summary @ H2 ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Xn
@ ( if @ nat @ ( Xn = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ Sn )
= ( none @ nat ) )
@ Xn
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi_lets_in_minNull
thf(fact_5243_del__x__mi__lets__in,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( X = Mi )
& ( ord_less @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Xn
@ ( if @ nat @ ( Xn = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
= ( none @ nat ) )
@ Xn
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
& ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi_lets_in
thf(fact_5244_del__x__mi,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L: nat] :
( ( ( X = Mi )
& ( ord_less @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Xn
@ ( if @ nat @ ( Xn = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
= ( none @ nat ) )
@ Xn
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
@ ( vEBT_vebt_delete @ Summary @ H2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi
thf(fact_5245_del__in__range,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less_eq @ nat @ Mi @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi ) @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
@ ( if @ nat
@ ( ( ( X = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi )
=> ( X = Ma ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( X = Mi ) @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi ) @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
@ ( if @ nat
@ ( ( ( X = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi )
=> ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).
% del_in_range
thf(fact_5246_succ__less__length__list,axiom,
! [Deg: nat,Mi: nat,X: nat,TreeList: list @ vEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% succ_less_length_list
thf(fact_5247_succ__greatereq__min,axiom,
! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% succ_greatereq_min
thf(fact_5248_pred__lesseq__max,axiom,
! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X @ Ma )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% pred_lesseq_max
thf(fact_5249_pred__less__length__list,axiom,
! [Deg: nat,X: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X @ Ma )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% pred_less_length_list
thf(fact_5250_Max_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Max.infinite
thf(fact_5251_Min_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Min.infinite
thf(fact_5252_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_5253_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ( ord_less @ nat @ X @ Mi )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) )
& ( ~ ( ( ord_less @ nat @ X @ Mi )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( ( ( X = Mi )
& ( X = Ma ) )
=> ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) )
& ( ~ ( ( X = Mi )
& ( X = Ma ) )
=> ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( plus_plus @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
thf(fact_5254_vebt__succ_Osimps_I6_J,axiom,
! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( some @ nat @ Mi ) ) )
& ( ~ ( ord_less @ nat @ X @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% vebt_succ.simps(6)
thf(fact_5255_vebt__pred_Osimps_I7_J,axiom,
! [Ma: nat,X: nat,Mi: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( some @ nat @ Ma ) ) )
& ( ~ ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% vebt_pred.simps(7)
thf(fact_5256_vebt__delete_Osimps_I7_J,axiom,
! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ( ord_less @ nat @ X @ Mi )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) )
& ( ~ ( ( ord_less @ nat @ X @ Mi )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( ( ( X = Mi )
& ( X = Ma ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) )
& ( ~ ( ( X = Mi )
& ( X = Ma ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi ) @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
@ ( if @ nat
@ ( ( ( X = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi )
=> ( X = Ma ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( X = Mi ) @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ ( suc @ ( suc @ Va2 ) )
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi ) @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
@ ( if @ nat
@ ( ( ( X = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi )
=> ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ ( suc @ ( suc @ Va2 ) )
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) ) ) ) ) ) ).
% vebt_delete.simps(7)
thf(fact_5257_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList @ S ) @ X )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_5258_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma @ X ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
thf(fact_5259_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
thf(fact_5260_vebt__delete_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_delete @ X @ Xa2 )
= Y )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ B4 ) ) ) )
=> ( ! [A4: $o] :
( ? [B4: $o] :
( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( vEBT_Leaf @ A4 @ $false ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ? [N2: nat] :
( Xa2
= ( suc @ ( suc @ N2 ) ) )
=> ( Y
!= ( vEBT_Leaf @ A4 @ B4 ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TrLst2: list @ vEBT_VEBT,Smry2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Tr2: list @ vEBT_VEBT,Sm2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ( ord_less @ nat @ Xa2 @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa2 ) )
=> ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( ord_less @ nat @ Xa2 @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa2 ) )
=> ( ( ( ( Xa2 = Mi2 )
& ( Xa2 = Ma2 ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( Xa2 = Mi2 )
& ( Xa2 = Ma2 ) )
=> ( Y
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa2 = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa2 != Mi2 )
=> ( Xa2 = Ma2 ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( Xa2 = Mi2 ) @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa2 = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa2 != Mi2 )
=> ( Xa2 = Ma2 ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_delete.elims
thf(fact_5261_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_V1232361888498592333_e_t_e @ X @ Xa2 )
= Y )
=> ( ( ? [A4: $o,B4: $o] :
( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,B4: $o] :
( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,B4: $o] :
( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ? [N2: nat] :
( Xa2
= ( suc @ ( suc @ N2 ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ( ord_less @ nat @ Xa2 @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa2 ) )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ( ord_less @ nat @ Xa2 @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa2 ) )
=> ( ( ( ( Xa2 = Mi2 )
& ( Xa2 = Ma2 ) )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ( Xa2 = Mi2 )
& ( Xa2 = Ma2 ) )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
thf(fact_5262_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V3: nat,TreeList: list @ vEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList @ Vd2 ) @ X )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_5263_vebt__succ_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: option @ nat] :
( ( ( vEBT_vebt_succ @ X @ Xa2 )
= Y )
=> ( ! [Uu: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ B4 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ~ ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ( ( ? [Uv: $o,Uw: $o] :
( X
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ( ? [N2: nat] :
( Xa2
= ( suc @ N2 ) )
=> ( Y
!= ( none @ nat ) ) ) )
=> ( ( ? [Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc2 @ Vd ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( Y
= ( some @ nat @ Mi2 ) ) )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.elims
thf(fact_5264_vebt__pred_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: option @ nat] :
( ( ( vEBT_vebt_pred @ X @ Xa2 )
= Y )
=> ( ( ? [Uu: $o,Uv: $o] :
( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
!= ( none @ nat ) ) ) )
=> ( ! [A4: $o] :
( ? [Uw: $o] :
( X
= ( vEBT_Leaf @ A4 @ Uw ) )
=> ( ( Xa2
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( ( A4
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ? [Va: nat] :
( Xa2
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( ( A4
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( Y
= ( none @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uy: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Vd: list @ vEBT_VEBT,Ve: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd @ Ve ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( Y
= ( some @ nat @ Ma2 ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi2 @ Xa2 ) @ ( some @ nat @ Mi2 ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.elims
thf(fact_5265_vebt__member_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( ( X != Mi )
=> ( ( X != Ma )
=> ( ~ ( ord_less @ nat @ X @ Mi )
& ( ~ ( ord_less @ nat @ X @ Mi )
=> ( ~ ( ord_less @ nat @ Ma @ X )
& ( ~ ( ord_less @ nat @ Ma @ X )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_5266_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi: nat,Ma: nat,V3: nat,TreeList: list @ vEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ V3 ) @ TreeList @ Vc ) @ X )
= ( ( X = Mi )
| ( X = Ma )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_5267_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_T_p_r_e_d @ X @ Xa2 )
= Y )
=> ( ( ? [Uu: $o,Uv: $o] :
( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,Uw: $o] :
( X
= ( vEBT_Leaf @ A4 @ Uw ) )
=> ( ( Xa2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ? [Va: nat] :
( Xa2
= ( suc @ ( suc @ Va ) ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B4 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ) )
=> ( ( ? [Uy: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Vd: list @ vEBT_VEBT,Ve: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd @ Ve ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa2 ) @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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_5268_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_T_s_u_c_c @ X @ Xa2 )
= Y )
=> ( ( ? [Uu: $o,B4: $o] :
( X
= ( vEBT_Leaf @ Uu @ B4 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ( ( ? [Uv: $o,Uw: $o] :
( X
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ( ? [N2: nat] :
( Xa2
= ( suc @ N2 ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc2 @ Vd ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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_5269_vebt__member__code_I3_J,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( ~ ( ( Deg
= ( zero_zero @ nat ) )
| ( Deg
= ( suc @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( Deg
= ( zero_zero @ nat ) )
| ( Deg
= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ( X != Mi )
=> ( ( X != Ma )
=> ( ~ ( ord_less @ nat @ X @ Mi )
& ( ~ ( ord_less @ nat @ X @ Mi )
=> ( ~ ( ord_less @ nat @ Ma @ X )
& ( ~ ( ord_less @ nat @ Ma @ X )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member_code(3)
thf(fact_5270_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ X @ Mi )
| ( ord_less @ nat @ Ma @ X ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( X = Mi )
& ( X = Ma ) )
@ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_m_i_n_t @ Summary ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_d_e_l_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( X = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi )
=> ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( X = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi )
=> ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
thf(fact_5271_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
= Y )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
= ( ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) )
=> Y )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S4: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) )
=> ( Y
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_5272_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S4: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_5273_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
!= ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S4: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_5274_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_5275_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_T_d_e_l_e_t_e @ X @ Xa2 )
= Y )
=> ( ( ? [A4: $o,B4: $o] :
( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,B4: $o] :
( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,B4: $o] :
( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ? [N2: nat] :
( Xa2
= ( suc @ ( suc @ N2 ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ Xa2 @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa2 ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( Xa2 = Mi2 )
& ( Xa2 = Ma2 ) )
@ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa2 = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa2 != Mi2 )
=> ( Xa2 = 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 @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa2 = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa2 != Mi2 )
=> ( Xa2 = Ma2 ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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_5276_invar__vebt_Ointros_I4_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_5277_vebt__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X @ Xa2 )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_5278_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ! [Uu: $o,Uv: $o] :
( X
!= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_5279_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
= Y )
=> ( ( ? [Uu: $o,Uv: $o] :
( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> Y )
=> ( ( ? [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) )
=> Y )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( Y
= ( ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
=> ( Y
= ( ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) )
=> ~ ! [V: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd ) )
=> ( Y
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_5280_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( X = Mi ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( X = Ma ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma @ X ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi @ X )
& ( ord_less @ nat @ X @ Ma ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( zero_zero @ nat ) )
@ ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
thf(fact_5281_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X = Mi )
| ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
thf(fact_5282_vebt__insert_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X = Mi )
| ( X = Ma ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) ) ).
% vebt_insert.simps(5)
thf(fact_5283_invar__vebt_Ointros_I5_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_5284_vebt__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa2 )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ! [V: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> ( ! [V: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_5285_vebt__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa2 )
= Y )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
= ( ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> Y )
=> ( ( ? [V: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> Y )
=> ( ( ? [V: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> Y )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
= ( ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_5286_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_T_m_e_m_b_e_r2 @ X @ Xa2 )
= Y )
=> ( ( ? [A4: $o,B4: $o] :
( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( Xa2 = Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( Xa2 = Ma2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa2 ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi2 @ Xa2 )
& ( ord_less @ nat @ Xa2 @ Ma2 ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( zero_zero @ nat ) )
@ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
thf(fact_5287_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X = Mi )
| ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
thf(fact_5288_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_T_i_n_s_e_r_t2 @ X @ Xa2 )
= Y )
=> ( ( ? [A4: $o,B4: $o] :
( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S4 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S4 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( 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 @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( 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_5289_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A13: vEBT_VEBT,A24: nat] :
( ( ? [A8: $o,B8: $o] :
( A13
= ( vEBT_Leaf @ A8 @ B8 ) )
& ( A24
= ( suc @ ( zero_zero @ nat ) ) ) )
| ? [TreeList4: list @ vEBT_VEBT,N6: nat,Summary3: vEBT_VEBT] :
( ( A13
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A24 @ TreeList4 @ Summary3 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ( vEBT_invar_vebt @ X4 @ N6 ) )
& ( vEBT_invar_vebt @ Summary3 @ N6 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList4 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) )
& ( A24
= ( plus_plus @ nat @ N6 @ N6 ) )
& ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
| ? [TreeList4: list @ vEBT_VEBT,N6: nat,Summary3: vEBT_VEBT] :
( ( A13
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A24 @ TreeList4 @ Summary3 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ( vEBT_invar_vebt @ X4 @ N6 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N6 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList4 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N6 ) ) )
& ( A24
= ( plus_plus @ nat @ N6 @ ( suc @ N6 ) ) )
& ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
| ? [TreeList4: list @ vEBT_VEBT,N6: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A13
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A24 @ TreeList4 @ Summary3 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ( vEBT_invar_vebt @ X4 @ N6 ) )
& ( vEBT_invar_vebt @ Summary3 @ N6 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList4 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) )
& ( A24
= ( plus_plus @ nat @ N6 @ N6 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A24 ) )
& ( ( Mi3 != Ma3 )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N6 )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N6 ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N6 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N6 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) )
| ? [TreeList4: list @ vEBT_VEBT,N6: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A13
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A24 @ TreeList4 @ Summary3 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ( vEBT_invar_vebt @ X4 @ N6 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N6 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList4 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N6 ) ) )
& ( A24
= ( plus_plus @ nat @ N6 @ ( suc @ N6 ) ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N6 ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A24 ) )
& ( ( Mi3 != Ma3 )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N6 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N6 )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N6 ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N6 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N6 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_5290_invar__vebt_Ocases,axiom,
! [A12: vEBT_VEBT,A23: nat] :
( ( vEBT_invar_vebt @ A12 @ A23 )
=> ( ( ? [A4: $o,B4: $o] :
( A12
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( A23
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3 = N2 )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M3 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3
= ( suc @ N2 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M3 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3 = N2 )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M3 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3
= ( suc @ N2 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M3 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_5291_vebt__insert_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa2 )
= Y )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
= ( vEBT_Leaf @ $true @ B4 ) ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A4 @ $true ) ) )
& ( ( Xa2
!= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S4 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S4 ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S4 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S4 ) ) )
=> ( ! [V: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% vebt_insert.elims
thf(fact_5292_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
! [Ma: nat,X: nat,Mi: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
thf(fact_5293_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X @ Mi )
=> ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ X @ Mi )
=> ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
thf(fact_5294_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_T_i_n_s_e_r_t @ X @ Xa2 )
= Y )
=> ( ( ? [A4: $o,B4: $o] :
( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( Xa2
= ( 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,S4: vEBT_VEBT] :
( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S4 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S4 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( 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 @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( 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_5295_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_T_p_r_e_d2 @ X @ Xa2 )
= Y )
=> ( ( ? [Uu: $o,Uv: $o] :
( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,Uw: $o] :
( X
= ( vEBT_Leaf @ A4 @ Uw ) )
=> ( ( Xa2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A4: $o,B4: $o] :
( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ? [Va: nat] :
( Xa2
= ( suc @ ( suc @ Va ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Uy: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Vd: list @ vEBT_VEBT,Ve: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd @ Ve ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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_5296_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_T_s_u_c_c2 @ X @ Xa2 )
= Y )
=> ( ( ? [Uu: $o,B4: $o] :
( X
= ( vEBT_Leaf @ Uu @ B4 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Uv: $o,Uw: $o] :
( X
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ( ? [N2: nat] :
( Xa2
= ( suc @ N2 ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc2 @ Vd ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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_5297_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N6: nat,TreeList4: list @ vEBT_VEBT,X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ X4 @ N6 ) ) @ ( vEBT_VEBT_low @ X4 @ N6 ) ) ) ) ).
% in_children_def
thf(fact_5298_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_T_d_e_l_e_t_e @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ! [N2: nat] :
( ( Xa2
= ( suc @ ( suc @ N2 ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ Xa2 @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa2 ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( Xa2 = Mi2 )
& ( Xa2 = Ma2 ) )
@ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa2 = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa2 != Mi2 )
=> ( Xa2 = 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 @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa2 = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa2 != Mi2 )
=> ( Xa2 = Ma2 ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
thf(fact_5299_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_T_s_u_c_c @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ B4 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ! [N2: nat] :
( ( Xa2
= ( suc @ N2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) ) ) ) )
=> ( ! [Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc2 @ Vd ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc2 @ Vd ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
thf(fact_5300_vebt__succ_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: option @ nat] :
( ( ( vEBT_vebt_succ @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ B4 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( Y
= ( none @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ! [N2: nat] :
( ( Xa2
= ( suc @ N2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) ) ) ) )
=> ( ! [Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc2 @ Vd ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc2 @ Vd ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( Y
= ( some @ nat @ Mi2 ) ) )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.pelims
thf(fact_5301_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_T_p_r_e_d @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A4: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ Uw ) )
=> ( ( Xa2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ! [Va: nat] :
( ( Xa2
= ( suc @ ( suc @ Va ) ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B4 @ ( 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 @ A4 @ B4 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
=> ( ! [Uy: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vd: list @ vEBT_VEBT,Ve: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd @ Ve ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd @ Ve ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa2 ) @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
thf(fact_5302_vebt__pred_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: option @ nat] :
( ( ( vEBT_vebt_pred @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A4: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ Uw ) )
=> ( ( Xa2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( ( A4
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( Y
= ( none @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ! [Va: nat] :
( ( Xa2
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( ( A4
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A4
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
=> ( ! [Uy: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vd: list @ vEBT_VEBT,Ve: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd @ Ve ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd @ Ve ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( Y
= ( some @ nat @ Ma2 ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi2 @ Xa2 ) @ ( some @ nat @ Mi2 ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.pelims
thf(fact_5303_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_V1232361888498592333_e_t_e @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ! [N2: nat] :
( ( Xa2
= ( suc @ ( suc @ N2 ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ( ord_less @ nat @ Xa2 @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa2 ) )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ( ord_less @ nat @ Xa2 @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa2 ) )
=> ( ( ( ( Xa2 = Mi2 )
& ( Xa2 = Ma2 ) )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ( Xa2 = Mi2 )
& ( Xa2 = Ma2 ) )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
thf(fact_5304_vebt__delete_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_delete @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ B4 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Xa2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( vEBT_Leaf @ A4 @ $false ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ! [N2: nat] :
( ( Xa2
= ( suc @ ( suc @ N2 ) ) )
=> ( ( Y
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TrLst2: list @ vEBT_VEBT,Smry2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Tr2: list @ vEBT_VEBT,Sm2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ( ord_less @ nat @ Xa2 @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa2 ) )
=> ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( ord_less @ nat @ Xa2 @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa2 ) )
=> ( ( ( ( Xa2 = Mi2 )
& ( Xa2 = Ma2 ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( Xa2 = Mi2 )
& ( Xa2 = Ma2 ) )
=> ( Y
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa2 = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa2 != Mi2 )
=> ( Xa2 = Ma2 ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( Xa2 = Mi2 ) @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa2 = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa2 != Mi2 )
=> ( Xa2 = Ma2 ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_delete.pelims
thf(fact_5305_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_T_s_u_c_c2 @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ B4 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ Uv @ Uw ) )
=> ! [N2: nat] :
( ( Xa2
= ( suc @ N2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) ) ) ) )
=> ( ! [Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc2 @ Vd ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc2 @ Vd ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
thf(fact_5306_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_T_i_n_s_e_r_t @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( Xa2
= ( 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 @ A4 @ B4 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S4 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S4 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S4 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S4 ) @ Xa2 ) ) ) )
=> ( ! [V: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( 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 @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
thf(fact_5307_vebt__insert_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
= ( vEBT_Leaf @ $true @ B4 ) ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A4 @ $true ) ) )
& ( ( Xa2
!= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S4 ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S4 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S4 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S4 ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S4 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S4 ) @ Xa2 ) ) ) )
=> ( ! [V: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_insert.pelims
thf(fact_5308_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_T_p_r_e_d2 @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A4: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ Uw ) )
=> ( ( Xa2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ! [Va: nat] :
( ( Xa2
= ( suc @ ( suc @ Va ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
=> ( ! [Uy: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vd: list @ vEBT_VEBT,Ve: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd @ Ve ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd @ Ve ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
thf(fact_5309_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_T_i_n_s_e_r_t2 @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S4 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts @ S4 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S4 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S4 ) @ Xa2 ) ) ) )
=> ( ! [V: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( 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 @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
thf(fact_5310_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_T_m_e_m_b_e_r2 @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( Xa2 = Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( Xa2 = Ma2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa2 ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi2 @ Xa2 )
& ( ord_less @ nat @ Xa2 @ Ma2 ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( zero_zero @ nat ) )
@ ( zero_zero @ nat ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
thf(fact_5311_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_T_m_e_m_b_e_r @ X @ Xa2 )
= Y )
=> ( ( ? [A4: $o,B4: $o] :
( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( Xa2
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ? [V: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( Xa2 = Ma2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
thf(fact_5312_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,X: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
thf(fact_5313_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,
! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz ) @ X )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
thf(fact_5314_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,
! [V3: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
thf(fact_5315_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( X
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
thf(fact_5316_member__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_m_e_m_b_e_r @ T2 @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ).
% member_bound_height
thf(fact_5317_member__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X: 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 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).
% member_bound_size_univ
thf(fact_5318_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( X = Mi ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( X = Ma ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma @ X ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
thf(fact_5319_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( ( vEBT_T_m_e_m_b_e_r @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( Xa2
= ( 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 @ A4 @ B4 ) @ Xa2 ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( Xa2 = Ma2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
thf(fact_5320_vebt__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) @ Xa2 ) ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_5321_vebt__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ Xa2 ) ) )
=> ( ! [V: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) @ Xa2 ) ) )
=> ( ! [V: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) )
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_5322_vebt__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) )
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_5323_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Xa2 ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V: nat,TreeList2: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) @ Xa2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_5324_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V: nat,TreeList2: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) @ Xa2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_5325_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
=> ( ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Xa2 ) ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V: nat,TreeList2: list @ vEBT_VEBT,S4: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) )
=> ( ( Y
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S4 ) @ Xa2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_5326_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) ) ) )
=> ( ! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( Y
= ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
=> ( ( Y
= ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [V: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd ) )
=> ( ( Y
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_5327_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) ) )
=> ( ! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) @ Xa2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd ) @ Xa2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_5328_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList2 @ Vd ) @ Xa2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_5329_Maclaurin__sin__expansion3,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less @ real @ T7 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_5330_Maclaurin__sin__expansion4,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_sin_expansion4
thf(fact_5331_Maclaurin__sin__expansion2,axiom,
! [X: real,N: nat] :
? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_sin_expansion2
thf(fact_5332_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_5333_pi__gt__zero,axiom,
ord_less @ real @ ( zero_zero @ real ) @ pi ).
% pi_gt_zero
thf(fact_5334_pi__not__less__zero,axiom,
~ ( ord_less @ real @ pi @ ( zero_zero @ real ) ) ).
% pi_not_less_zero
thf(fact_5335_pi__ge__zero,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ pi ).
% pi_ge_zero
thf(fact_5336_sin__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ pi )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_gt_zero
thf(fact_5337_sin__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_ge_zero
thf(fact_5338_pi__less__4,axiom,
ord_less @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ).
% pi_less_4
thf(fact_5339_pi__ge__two,axiom,
ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ).
% pi_ge_two
thf(fact_5340_sin__eq__0__pi,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( ord_less @ real @ X @ pi )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ) ).
% sin_eq_0_pi
thf(fact_5341_sin__zero__pi__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ pi )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% sin_zero_pi_iff
thf(fact_5342_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_5343_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_5344_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_5345_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_5346_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_5347_arctan__ubound,axiom,
! [Y: real] : ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arctan_ubound
thf(fact_5348_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_5349_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_5350_sin__gt__zero2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_gt_zero2
thf(fact_5351_sin__lt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ pi @ X )
=> ( ( ord_less @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less @ real @ ( sin @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% sin_lt_zero
thf(fact_5352_sin__inj__pi,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( sin @ real @ X )
= ( sin @ real @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% sin_inj_pi
thf(fact_5353_sin__mono__le__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ) ) ).
% sin_mono_le_eq
thf(fact_5354_sin__monotone__2pi__le,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sin @ real @ Y ) @ ( sin @ real @ X ) ) ) ) ) ).
% sin_monotone_2pi_le
thf(fact_5355_arctan__bounded,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) )
& ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_bounded
thf(fact_5356_arctan__lbound,axiom,
! [Y: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) ) ).
% arctan_lbound
thf(fact_5357_sin__le__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ pi @ X )
=> ( ( ord_less @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% sin_le_zero
thf(fact_5358_sin__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( sin @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% sin_less_zero
thf(fact_5359_sin__mono__less__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ) ) ).
% sin_mono_less_eq
thf(fact_5360_sin__monotone__2pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sin @ real @ Y ) @ ( sin @ real @ X ) ) ) ) ) ).
% sin_monotone_2pi
thf(fact_5361_sin__total,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X3 )
= Y )
& ! [Y4: real] :
( ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y4 )
& ( ord_less_eq @ real @ Y4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ Y4 )
= Y ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% sin_total
thf(fact_5362_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_5363_Maclaurin__cos__expansion2,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less @ real @ T7 @ X )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_5364_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ? [T7: real] :
( ( ord_less @ real @ X @ T7 )
& ( ord_less @ real @ T7 @ ( zero_zero @ real ) )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_5365_monoseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( topological_monoseq @ real
@ ^ [N6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_5366_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_5367_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_5368_cos__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( one_one @ real ) ) ).
% cos_le_one
thf(fact_5369_cos__one__sin__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
= ( one_one @ A ) )
=> ( ( sin @ A @ X )
= ( zero_zero @ A ) ) ) ) ).
% cos_one_sin_zero
thf(fact_5370_cos__monotone__0__pi__le,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi_le
thf(fact_5371_cos__mono__le__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ pi )
=> ( ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) )
= ( ord_less_eq @ real @ Y @ X ) ) ) ) ) ) ).
% cos_mono_le_eq
thf(fact_5372_cos__inj__pi,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ pi )
=> ( ( ( cos @ real @ X )
= ( cos @ real @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% cos_inj_pi
thf(fact_5373_cos__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( cos @ real @ X ) ) ).
% cos_ge_minus_one
thf(fact_5374_abs__cos__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( cos @ real @ X ) ) @ ( one_one @ real ) ) ).
% abs_cos_le_one
thf(fact_5375_mono__SucI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X6: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
=> ( topological_monoseq @ A @ X6 ) ) ) ).
% mono_SucI1
thf(fact_5376_mono__SucI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X6: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) )
=> ( topological_monoseq @ A @ X6 ) ) ) ).
% mono_SucI2
thf(fact_5377_monoseq__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X8: nat > A] :
( ! [N6: nat] : ( ord_less_eq @ A @ ( X8 @ N6 ) @ ( X8 @ ( suc @ N6 ) ) )
| ! [N6: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N6 ) ) @ ( X8 @ N6 ) ) ) ) ) ) ).
% monoseq_Suc
thf(fact_5378_monoseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X8: nat > A] :
( ! [M5: nat,N6: nat] :
( ( ord_less_eq @ nat @ M5 @ N6 )
=> ( ord_less_eq @ A @ ( X8 @ M5 ) @ ( X8 @ N6 ) ) )
| ! [M5: nat,N6: nat] :
( ( ord_less_eq @ nat @ M5 @ N6 )
=> ( ord_less_eq @ A @ ( X8 @ N6 ) @ ( X8 @ M5 ) ) ) ) ) ) ) ).
% monoseq_def
thf(fact_5379_monoI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X6: nat > A] :
( ! [M3: nat,N2: nat] :
( ( ord_less_eq @ nat @ M3 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( X6 @ M3 ) ) )
=> ( topological_monoseq @ A @ X6 ) ) ) ).
% monoI2
thf(fact_5380_monoI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X6: nat > A] :
( ! [M3: nat,N2: nat] :
( ( ord_less_eq @ nat @ M3 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) )
=> ( topological_monoseq @ A @ X6 ) ) ) ).
% monoI1
thf(fact_5381_sin__zero__norm__cos__one,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sin @ A @ X )
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( cos @ A @ X ) )
= ( one_one @ real ) ) ) ) ).
% sin_zero_norm_cos_one
thf(fact_5382_cos__mono__less__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ pi )
=> ( ( ord_less @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) )
= ( ord_less @ real @ Y @ X ) ) ) ) ) ) ).
% cos_mono_less_eq
thf(fact_5383_cos__monotone__0__pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi
thf(fact_5384_cos__monotone__minus__pi__0_H,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( cos @ real @ Y ) @ ( cos @ real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0'
thf(fact_5385_cos__two__less__zero,axiom,
ord_less @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).
% cos_two_less_zero
thf(fact_5386_cos__is__zero,axiom,
? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X3 )
= ( zero_zero @ real ) )
& ! [Y4: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y4 )
& ( ord_less_eq @ real @ Y4 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ Y4 )
= ( zero_zero @ real ) ) )
=> ( Y4 = X3 ) ) ) ).
% cos_is_zero
thf(fact_5387_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_5388_cos__monotone__minus__pi__0,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cos @ real @ Y ) @ ( cos @ real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0
thf(fact_5389_cos__total,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ pi )
& ( ( cos @ real @ X3 )
= Y )
& ! [Y4: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y4 )
& ( ord_less_eq @ real @ Y4 @ pi )
& ( ( cos @ real @ Y4 )
= Y ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% cos_total
thf(fact_5390_sincos__principal__value,axiom,
! [X: real] :
? [Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ Y3 )
& ( ord_less_eq @ real @ Y3 @ pi )
& ( ( sin @ real @ Y3 )
= ( sin @ real @ X ) )
& ( ( cos @ real @ Y3 )
= ( cos @ real @ X ) ) ) ).
% sincos_principal_value
thf(fact_5391_monoseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( topological_monoseq @ real @ ( power_power @ real @ X ) ) ) ) ).
% monoseq_realpow
thf(fact_5392_sin__cos__le1,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( plus_plus @ real @ ( times_times @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y ) ) @ ( times_times @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) ) ) ) @ ( one_one @ real ) ) ).
% sin_cos_le1
thf(fact_5393_cos__double__less__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) ) @ ( one_one @ real ) ) ) ) ).
% cos_double_less_one
thf(fact_5394_cos__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X ) ) ) ) ).
% cos_gt_zero
thf(fact_5395_cos__gt__zero__pi,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X ) ) ) ) ).
% cos_gt_zero_pi
thf(fact_5396_cos__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cos @ real @ X ) ) ) ) ).
% cos_ge_zero
thf(fact_5397_pi__half,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
= ( the @ real
@ ^ [X4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less_eq @ real @ X4 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X4 )
= ( zero_zero @ real ) ) ) ) ) ).
% pi_half
thf(fact_5398_pi__def,axiom,
( pi
= ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) )
@ ( the @ real
@ ^ [X4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less_eq @ real @ X4 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X4 )
= ( zero_zero @ real ) ) ) ) ) ) ).
% pi_def
thf(fact_5399_sincos__total__pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ pi )
& ( X
= ( cos @ real @ T7 ) )
& ( Y
= ( sin @ real @ T7 ) ) ) ) ) ).
% sincos_total_pi
thf(fact_5400_sin__expansion__lemma,axiom,
! [X: real,M: nat] :
( ( sin @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( cos @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_expansion_lemma
thf(fact_5401_cos__expansion__lemma,axiom,
! [X: real,M: nat] :
( ( cos @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( uminus_uminus @ real @ ( sin @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_expansion_lemma
thf(fact_5402_sincos__total__pi__half,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( X
= ( cos @ real @ T7 ) )
& ( Y
= ( sin @ real @ T7 ) ) ) ) ) ) ).
% sincos_total_pi_half
thf(fact_5403_sincos__total__2pi__le,axiom,
! [X: real,Y: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
& ( X
= ( cos @ real @ T7 ) )
& ( Y
= ( sin @ real @ T7 ) ) ) ) ).
% sincos_total_2pi_le
thf(fact_5404_sincos__total__2pi,axiom,
! [X: real,Y: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ~ ! [T7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
=> ( ( ord_less @ real @ T7 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ( X
= ( cos @ real @ T7 ) )
=> ( Y
!= ( sin @ real @ T7 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_5405_Maclaurin__cos__expansion,axiom,
! [X: real,N: nat] :
? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_cos_expansion
thf(fact_5406_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_5407_lowi__hT,axiom,
! [X: nat,N: nat] :
( time_htt @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_lowi @ X @ N )
@ ^ [R5: nat] :
( pure_assn
@ ( R5
= ( vEBT_VEBT_low @ X @ N ) ) )
@ ( one_one @ nat ) ) ).
% lowi_hT
thf(fact_5408_highi__hT,axiom,
! [X: nat,N: nat] :
( time_htt @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_highi @ X @ N )
@ ^ [R5: nat] :
( pure_assn
@ ( R5
= ( vEBT_VEBT_high @ X @ N ) ) )
@ ( one_one @ nat ) ) ).
% highi_hT
thf(fact_5409_TBOUND__highi,axiom,
! [X: nat,N: nat] : ( time_TBOUND @ nat @ ( vEBT_VEBT_highi @ X @ N ) @ ( one_one @ nat ) ) ).
% TBOUND_highi
thf(fact_5410_TBOUND__lowi,axiom,
! [X: nat,N: nat] : ( time_TBOUND @ nat @ ( vEBT_VEBT_lowi @ X @ N ) @ ( one_one @ nat ) ) ).
% TBOUND_lowi
thf(fact_5411_highi__h,axiom,
! [X: nat,N: nat] :
( hoare_hoare_triple @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_highi @ X @ N )
@ ^ [R5: nat] :
( pure_assn
@ ( R5
= ( vEBT_VEBT_high @ X @ N ) ) ) ) ).
% highi_h
thf(fact_5412_lowi__h,axiom,
! [X: nat,N: nat] :
( hoare_hoare_triple @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_lowi @ X @ N )
@ ^ [R5: nat] :
( pure_assn
@ ( R5
= ( vEBT_VEBT_low @ X @ N ) ) ) ) ).
% lowi_h
thf(fact_5413_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_5414_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_5415_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_5416_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_5417_set__encode__insert,axiom,
! [A5: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ A5 )
=> ( ~ ( member @ nat @ N @ A5 )
=> ( ( nat_set_encode @ ( insert @ nat @ N @ A5 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A5 ) ) ) ) ) ).
% set_encode_insert
thf(fact_5418_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N6: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) @ ( zero_zero @ real ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N6 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N6 ) ) ) ) ) ).
% sin_coeff_def
thf(fact_5419_ln__series,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ( ln_ln @ real @ X )
= ( suminf @ real
@ ^ [N6: nat] : ( times_times @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N6 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ N6 @ ( one_one @ nat ) ) ) ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ ( one_one @ real ) ) @ ( suc @ N6 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_5420_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_5421_dvd__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ A2 @ ( zero_zero @ A ) ) ) ).
% dvd_0_right
thf(fact_5422_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_5423_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_5424_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_5425_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C3 ) )
= ( dvd_dvd @ A @ B2 @ C3 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_5426_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C3 @ A2 ) )
= ( dvd_dvd @ A @ B2 @ C3 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_5427_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_5428_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% dvd_1_left
thf(fact_5429_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_5430_dvd__prodI,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A5: set @ B,A2: B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( member @ B @ A2 @ A5 )
=> ( dvd_dvd @ A @ ( F3 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) ) ) ) ) ).
% dvd_prodI
thf(fact_5431_dvd__prod__eqI,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A5: set @ B,A2: B,B2: A,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( member @ B @ A2 @ A5 )
=> ( ( B2
= ( F3 @ A2 ) )
=> ( dvd_dvd @ A @ B2 @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_5432_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_5433_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_5434_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D6: nat] : ( dvd_dvd @ nat @ D6 @ N ) ) )
= N ) ) ).
% Max_divisors_self_nat
thf(fact_5435_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_5436_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_5437_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_5438_powser__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F3: nat > A] :
( ( suminf @ A
@ ^ [N6: nat] : ( times_times @ A @ ( F3 @ N6 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N6 ) ) )
= ( F3 @ ( zero_zero @ nat ) ) ) ) ).
% powser_zero
thf(fact_5439_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_5440_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_5441_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_5442_zero__le__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W2: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_5443_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W2: num] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_5444_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_5445_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_5446_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_5447_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_5448_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W2: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) )
= ( ( ( numeral_numeral @ nat @ W2 )
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A2
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_5449_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W2: num] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_5450_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_5451_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_5452_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_5453_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z3: B] :
! [X5: B] :
( ( ord_less @ B @ X5 @ Z3 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S ) ) ) ) ) ) ).
% minf(10)
thf(fact_5454_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z3: B] :
! [X5: B] :
( ( ord_less @ B @ X5 @ Z3 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S ) ) ) ) ) ).
% minf(9)
thf(fact_5455_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z3: B] :
! [X5: B] :
( ( ord_less @ B @ Z3 @ X5 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S ) ) ) ) ) ) ).
% pinf(10)
thf(fact_5456_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z3: B] :
! [X5: B] :
( ( ord_less @ B @ Z3 @ X5 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X5 @ S ) ) ) ) ) ).
% pinf(9)
thf(fact_5457_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_5458_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ ( set @ A )
@ ( collect @ A
@ ^ [C7: A] : ( dvd_dvd @ A @ C7 @ A2 ) )
@ ( collect @ A
@ ^ [C7: A] : ( dvd_dvd @ A @ C7 @ B2 ) ) )
= ( ( dvd_dvd @ A @ A2 @ B2 )
& ~ ( dvd_dvd @ A @ B2 @ A2 ) ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_5459_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_5460_dvd__field__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A8: A,B8: A] :
( ( A8
= ( zero_zero @ A ) )
=> ( B8
= ( zero_zero @ A ) ) ) ) ) ) ).
% dvd_field_iff
thf(fact_5461_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_5462_gcd__nat_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_5463_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_5464_gcd__nat_Oextremum__unique,axiom,
! [A2: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_unique
thf(fact_5465_gcd__nat_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
& ( ( zero_zero @ nat )
!= A2 ) ) ).
% gcd_nat.extremum_strict
thf(fact_5466_gcd__nat_Oextremum,axiom,
! [A2: nat] : ( dvd_dvd @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% gcd_nat.extremum
thf(fact_5467_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ M @ N )
=> ( ( dvd_dvd @ nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_5468_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C7: A] : ( dvd_dvd @ A @ C7 @ A2 ) )
@ ( collect @ A
@ ^ [C7: A] : ( dvd_dvd @ A @ C7 @ B2 ) ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% subset_divisors_dvd
thf(fact_5469_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_5470_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A8: A,B8: A] :
( ( modulo_modulo @ A @ B8 @ A8 )
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_5471_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_5472_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_5473_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_5474_dvd__power__le,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A,N: nat,M: nat] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ M ) ) ) ) ) ).
% dvd_power_le
thf(fact_5475_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_5476_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_5477_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_5478_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_5479_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_5480_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_5481_set__encode__eq,axiom,
! [A5: set @ nat,B5: set @ nat] :
( ( finite_finite2 @ nat @ A5 )
=> ( ( finite_finite2 @ nat @ B5 )
=> ( ( ( nat_set_encode @ A5 )
= ( nat_set_encode @ B5 ) )
= ( A5 = B5 ) ) ) ) ).
% set_encode_eq
thf(fact_5482_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_5483_dvd__Gcd__fin__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A,B2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( dvd_dvd @ A @ B2 @ ( semiring_gcd_Gcd_fin @ A @ A5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( dvd_dvd @ A @ B2 @ X4 ) ) ) ) ) ) ).
% dvd_Gcd_fin_iff
thf(fact_5484_Gcd__fin__greatest,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ A5 )
=> ( dvd_dvd @ A @ A2 @ B4 ) )
=> ( dvd_dvd @ A @ A2 @ ( semiring_gcd_Gcd_fin @ A @ A5 ) ) ) ) ) ).
% Gcd_fin_greatest
thf(fact_5485_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P2: A > $o,L: A] :
( ( ? [X4: A] : ( P2 @ ( times_times @ A @ L @ X4 ) ) )
= ( ? [X4: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X4 @ ( zero_zero @ A ) ) )
& ( P2 @ X4 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_5486_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ~ ( ( A2
!= ( zero_zero @ A ) )
=> ! [C2: A] :
( B2
!= ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% unit_dvdE
thf(fact_5487_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_5488_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C3: A,B2: A,D2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ C3 @ D2 )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= ( divide_divide @ A @ D2 @ C3 ) )
= ( ( times_times @ A @ B2 @ C3 )
= ( times_times @ A @ A2 @ D2 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_5489_dvd__div__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_5490_div__dvd__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C3 )
= ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_5491_dvd__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= C3 )
= ( B2
= ( times_times @ A @ C3 @ A2 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_5492_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_5493_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_5494_even__set__encode__iff,axiom,
! [A5: set @ nat] :
( ( finite_finite2 @ nat @ A5 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A5 ) )
= ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A5 ) ) ) ) ).
% even_set_encode_iff
thf(fact_5495_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_5496_prod__dvd__prod__subset2,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [B5: set @ B,A5: set @ B,F3: B > A,G3: B > A] :
( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ A5 )
=> ( dvd_dvd @ A @ ( F3 @ A4 ) @ ( G3 @ A4 ) ) )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B5 ) ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_5497_prod__dvd__prod__subset,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B5: set @ B,A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B5 )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_5498_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_5499_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_5500_bezout__add__strong__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ? [D3: nat,X3: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D3 @ A2 )
& ( dvd_dvd @ nat @ D3 @ B2 )
& ( ( times_times @ nat @ A2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ D3 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_5501_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_5502_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_5503_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_5504_even__sum__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( finite_card @ B
@ ( collect @ B
@ ^ [A8: B] :
( ( member @ B @ A8 @ A5 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F3 @ A8 ) ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_5505_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [D6: nat] : ( dvd_dvd @ nat @ D6 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_5506_gcd__nat_Oordering__top__axioms,axiom,
( ordering_top @ nat @ ( dvd_dvd @ nat )
@ ^ [M5: nat,N6: nat] :
( ( dvd_dvd @ nat @ M5 @ N6 )
& ( M5 != N6 ) )
@ ( zero_zero @ nat ) ) ).
% gcd_nat.ordering_top_axioms
thf(fact_5507_set__encode__inf,axiom,
! [A5: set @ nat] :
( ~ ( finite_finite2 @ nat @ A5 )
=> ( ( nat_set_encode @ A5 )
= ( zero_zero @ nat ) ) ) ).
% set_encode_inf
thf(fact_5508_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_5509_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C3: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ~ ( ( A2
!= ( zero_zero @ A ) )
=> ! [B4: A] :
( ( B4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B4 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
= B4 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B4 )
= A2 )
=> ( ( ( times_times @ A @ A2 @ B4 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C3 @ A2 )
!= ( times_times @ A @ C3 @ B4 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_5510_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_5511_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_5512_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,M: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ N ) )
= ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M @ N ) ) ) ) ) ).
% dvd_power_iff
thf(fact_5513_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat,X: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
| ( X
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X @ ( power_power @ A @ X @ N ) ) ) ) ).
% dvd_power
thf(fact_5514_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_5515_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_5516_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_5517_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_5518_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_5519_diff__mod__le,axiom,
! [A2: nat,D2: nat,B2: nat] :
( ( ord_less @ nat @ A2 @ D2 )
=> ( ( dvd_dvd @ nat @ B2 @ D2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ A2 @ ( modulo_modulo @ nat @ A2 @ B2 ) ) @ ( minus_minus @ nat @ D2 @ B2 ) ) ) ) ).
% diff_mod_le
thf(fact_5520_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_5521_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_5522_even__prod__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_5523_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_5524_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_5525_odd__card__imp__not__empty,axiom,
! [A: $tType,A5: set @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A5 ) )
=> ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% odd_card_imp_not_empty
thf(fact_5526_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_5527_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_5528_dvd__partition,axiom,
! [A: $tType,C4: set @ ( set @ A ),K: nat] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C4 )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ X3 ) ) )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C4 )
=> ! [Xa4: set @ A] :
( ( member @ ( set @ A ) @ Xa4 @ C4 )
=> ( ( X3 != Xa4 )
=> ( ( inf_inf @ ( set @ A ) @ X3 @ Xa4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_5529_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A5: set @ B,F3: B > A,B2: A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ B2 )
= ( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [A8: B] : ( divide_divide @ A @ ( F3 @ A8 ) @ B2 )
@ ( inf_inf @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [A8: B] : ( dvd_dvd @ A @ B2 @ ( F3 @ A8 ) ) ) ) )
@ ( divide_divide @ A
@ ( groups7311177749621191930dd_sum @ B @ A @ F3
@ ( inf_inf @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [A8: B] :
~ ( dvd_dvd @ A @ B2 @ ( F3 @ A8 ) ) ) ) )
@ B2 ) ) ) ) ) ).
% sum_div_partition
thf(fact_5530_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_5531_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_5532_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_5533_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_5534_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_5535_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_5536_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_5537_Gcd__eq__Max,axiom,
! [M7: set @ nat] :
( ( finite_finite2 @ nat @ M7 )
=> ( ( M7
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( gcd_Gcd @ nat @ M7 )
= ( lattic643756798349783984er_Max @ nat
@ ( complete_Inf_Inf @ ( set @ nat )
@ ( image @ nat @ ( set @ nat )
@ ^ [M5: nat] :
( collect @ nat
@ ^ [D6: nat] : ( dvd_dvd @ nat @ D6 @ M5 ) )
@ M7 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_5538_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_5539_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_5540_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_5541_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_5542_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_5543_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_5544_Bernoulli__inequality__even,axiom,
! [N: nat,X: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ N ) ) ) ).
% Bernoulli_inequality_even
thf(fact_5545_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
! [N: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N ) ) )
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N ) ) )
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi.simps(3)
thf(fact_5546_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_5547_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_5548_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V441764108873111860ildupi @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ ( suc @ N2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi.elims
thf(fact_5549_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
! [N: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb'.simps(3)
thf(fact_5550_VEBT__internal_OTb_H_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_VEBT_Tb2 @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ ( suc @ N2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb'.elims
thf(fact_5551_cos__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
=> ? [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_lemma
thf(fact_5552_sin__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
=> ? [N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_lemma
thf(fact_5553_VEBT__internal_OTb_Osimps_I3_J,axiom,
! [N: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N ) ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N ) ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb.simps(3)
thf(fact_5554_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
! [N: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N ) ) )
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N ) ) )
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi'.simps(3)
thf(fact_5555_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
! [Va2: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va2 ) ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va2 ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
thf(fact_5556_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
! [Va2: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va2 ) ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va2 ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
thf(fact_5557_vebt__buildup_Oelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.elims
thf(fact_5558_VEBT__internal_OTb_Oelims,axiom,
! [X: nat,Y: int] :
( ( ( vEBT_VEBT_Tb @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ ( suc @ N2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb.elims
thf(fact_5559_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8346862874174094_d_u_p @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
thf(fact_5560_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8646137997579335489_i_l_d @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
thf(fact_5561_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
! [X: nat,Y: int] :
( ( ( vEBT_V9176841429113362141ildupi @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ int ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( one_one @ int ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ ( suc @ N2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi'.elims
thf(fact_5562_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
@ ^ [I3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I3 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_even_sum
thf(fact_5563_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
@ ^ [I3: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N @ I3 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_odd_sum
thf(fact_5564_arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( arctan @ X )
= ( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_5565_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
! [X: nat,Y: int] :
( ( ( vEBT_V9176841429113362141ildupi @ X )
= Y )
=> ( ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ int ) )
=> ~ ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( one_one @ int ) )
=> ~ ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ ( suc @ N2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi'.pelims
thf(fact_5566_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8646137997579335489_i_l_d @ X )
= Y )
=> ( ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
thf(fact_5567_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8346862874174094_d_u_p @ X )
= Y )
=> ( ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
thf(fact_5568_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_5569_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_5570_abs__div,axiom,
! [Y: int,X: int] :
( ( dvd_dvd @ int @ Y @ X )
=> ( ( abs_abs @ int @ ( divide_divide @ int @ X @ Y ) )
= ( divide_divide @ int @ ( abs_abs @ int @ X ) @ ( abs_abs @ int @ Y ) ) ) ) ).
% abs_div
thf(fact_5571_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_5572_dvd__imp__le__int,axiom,
! [I: int,D2: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ D2 @ I )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ D2 ) @ ( abs_abs @ int @ I ) ) ) ) ).
% dvd_imp_le_int
thf(fact_5573_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_5574_int__div__sub__1,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ M )
=> ( ( ( dvd_dvd @ int @ M @ N )
=> ( ( divide_divide @ int @ ( minus_minus @ int @ N @ ( one_one @ int ) ) @ M )
= ( minus_minus @ int @ ( divide_divide @ int @ N @ M ) @ ( one_one @ int ) ) ) )
& ( ~ ( dvd_dvd @ int @ M @ N )
=> ( ( divide_divide @ int @ ( minus_minus @ int @ N @ ( one_one @ int ) ) @ M )
= ( divide_divide @ int @ N @ M ) ) ) ) ) ).
% int_div_sub_1
thf(fact_5575_div__dvd__sgn__abs,axiom,
! [L: int,K: int] :
( ( dvd_dvd @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( times_times @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( sgn_sgn @ int @ L ) ) @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ) ).
% div_dvd_sgn_abs
thf(fact_5576_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_5577_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_5578_eme1p,axiom,
! [N: int,D2: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ D2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N ) @ D2 )
= ( plus_plus @ int @ ( one_one @ int ) @ ( modulo_modulo @ int @ N @ D2 ) ) ) ) ) ) ).
% eme1p
thf(fact_5579_emep1,axiom,
! [N: int,D2: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ D2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ N @ ( one_one @ int ) ) @ D2 )
= ( plus_plus @ int @ ( modulo_modulo @ int @ N @ D2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% emep1
thf(fact_5580_vebt__buildup_Opelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( accp @ nat @ vEBT_v4011308405150292612up_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.pelims
thf(fact_5581_VEBT__internal_OTb_Opelims,axiom,
! [X: nat,Y: int] :
( ( ( vEBT_VEBT_Tb @ X )
= Y )
=> ( ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ ( suc @ N2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb.pelims
thf(fact_5582_VEBT__internal_OTb_H_Opelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_VEBT_Tb2 @ X )
= Y )
=> ( ( accp @ nat @ vEBT_VEBT_Tb_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ ( suc @ N2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb'.pelims
thf(fact_5583_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V441764108873111860ildupi @ X )
= Y )
=> ( ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ ( suc @ N2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi.pelims
thf(fact_5584_suminf__geometric,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C3 ) @ ( one_one @ real ) )
=> ( ( suminf @ A @ ( power_power @ A @ C3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C3 ) ) ) ) ) ).
% suminf_geometric
thf(fact_5585_suminf__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ( ( suminf @ A
@ ^ [N6: nat] : ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% suminf_zero
thf(fact_5586_suminf__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [N8: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ N8 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N8 )
=> ( ( F3 @ N2 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A @ F3 )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ N8 ) ) ) ) ) ).
% suminf_finite
thf(fact_5587_vebt__delete__code_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
= ( case_option @ vEBT_VEBT @ ( product_prod @ nat @ nat ) @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary )
@ ^ [Mima: product_prod @ nat @ nat] :
( if @ vEBT_VEBT @ ( ord_less_eq @ nat @ Deg @ ( one_one @ nat ) ) @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary )
@ ( product_case_prod @ nat @ nat @ vEBT_VEBT
@ ^ [Mi3: nat,Ma3: nat] :
( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ X @ Mi3 )
| ( ord_less @ nat @ Ma3 @ X ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ Deg @ TreeList @ Summary )
@ ( if @ vEBT_VEBT
@ ( ( X = Mi3 )
& ( X = Ma3 ) )
@ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary )
@ ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi3 ) @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi3 )
@ ( if @ nat
@ ( ( ( X = Mi3 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma3 ) )
& ( ( X != Mi3 )
=> ( X = Ma3 ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( X = Mi3 ) @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi3 )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma3 ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi3 ) @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi3 )
@ ( if @ nat
@ ( ( ( X = Mi3 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma3 ) )
& ( ( X != Mi3 )
=> ( X = Ma3 ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma3 ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ Deg @ TreeList @ Summary ) ) ) )
@ Mima ) )
@ Info ) ) ).
% vebt_delete_code(2)
thf(fact_5588_round__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: int] :
( ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y ) )
=> ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
=> ( ( archimedean_round @ A @ X )
= Y ) ) ) ) ).
% round_unique
thf(fact_5589_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_5590_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A5: set @ ( product_prod @ A @ B ),F3: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ A5 )
=> ( member @ C @ ( F3 @ A2 @ B2 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F3 ) @ A5 ) ) ) ).
% pair_imageI
thf(fact_5591_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F3: B > C > A,A2: B,B2: C] :
( ( product_case_prod @ B @ C @ A @ F3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( F3 @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_5592_round__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% round_0
thf(fact_5593_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_5594_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_5595_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_5596_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_5597_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_5598_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q3: product_prod @ A @ B,F3: A > B > C,G3: A > B > C,P3: product_prod @ A @ B] :
( ! [X3: A,Y3: B] :
( ( ( product_Pair @ A @ B @ X3 @ Y3 )
= Q3 )
=> ( ( F3 @ X3 @ Y3 )
= ( G3 @ X3 @ Y3 ) ) )
=> ( ( P3 = Q3 )
=> ( ( product_case_prod @ A @ B @ C @ F3 @ P3 )
= ( product_case_prod @ A @ B @ C @ G3 @ Q3 ) ) ) ) ).
% split_cong
thf(fact_5599_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F3: A > B > C,X1: A,X2: B] :
( ( product_case_prod @ A @ B @ C @ F3 @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= ( F3 @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_5600_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B > C,G3: ( product_prod @ A @ B ) > C] :
( ! [X3: A,Y3: B] :
( ( F3 @ X3 @ Y3 )
= ( G3 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F3 )
= G3 ) ) ).
% cond_case_prod_eta
thf(fact_5601_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X4: A,Y5: B] : ( F3 @ ( product_Pair @ A @ B @ X4 @ Y5 ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_5602_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q2: A > $o,P2: B > C > A,Z: product_prod @ B @ C] :
( ( Q2 @ ( product_case_prod @ B @ C @ A @ P2 @ Z ) )
=> ~ ! [X3: B,Y3: C] :
( ( Z
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( Q2 @ ( P2 @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_5603_nested__case__prod__simp,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F4: B > C > D > A,X4: product_prod @ B @ C,Y5: D] :
( product_case_prod @ B @ C @ A
@ ^ [A8: B,B8: C] : ( F4 @ A8 @ B8 @ Y5 )
@ X4 ) ) ) ).
% nested_case_prod_simp
thf(fact_5604_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H2: C > D,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( H2 @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X16: A,X24: B] : ( H2 @ ( F3 @ X16 @ X24 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5605_uncurry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( uncurry @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% uncurry_def
thf(fact_5606_internal__case__prod__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( produc5280177257484947105e_prod @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% internal_case_prod_def
thf(fact_5607_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G3 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G3 @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_5608_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G3 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G3 @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_5609_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G3 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G3 @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.triangle_reindex
thf(fact_5610_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G3 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G3 @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.triangle_reindex
thf(fact_5611_round__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y ) ) ) ) ).
% round_mono
thf(fact_5612_floor__le__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archimedean_round @ A @ X ) ) ) ).
% floor_le_round
thf(fact_5613_ceiling__ge__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% ceiling_ge_round
thf(fact_5614_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F3: ( product_prod @ B @ C ) > A,G3: D > B,Xs: list @ D,Ys: list @ C] :
( ( map @ ( product_prod @ B @ C ) @ A @ F3 @ ( zip @ B @ C @ ( map @ D @ B @ G3 @ Xs ) @ Ys ) )
= ( map @ ( product_prod @ D @ C ) @ A
@ ( product_case_prod @ D @ C @ A
@ ^ [X4: D,Y5: C] : ( F3 @ ( product_Pair @ B @ C @ ( G3 @ X4 ) @ Y5 ) ) )
@ ( zip @ D @ C @ Xs @ Ys ) ) ) ).
% map_zip_map
thf(fact_5615_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F3: ( product_prod @ B @ C ) > A,Xs: list @ B,G3: D > C,Ys: list @ D] :
( ( map @ ( product_prod @ B @ C ) @ A @ F3 @ ( zip @ B @ C @ Xs @ ( map @ D @ C @ G3 @ Ys ) ) )
= ( map @ ( product_prod @ B @ D ) @ A
@ ( product_case_prod @ B @ D @ A
@ ^ [X4: B,Y5: D] : ( F3 @ ( product_Pair @ B @ C @ X4 @ ( G3 @ Y5 ) ) ) )
@ ( zip @ B @ D @ Xs @ Ys ) ) ) ).
% map_zip_map2
thf(fact_5616_foldl__snd__zip,axiom,
! [B: $tType,C: $tType,A: $tType,Ys: list @ A,Xs: list @ B,F3: C > A > C,B2: C] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( foldl @ C @ ( product_prod @ B @ A )
@ ^ [B8: C] :
( product_case_prod @ B @ A @ C
@ ^ [X4: B] : ( F3 @ B8 ) )
@ B2
@ ( zip @ B @ A @ Xs @ Ys ) )
= ( foldl @ C @ A @ F3 @ B2 @ Ys ) ) ) ).
% foldl_snd_zip
thf(fact_5617_divmod__int__def,axiom,
( ( unique8689654367752047608divmod @ int )
= ( ^ [M5: num,N6: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N6 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N6 ) ) ) ) ) ).
% divmod_int_def
thf(fact_5618_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M5: num,N6: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M5 ) @ ( numeral_numeral @ nat @ N6 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M5 ) @ ( numeral_numeral @ nat @ N6 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_5619_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M5: num,N6: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M5 ) @ ( numeral_numeral @ A @ N6 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M5 ) @ ( numeral_numeral @ A @ N6 ) ) ) ) ) ) ).
% divmod_def
thf(fact_5620_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_5621_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( ^ [A6: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y5: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( ord_max @ A @ X4 ) @ Y5 ) )
@ ( none @ A )
@ A6 ) ) ) ) ) ).
% Max.eq_fold'
thf(fact_5622_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M5: num,N6: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M5 @ N6 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M5 ) ) @ ( unique1321980374590559556d_step @ A @ N6 @ ( unique8689654367752047608divmod @ A @ M5 @ ( bit0 @ N6 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_5623_vebt__insert__code_I2_J,axiom,
! [Deg: nat,Info: option @ ( product_prod @ nat @ nat ),TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ( ord_less_eq @ nat @ Deg @ ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) )
& ( ~ ( ord_less_eq @ nat @ Deg @ ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
= ( case_option @ vEBT_VEBT @ ( product_prod @ nat @ nat ) @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ X ) ) @ Deg @ TreeList @ Summary )
@ ( product_case_prod @ nat @ nat @ vEBT_VEBT
@ ^ [Mi3: nat,Ma3: nat] :
( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X = Mi3 )
| ( X = Ma3 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ X @ Mi3 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ Ma3 ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ Deg @ TreeList @ Summary ) ) )
@ Info ) ) ) ) ).
% vebt_insert_code(2)
thf(fact_5624_of__int__round__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% of_int_round_le
thf(fact_5625_of__int__round__ge,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% of_int_round_ge
thf(fact_5626_of__int__round__gt,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% of_int_round_gt
thf(fact_5627_of__int__round__abs__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ X ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_int_round_abs_le
thf(fact_5628_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,N: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ N ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X )
= N ) ) ) ).
% round_unique'
thf(fact_5629_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X4: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X4 ) ) @ ( archimedean_ceiling @ A @ X4 ) @ ( archim6421214686448440834_floor @ A @ X4 ) ) ) ) ) ).
% round_altdef
thf(fact_5630_of__nat__code__if,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N6: nat] :
( if @ A
@ ( N6
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [M5: nat,Q8: nat] :
( if @ A
@ ( Q8
= ( zero_zero @ nat ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M5 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M5 ) ) @ ( one_one @ A ) ) )
@ ( divmod_nat @ N6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_5631_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_5632_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_5633_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P3: product_prod @ A @ B,Z: C,C3: A > B > ( set @ C )] :
( ! [A4: A,B4: B] :
( ( P3
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( member @ C @ Z @ ( C3 @ A4 @ B4 ) ) )
=> ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P3 ) ) ) ).
% mem_case_prodI2
thf(fact_5634_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),A2: B,B2: C] :
( ( member @ A @ Z @ ( C3 @ A2 @ B2 ) )
=> ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_5635_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P3: product_prod @ A @ B,C3: A > B > C > $o,X: C] :
( ! [A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A4 @ B4 )
= P3 )
=> ( C3 @ A4 @ B4 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P3 @ X ) ) ).
% case_prodI2'
thf(fact_5636_split__part,axiom,
! [B: $tType,A: $tType,P2: $o,Q2: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A8: A,B8: B] :
( P2
& ( Q2 @ A8 @ B8 ) ) )
= ( ^ [Ab2: product_prod @ A @ B] :
( P2
& ( product_case_prod @ A @ B @ $o @ Q2 @ Ab2 ) ) ) ) ).
% split_part
thf(fact_5637_The__split__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X17: A,Y9: B] :
( ( X = X17 )
& ( Y = Y9 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% The_split_eq
thf(fact_5638_case__prodI2,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B,C3: A > B > $o] :
( ! [A4: A,B4: B] :
( ( P3
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( C3 @ A4 @ B4 ) )
=> ( product_case_prod @ A @ B @ $o @ C3 @ P3 ) ) ).
% case_prodI2
thf(fact_5639_case__prodI,axiom,
! [A: $tType,B: $tType,F3: A > B > $o,A2: A,B2: B] :
( ( F3 @ A2 @ B2 )
=> ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_5640_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P2: $o] :
( ( P2
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A8: A,B8: B] : P2 ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P2
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A8: A,B8: B] : P2 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_5641_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_5642_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_5643_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 )
@ ^ [Q8: A,R5: A] : ( product_Pair @ A @ A @ Q8 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_5644_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 )
@ ^ [Q8: A,R5: A] : ( product_Pair @ A @ A @ Q8 @ ( 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_5645_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A5: A > B > $o,B5: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A5 @ B5 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A5 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B5 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_5646_case__prodE,axiom,
! [A: $tType,B: $tType,C3: A > B > $o,P3: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C3 @ P3 )
=> ~ ! [X3: A,Y3: B] :
( ( P3
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C3 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_5647_case__prodD,axiom,
! [A: $tType,B: $tType,F3: A > B > $o,A2: A,B2: B] :
( ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( F3 @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_5648_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P5: A > $o,R4: A > ( set @ ( product_prod @ B @ B ) )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [X17: A,Y9: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y5: B] :
( ( X17 = X4 )
& ( P5 @ X4 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y9 @ Y5 ) @ ( R4 @ X4 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_5649_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),P3: product_prod @ B @ C] :
( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P3 ) )
=> ~ ! [X3: B,Y3: C] :
( ( P3
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( member @ A @ Z @ ( C3 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5650_lex__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( lex_prod @ A @ B )
= ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [A8: A,B8: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A16: A,B13: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ A16 ) @ Ra )
| ( ( A8 = A16 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B8 @ B13 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_5651_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P3: product_prod @ A @ B,Z: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P3 @ Z )
=> ~ ! [X3: A,Y3: B] :
( ( P3
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C3 @ X3 @ Y3 @ Z ) ) ) ).
% case_prodE'
thf(fact_5652_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R3: A > B > C > $o,A2: A,B2: B,C3: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R3 @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C3 )
=> ( R3 @ A2 @ B2 @ C3 ) ) ).
% case_prodD'
thf(fact_5653_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu3: A,Uv3: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_5654_zip__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs ) )
= ( map @ ( product_prod @ ( product_prod @ A @ B ) @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ ( product_prod @ B @ C ) ) )
@ ^ [X4: A,Y5: B,Z2: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X4 @ ( product_Pair @ B @ C @ Y5 @ Z2 ) ) ) )
@ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs @ Ys ) @ Zs ) ) ) ).
% zip_assoc
thf(fact_5655_zip__left__commute,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs ) )
= ( map @ ( product_prod @ B @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ B @ ( product_prod @ A @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [Y5: B] :
( product_case_prod @ A @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [X4: A,Z2: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X4 @ ( product_Pair @ B @ C @ Y5 @ Z2 ) ) ) )
@ ( zip @ B @ ( product_prod @ A @ C ) @ Ys @ ( zip @ A @ C @ Xs @ Zs ) ) ) ) ).
% zip_left_commute
thf(fact_5656_TBOUND__prod__case,axiom,
! [C: $tType,B: $tType,A: $tType,T2: product_prod @ A @ B,F3: A > B > ( heap_Time_Heap @ C ),Bnd: A > B > nat] :
( ! [A4: A,B4: B] :
( ( T2
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( time_TBOUND @ C @ ( F3 @ A4 @ B4 ) @ ( Bnd @ A4 @ B4 ) ) )
=> ( time_TBOUND @ C @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F3 @ T2 ) @ ( product_case_prod @ A @ B @ nat @ Bnd @ T2 ) ) ) ).
% TBOUND_prod_case
thf(fact_5657_zip__commute,axiom,
! [B: $tType,A: $tType] :
( ( zip @ A @ B )
= ( ^ [Xs2: list @ A,Ys3: list @ B] :
( map @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X4: B,Y5: A] : ( product_Pair @ A @ B @ Y5 @ X4 ) )
@ ( zip @ B @ A @ Ys3 @ Xs2 ) ) ) ) ).
% zip_commute
thf(fact_5658_case__prod__rule,axiom,
! [A: $tType,B: $tType,C: $tType,X: product_prod @ A @ B,P2: assn,F3: A > B > ( heap_Time_Heap @ C ),Q2: C > assn] :
( ! [A4: A,B4: B] :
( ( X
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( hoare_hoare_triple @ C @ P2 @ ( F3 @ A4 @ B4 ) @ Q2 ) )
=> ( hoare_hoare_triple @ C @ P2 @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F3 @ X ) @ Q2 ) ) ).
% case_prod_rule
thf(fact_5659_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,F3: C > B,Ys: list @ C] :
( ( zip @ A @ B @ Xs @ ( map @ C @ B @ F3 @ Ys ) )
= ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [X4: A,Y5: C] : ( product_Pair @ A @ B @ X4 @ ( F3 @ Y5 ) ) )
@ ( zip @ A @ C @ Xs @ Ys ) ) ) ).
% zip_map2
thf(fact_5660_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,Xs: list @ C,Ys: list @ B] :
( ( zip @ A @ B @ ( map @ C @ A @ F3 @ Xs ) @ Ys )
= ( map @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ B @ ( product_prod @ A @ B )
@ ^ [X4: C] : ( product_Pair @ A @ B @ ( F3 @ X4 ) ) )
@ ( zip @ C @ B @ Xs @ Ys ) ) ) ).
% zip_map1
thf(fact_5661_map__prod__fun__zip,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F3: C > A,G3: D > B,Xs: list @ C,Ys: list @ D] :
( ( map @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X4: C,Y5: D] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G3 @ Y5 ) ) )
@ ( zip @ C @ D @ Xs @ Ys ) )
= ( zip @ A @ B @ ( map @ C @ A @ F3 @ Xs ) @ ( map @ D @ B @ G3 @ Ys ) ) ) ).
% map_prod_fun_zip
thf(fact_5662_less__eq__option__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less_eq @ ( option @ A ) )
= ( ^ [X4: option @ A,Y5: option @ A] :
( case_option @ $o @ A @ $true
@ ^ [Z2: A] : ( case_option @ $o @ A @ $false @ ( ord_less_eq @ A @ Z2 ) @ Y5 )
@ X4 ) ) ) ) ).
% less_eq_option_def
thf(fact_5663_less__option__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ ( option @ A ) )
= ( ^ [X4: option @ A] :
( case_option @ $o @ A @ $false
@ ^ [Y5: A] :
( case_option @ $o @ A @ $true
@ ^ [Z2: A] : ( ord_less @ A @ Z2 @ Y5 )
@ X4 ) ) ) ) ) ).
% less_option_def
thf(fact_5664_comp__fun__commute__relcomp__fold,axiom,
! [A: $tType,B: $tType,C: $tType,S3: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 )
=> ( finite6289374366891150609ommute @ ( product_prod @ C @ A ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ C @ A @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [X4: C,Y5: A,A6: set @ ( product_prod @ C @ B )] :
( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z2: B,A11: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y5 = W3 ) @ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X4 @ Z2 ) @ A11 ) @ A11 ) )
@ A6
@ S3 ) ) ) ) ).
% comp_fun_commute_relcomp_fold
thf(fact_5665_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S6: set @ ( A > B > $o ),X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S6 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_5666_finite__psubset__def,axiom,
! [A: $tType] :
( ( finite_psubset @ A )
= ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A6: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B7 )
& ( finite_finite2 @ A @ B7 ) ) ) ) ) ).
% finite_psubset_def
thf(fact_5667_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf_Inf @ ( A > B > $o ) )
= ( ^ [S6: set @ ( A > B > $o ),X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S6 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_5668_rel__of__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_of @ A @ B )
= ( ^ [M5: A > ( option @ B ),P5: ( product_prod @ A @ B ) > $o] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K3: A,V4: B] :
( ( ( M5 @ K3 )
= ( some @ B @ V4 ) )
& ( P5 @ ( product_Pair @ A @ B @ K3 @ V4 ) ) ) ) ) ) ) ).
% rel_of_def
thf(fact_5669_trancl__insert2,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y5: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ A2 ) @ ( transitive_trancl @ A @ R2 ) )
| ( X4 = A2 ) )
& ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y5 ) @ ( transitive_trancl @ A @ R2 ) )
| ( Y5 = B2 ) ) ) ) ) ) ) ).
% trancl_insert2
thf(fact_5670_foldr__snd__zip,axiom,
! [B: $tType,A: $tType,C: $tType,Ys: list @ A,Xs: list @ B,F3: A > C > C,B2: C] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( foldr @ ( product_prod @ B @ A ) @ C
@ ( product_case_prod @ B @ A @ ( C > C )
@ ^ [X4: B] : F3 )
@ ( zip @ B @ A @ Xs @ Ys )
@ B2 )
= ( foldr @ A @ C @ F3 @ Ys @ B2 ) ) ) ).
% foldr_snd_zip
thf(fact_5671_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
@ ^ [Xs2: list @ A,Ys3: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys3 ) )
| ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys3 ) @ ( lex @ A @ R5 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_5672_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M5: nat,N6: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N6
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M5 @ N6 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M5 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q8: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q8 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M5 @ N6 ) @ N6 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_5673_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L4: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q8: code_integer,R5: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L4 ) @ R5 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q8 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R5 @ ( numeral_numeral @ code_integer @ L4 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q8 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_5674_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L4: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q8: nat,R5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L4 ) @ R5 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q8 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L4 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q8 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_5675_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L4: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q8: int,R5: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L4 ) @ R5 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q8 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L4 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q8 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_5676_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L4: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q8: A,R5: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L4 ) @ R5 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q8 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L4 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q8 ) @ R5 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_5677_vebt__succ__code_I2_J,axiom,
! [Deg: nat,Info: option @ ( product_prod @ nat @ nat ),TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ( ord_less_eq @ nat @ Deg @ ( one_one @ nat ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
= ( none @ nat ) ) )
& ( ~ ( ord_less_eq @ nat @ Deg @ ( one_one @ nat ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
= ( case_option @ ( option @ nat ) @ ( product_prod @ nat @ nat ) @ ( none @ nat )
@ ( product_case_prod @ nat @ nat @ ( option @ nat )
@ ^ [Mi3: nat,Ma3: nat] :
( if @ ( option @ nat ) @ ( ord_less @ nat @ X @ Mi3 ) @ ( some @ nat @ Mi3 )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) )
@ Info ) ) ) ) ).
% vebt_succ_code(2)
thf(fact_5678_quickcheck__narrowing__samples_Onarrowing__samples_Osimps,axiom,
! [A: $tType] :
( ( ( code_term_of @ A )
& ( quickc6926020345158392990erm_of @ A ) )
=> ( ( code_T4080844693773952564amples @ A )
= ( ^ [A_of_integer: code_integer > ( product_prod @ A @ A ),Zero: A,I3: code_integer] :
( if @ ( list @ A ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ I3 )
@ ( product_case_prod @ A @ A @ ( list @ A )
@ ^ [A8: A,A16: A] : ( append @ A @ ( code_T4080844693773952564amples @ A @ A_of_integer @ Zero @ ( minus_minus @ code_integer @ I3 @ ( one_one @ code_integer ) ) ) @ ( cons @ A @ A8 @ ( cons @ A @ A16 @ ( nil @ A ) ) ) )
@ ( A_of_integer @ I3 ) )
@ ( cons @ A @ Zero @ ( nil @ A ) ) ) ) ) ) ).
% quickcheck_narrowing_samples.narrowing_samples.simps
thf(fact_5679_quickcheck__narrowing__samples_Onarrowing__samples_Oelims,axiom,
! [A: $tType] :
( ( ( code_term_of @ A )
& ( quickc6926020345158392990erm_of @ A ) )
=> ! [A_of_integer2: code_integer > ( product_prod @ A @ A ),Zero2: A,X: code_integer,Y: list @ A] :
( ( ( code_T4080844693773952564amples @ A @ A_of_integer2 @ Zero2 @ X )
= Y )
=> ( ( ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ X )
=> ( Y
= ( product_case_prod @ A @ A @ ( list @ A )
@ ^ [A8: A,A16: A] : ( append @ A @ ( code_T4080844693773952564amples @ A @ A_of_integer2 @ Zero2 @ ( minus_minus @ code_integer @ X @ ( one_one @ code_integer ) ) ) @ ( cons @ A @ A8 @ ( cons @ A @ A16 @ ( nil @ A ) ) ) )
@ ( A_of_integer2 @ X ) ) ) )
& ( ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ X )
=> ( Y
= ( cons @ A @ Zero2 @ ( nil @ A ) ) ) ) ) ) ) ).
% quickcheck_narrowing_samples.narrowing_samples.elims
thf(fact_5680_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_5681_integer__of__int__eq__of__int,axiom,
( code_integer_of_int
= ( ring_1_of_int @ code_integer ) ) ).
% integer_of_int_eq_of_int
thf(fact_5682_divmod__integer_H__def,axiom,
( ( unique8689654367752047608divmod @ code_integer )
= ( ^ [M5: num,N6: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M5 ) @ ( numeral_numeral @ code_integer @ N6 ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M5 ) @ ( numeral_numeral @ code_integer @ N6 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_5683_one__integer__def,axiom,
( ( one_one @ code_integer )
= ( code_integer_of_int @ ( one_one @ int ) ) ) ).
% one_integer_def
thf(fact_5684_less__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( ord_less @ code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( ord_less @ int @ Xa2 @ X ) ) ).
% less_integer.abs_eq
thf(fact_5685_divide__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( divide_divide @ code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( divide_divide @ int @ Xa2 @ X ) ) ) ).
% divide_integer.abs_eq
thf(fact_5686_modulo__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( modulo_modulo @ code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( modulo_modulo @ int @ Xa2 @ X ) ) ) ).
% modulo_integer.abs_eq
thf(fact_5687_integer__of__int__inject,axiom,
! [X: int,Y: int] :
( ( member @ int @ X @ ( top_top @ ( set @ int ) ) )
=> ( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
=> ( ( ( code_integer_of_int @ X )
= ( code_integer_of_int @ Y ) )
= ( X = Y ) ) ) ) ).
% integer_of_int_inject
thf(fact_5688_integer__of__int__induct,axiom,
! [P2: code_integer > $o,X: code_integer] :
( ! [Y3: int] :
( ( member @ int @ Y3 @ ( top_top @ ( set @ int ) ) )
=> ( P2 @ ( code_integer_of_int @ Y3 ) ) )
=> ( P2 @ X ) ) ).
% integer_of_int_induct
thf(fact_5689_integer__of__int__cases,axiom,
! [X: code_integer] :
~ ! [Y3: int] :
( ( X
= ( code_integer_of_int @ Y3 ) )
=> ~ ( member @ int @ Y3 @ ( top_top @ ( set @ int ) ) ) ) ).
% integer_of_int_cases
thf(fact_5690_zero__integer__def,axiom,
( ( zero_zero @ code_integer )
= ( code_integer_of_int @ ( zero_zero @ int ) ) ) ).
% zero_integer_def
thf(fact_5691_Code__Numeral_Oset__bit__integer_Oabs__eq,axiom,
! [Xa2: nat,X: int] :
( ( bit_se5668285175392031749et_bit @ code_integer @ Xa2 @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se5668285175392031749et_bit @ int @ Xa2 @ X ) ) ) ).
% Code_Numeral.set_bit_integer.abs_eq
thf(fact_5692_uminus__integer_Oabs__eq,axiom,
! [X: int] :
( ( uminus_uminus @ code_integer @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( uminus_uminus @ int @ X ) ) ) ).
% uminus_integer.abs_eq
thf(fact_5693_less__eq__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( ord_less_eq @ code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( ord_less_eq @ int @ Xa2 @ X ) ) ).
% less_eq_integer.abs_eq
thf(fact_5694_plus__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( plus_plus @ code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( plus_plus @ int @ Xa2 @ X ) ) ) ).
% plus_integer.abs_eq
thf(fact_5695_minus__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( minus_minus @ code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( minus_minus @ int @ Xa2 @ X ) ) ) ).
% minus_integer.abs_eq
thf(fact_5696_times__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( times_times @ code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( times_times @ int @ Xa2 @ X ) ) ) ).
% times_integer.abs_eq
thf(fact_5697_abs__integer_Oabs__eq,axiom,
! [X: int] :
( ( abs_abs @ code_integer @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( abs_abs @ int @ X ) ) ) ).
% abs_integer.abs_eq
thf(fact_5698_sgn__integer_Oabs__eq,axiom,
! [X: int] :
( ( sgn_sgn @ code_integer @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( sgn_sgn @ int @ X ) ) ) ).
% sgn_integer.abs_eq
thf(fact_5699_quickcheck__narrowing__samples_Onarrowing__samples_Opelims,axiom,
! [A: $tType] :
( ( ( code_term_of @ A )
& ( quickc6926020345158392990erm_of @ A ) )
=> ! [A_of_integer2: code_integer > ( product_prod @ A @ A ),Zero2: A,X: code_integer,Y: list @ A] :
( ( ( code_T4080844693773952564amples @ A @ A_of_integer2 @ Zero2 @ X )
= Y )
=> ( ( accp @ code_integer @ ( code_T1710151556404007877es_rel @ A @ A_of_integer2 ) @ X )
=> ~ ( ( ( ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ X )
=> ( Y
= ( product_case_prod @ A @ A @ ( list @ A )
@ ^ [A8: A,A16: A] : ( append @ A @ ( code_T4080844693773952564amples @ A @ A_of_integer2 @ Zero2 @ ( minus_minus @ code_integer @ X @ ( one_one @ code_integer ) ) ) @ ( cons @ A @ A8 @ ( cons @ A @ A16 @ ( nil @ A ) ) ) )
@ ( A_of_integer2 @ X ) ) ) )
& ( ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ X )
=> ( Y
= ( cons @ A @ Zero2 @ ( nil @ A ) ) ) ) )
=> ~ ( accp @ code_integer @ ( code_T1710151556404007877es_rel @ A @ A_of_integer2 ) @ X ) ) ) ) ) ).
% quickcheck_narrowing_samples.narrowing_samples.pelims
thf(fact_5700_quickcheck__narrowing__samples_Onarrowing__samples_Opsimps,axiom,
! [A: $tType] :
( ( ( code_term_of @ A )
& ( quickc6926020345158392990erm_of @ A ) )
=> ! [A_of_integer2: code_integer > ( product_prod @ A @ A ),I: code_integer,Zero2: A] :
( ( accp @ code_integer @ ( code_T1710151556404007877es_rel @ A @ A_of_integer2 ) @ I )
=> ( ( ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ I )
=> ( ( code_T4080844693773952564amples @ A @ A_of_integer2 @ Zero2 @ I )
= ( product_case_prod @ A @ A @ ( list @ A )
@ ^ [A8: A,A16: A] : ( append @ A @ ( code_T4080844693773952564amples @ A @ A_of_integer2 @ Zero2 @ ( minus_minus @ code_integer @ I @ ( one_one @ code_integer ) ) ) @ ( cons @ A @ A8 @ ( cons @ A @ A16 @ ( nil @ A ) ) ) )
@ ( A_of_integer2 @ I ) ) ) )
& ( ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ I )
=> ( ( code_T4080844693773952564amples @ A @ A_of_integer2 @ Zero2 @ I )
= ( cons @ A @ Zero2 @ ( nil @ A ) ) ) ) ) ) ) ).
% quickcheck_narrowing_samples.narrowing_samples.psimps
thf(fact_5701_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N6: nat,A8: A] :
( if @ A
@ ( N6
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N6 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_5702_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_5703_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_5704_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_5705_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_5706_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_5707_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_5708_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_5709_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_5710_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_5711_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_5712_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_5713_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_5714_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_5715_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_5716_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_5717_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_5718_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_5719_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_5720_quickcheck__narrowing__samples_Onarrowing__samples_Opinduct,axiom,
! [A: $tType] :
( ( ( code_term_of @ A )
& ( quickc6926020345158392990erm_of @ A ) )
=> ! [A_of_integer2: code_integer > ( product_prod @ A @ A ),A0: code_integer,P2: code_integer > $o] :
( ( accp @ code_integer @ ( code_T1710151556404007877es_rel @ A @ A_of_integer2 ) @ A0 )
=> ( ! [I2: code_integer] :
( ( accp @ code_integer @ ( code_T1710151556404007877es_rel @ A @ A_of_integer2 ) @ I2 )
=> ( ! [X5: product_prod @ A @ A,Xa: A,Y4: A] :
( ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ I2 )
=> ( ( X5
= ( A_of_integer2 @ I2 ) )
=> ( ( ( product_Pair @ A @ A @ Xa @ Y4 )
= X5 )
=> ( P2 @ ( minus_minus @ code_integer @ I2 @ ( one_one @ code_integer ) ) ) ) ) )
=> ( P2 @ I2 ) ) )
=> ( P2 @ A0 ) ) ) ) ).
% quickcheck_narrowing_samples.narrowing_samples.pinduct
thf(fact_5721_lowi__def,axiom,
( vEBT_VEBT_lowi
= ( ^ [X4: nat,N6: nat] : ( heap_Time_return @ nat @ ( modulo_modulo @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) ) ) ) ) ).
% lowi_def
thf(fact_5722_summable__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_5723_highi__def,axiom,
( vEBT_VEBT_highi
= ( ^ [X4: nat,N6: nat] : ( heap_Time_return @ nat @ ( divide_divide @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) ) ) ) ) ).
% highi_def
thf(fact_5724_highsimp,axiom,
! [X: nat,N: nat] :
( ( heap_Time_return @ nat @ ( vEBT_VEBT_high @ X @ N ) )
= ( vEBT_VEBT_highi @ X @ N ) ) ).
% highsimp
thf(fact_5725_lowsimp,axiom,
! [X: nat,N: nat] :
( ( heap_Time_return @ nat @ ( vEBT_VEBT_low @ X @ N ) )
= ( vEBT_VEBT_lowi @ X @ N ) ) ).
% lowsimp
thf(fact_5726_summable__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I: nat,F3: nat > A] :
( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( R5 = I ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) ) ) ) ).
% summable_single
thf(fact_5727_summable__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( summable @ A
@ ^ [N6: nat] : ( zero_zero @ A ) ) ) ).
% summable_zero
thf(fact_5728_summable__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F3: nat > A] :
( ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ C3 @ ( F3 @ N6 ) ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( summable @ A @ F3 ) ) ) ) ).
% summable_cmult_iff
thf(fact_5729_summable__divide__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: nat > A,C3: A] :
( ( summable @ A
@ ^ [N6: nat] : ( divide_divide @ A @ ( F3 @ N6 ) @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( summable @ A @ F3 ) ) ) ) ).
% summable_divide_iff
thf(fact_5730_summable__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P2: nat > $o,F3: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P2 ) )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( P2 @ R5 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite
thf(fact_5731_summable__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A5: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ A5 )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A5 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite_set
thf(fact_5732_summable__geometric__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A] :
( ( summable @ A @ ( power_power @ A @ C3 ) )
= ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C3 ) @ ( one_one @ real ) ) ) ) ).
% summable_geometric_iff
thf(fact_5733_length__corresp,axiom,
! [B: $tType,A: $tType] :
( ( heap @ A )
=> ! [Tree_array: array @ A,Tree_is: list @ B] :
( ( ( ex_assn @ ( list @ A ) @ ( snga_assn @ A @ Tree_array ) )
= ( top_top @ assn ) )
=> ( ( heap_Time_return @ nat @ ( size_size @ ( list @ B ) @ Tree_is ) )
= ( array_len @ A @ Tree_array ) ) ) ) ).
% length_corresp
thf(fact_5734_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_5735_summable__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N8: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ N8 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N8 )
=> ( ( F3 @ N2 )
= ( zero_zero @ A ) ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_finite
thf(fact_5736_summable__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F3: nat > A] :
( ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ C3 @ ( F3 @ N6 ) ) )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_mult_D
thf(fact_5737_summable__comparison__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,G3: nat > real] :
( ? [N10: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N10 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) @ ( G3 @ N2 ) ) )
=> ( ( summable @ real @ G3 )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_comparison_test
thf(fact_5738_summable__comparison__test_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G3: nat > real,N8: nat,F3: nat > A] :
( ( summable @ real @ G3 )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N8 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) @ ( G3 @ N2 ) ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_comparison_test'
thf(fact_5739_summable__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A
@ ^ [N6: nat] : ( F3 @ ( suc @ N6 ) ) )
= ( summable @ A @ F3 ) ) ) ).
% summable_Suc_iff
thf(fact_5740_summable__const__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C3: A] :
( ( summable @ A
@ ^ [Uu3: nat] : C3 )
= ( C3
= ( zero_zero @ A ) ) ) ) ).
% summable_const_iff
thf(fact_5741_suminf__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,G3: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F3 @ N2 ) @ ( G3 @ N2 ) )
=> ( ( summable @ A @ F3 )
=> ( ( summable @ A @ G3 )
=> ( ord_less_eq @ A @ ( suminf @ A @ F3 ) @ ( suminf @ A @ G3 ) ) ) ) ) ) ).
% suminf_le
thf(fact_5742_powser__insidea,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A,X: A,Z: A] :
( ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( F3 @ N6 ) @ ( power_power @ A @ X @ N6 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ real
@ ^ [N6: nat] : ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( F3 @ N6 ) @ ( power_power @ A @ Z @ N6 ) ) ) ) ) ) ) ).
% powser_insidea
thf(fact_5743_suminf__eq__zero__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) )
=> ( ( ( suminf @ A @ F3 )
= ( zero_zero @ A ) )
= ( ! [N6: nat] :
( ( F3 @ N6 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_5744_suminf__nonneg,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_nonneg
thf(fact_5745_suminf__pos,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N2: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_pos
thf(fact_5746_summable__0__powser,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A] :
( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( F3 @ N6 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N6 ) ) ) ) ).
% summable_0_powser
thf(fact_5747_summable__zero__power_H,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A] :
( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( F3 @ N6 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N6 ) ) ) ) ).
% summable_zero_power'
thf(fact_5748_summable__powser__split__head,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A,Z: A] :
( ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( F3 @ ( suc @ N6 ) ) @ ( power_power @ A @ Z @ N6 ) ) )
= ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( F3 @ N6 ) @ ( power_power @ A @ Z @ N6 ) ) ) ) ) ).
% summable_powser_split_head
thf(fact_5749_powser__split__head_I3_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z: A] :
( ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( F3 @ N6 ) @ ( power_power @ A @ Z @ N6 ) ) )
=> ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( F3 @ ( suc @ N6 ) ) @ ( power_power @ A @ Z @ N6 ) ) ) ) ) ).
% powser_split_head(3)
thf(fact_5750_summable__norm__comparison__test,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G3: nat > real] :
( ? [N10: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N10 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) @ ( G3 @ N2 ) ) )
=> ( ( summable @ real @ G3 )
=> ( summable @ real
@ ^ [N6: nat] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ N6 ) ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_5751_summable__rabs__comparison__test,axiom,
! [F3: nat > real,G3: nat > real] :
( ? [N10: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N10 @ N2 )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( F3 @ N2 ) ) @ ( G3 @ N2 ) ) )
=> ( ( summable @ real @ G3 )
=> ( summable @ real
@ ^ [N6: nat] : ( abs_abs @ real @ ( F3 @ N6 ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_5752_summable__rabs,axiom,
! [F3: nat > real] :
( ( summable @ real
@ ^ [N6: nat] : ( abs_abs @ real @ ( F3 @ N6 ) ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( suminf @ real @ F3 ) )
@ ( suminf @ real
@ ^ [N6: nat] : ( abs_abs @ real @ ( F3 @ N6 ) ) ) ) ) ).
% summable_rabs
thf(fact_5753_suminf__pos__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) )
= ( ? [I3: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_5754_suminf__pos2,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,I: nat] :
( ( summable @ A @ F3 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ).
% suminf_pos2
thf(fact_5755_suminf__le__const,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,X: A] :
( ( summable @ A @ F3 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N2 ) ) @ X )
=> ( ord_less_eq @ A @ ( suminf @ A @ F3 ) @ X ) ) ) ) ).
% suminf_le_const
thf(fact_5756_powser__inside,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,X: A,Z: A] :
( ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( F3 @ N6 ) @ ( power_power @ A @ X @ N6 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( F3 @ N6 ) @ ( power_power @ A @ Z @ N6 ) ) ) ) ) ) ).
% powser_inside
thf(fact_5757_suminf__split__head,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A
@ ^ [N6: nat] : ( F3 @ ( suc @ N6 ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F3 ) @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% suminf_split_head
thf(fact_5758_summableI__nonneg__bounded,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,X: A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N2 ) ) @ X )
=> ( summable @ A @ F3 ) ) ) ) ).
% summableI_nonneg_bounded
thf(fact_5759_bounded__imp__summable,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linord2810124833399127020strict @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A2: nat > A,B5: 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 ) ) @ B5 )
=> ( summable @ A @ A2 ) ) ) ) ).
% bounded_imp_summable
thf(fact_5760_summable__geometric,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C3 ) @ ( one_one @ real ) )
=> ( summable @ A @ ( power_power @ A @ C3 ) ) ) ) ).
% summable_geometric
thf(fact_5761_complete__algebra__summable__geometric,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( summable @ A @ ( power_power @ A @ X ) ) ) ) ).
% complete_algebra_summable_geometric
thf(fact_5762_summable__norm,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A] :
( ( summable @ real
@ ^ [N6: nat] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ N6 ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( suminf @ A @ F3 ) )
@ ( suminf @ real
@ ^ [N6: nat] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ N6 ) ) ) ) ) ) ).
% summable_norm
thf(fact_5763_sum__le__suminf,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,I6: set @ nat] :
( ( summable @ A @ F3 )
=> ( ( finite_finite2 @ nat @ I6 )
=> ( ! [N2: nat] :
( ( member @ nat @ N2 @ ( uminus_uminus @ ( set @ nat ) @ I6 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ I6 ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ).
% sum_le_suminf
thf(fact_5764_sum__less__suminf,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,N: nat] :
( ( summable @ A @ F3 )
=> ( ! [M3: nat] :
( ( ord_less_eq @ nat @ N @ M3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ M3 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% sum_less_suminf
thf(fact_5765_summable__Cauchy,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ( ( summable @ A )
= ( ^ [F4: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N7: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ N7 @ M5 )
=> ! [N6: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F4 @ ( set_or7035219750837199246ssThan @ nat @ M5 @ N6 ) ) ) @ E4 ) ) ) ) ) ) ).
% summable_Cauchy
thf(fact_5766_powser__split__head_I1_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z: A] :
( ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( F3 @ N6 ) @ ( power_power @ A @ Z @ N6 ) ) )
=> ( ( suminf @ A
@ ^ [N6: nat] : ( times_times @ A @ ( F3 @ N6 ) @ ( power_power @ A @ Z @ N6 ) ) )
= ( plus_plus @ A @ ( F3 @ ( zero_zero @ nat ) )
@ ( times_times @ A
@ ( suminf @ A
@ ^ [N6: nat] : ( times_times @ A @ ( F3 @ ( suc @ N6 ) ) @ ( power_power @ A @ Z @ N6 ) ) )
@ Z ) ) ) ) ) ).
% powser_split_head(1)
thf(fact_5767_powser__split__head_I2_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z: A] :
( ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( F3 @ N6 ) @ ( power_power @ A @ Z @ N6 ) ) )
=> ( ( times_times @ A
@ ( suminf @ A
@ ^ [N6: nat] : ( times_times @ A @ ( F3 @ ( suc @ N6 ) ) @ ( power_power @ A @ Z @ N6 ) ) )
@ Z )
= ( minus_minus @ A
@ ( suminf @ A
@ ^ [N6: nat] : ( times_times @ A @ ( F3 @ N6 ) @ ( power_power @ A @ Z @ N6 ) ) )
@ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% powser_split_head(2)
thf(fact_5768_suminf__exist__split,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real,F3: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( summable @ A @ F3 )
=> ? [N11: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ N11 @ N9 )
=> ( ord_less @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( suminf @ A
@ ^ [I3: nat] : ( F3 @ ( plus_plus @ nat @ I3 @ N9 ) ) ) )
@ R2 ) ) ) ) ) ).
% suminf_exist_split
thf(fact_5769_summable__partial__sum__bound,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,E3: real] :
( ( summable @ A @ F3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ~ ! [N11: nat] :
~ ! [M2: nat] :
( ( ord_less_eq @ nat @ N11 @ M2 )
=> ! [N9: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N9 ) ) ) @ E3 ) ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_5770_summable__power__series,axiom,
! [F3: nat > real,Z: real] :
( ! [I2: nat] : ( ord_less_eq @ real @ ( F3 @ I2 ) @ ( one_one @ real ) )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ I2 ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Z )
=> ( ( ord_less @ real @ Z @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [I3: nat] : ( times_times @ real @ ( F3 @ I3 ) @ ( power_power @ real @ Z @ I3 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_5771_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
@ ^ [N6: nat] : ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A2 @ N6 ) ) @ ( power_power @ real @ R2 @ N6 ) ) ) ) ) ) ) ).
% Abel_lemma
thf(fact_5772_summable__ratio__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [C3: real,N8: nat,F3: nat > A] :
( ( ord_less @ real @ C3 @ ( one_one @ real ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N8 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ ( suc @ N2 ) ) ) @ ( times_times @ real @ C3 @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) ) ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_ratio_test
thf(fact_5773_sum__less__suminf2,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,N: nat,I: nat] :
( ( summable @ A @ F3 )
=> ( ! [M3: nat] :
( ( ord_less_eq @ nat @ N @ M3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ M3 ) ) )
=> ( ( ord_less_eq @ nat @ N @ I )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_5774_suminf__eq__SUP__real,axiom,
! [X6: nat > real] :
( ( summable @ real @ X6 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( X6 @ I2 ) )
=> ( ( suminf @ real @ X6 )
= ( complete_Sup_Sup @ real
@ ( image @ nat @ real
@ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ real @ X6 @ ( set_ord_lessThan @ nat @ I3 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% suminf_eq_SUP_real
thf(fact_5775_sum__pos__lt__pair,axiom,
! [F3: nat > real,K: nat] :
( ( summable @ real @ F3 )
=> ( ! [D3: nat] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( F3 @ ( plus_plus @ nat @ K @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D3 ) ) ) @ ( F3 @ ( plus_plus @ nat @ K @ ( plus_plus @ nat @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D3 ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ord_less @ real @ ( groups7311177749621191930dd_sum @ nat @ real @ F3 @ ( set_ord_lessThan @ nat @ K ) ) @ ( suminf @ real @ F3 ) ) ) ) ).
% sum_pos_lt_pair
thf(fact_5776_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_5777_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_5778_tanh__ln__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( tanh @ real @ ( ln_ln @ real @ X ) )
= ( divide_divide @ real @ ( minus_minus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% tanh_ln_real
thf(fact_5779_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_5780_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_5781_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_5782_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_5783_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_5784_tanh__real__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( tanh @ real @ X ) @ ( tanh @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% tanh_real_less_iff
thf(fact_5785_tanh__real__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( tanh @ real @ X ) @ ( tanh @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% tanh_real_le_iff
thf(fact_5786_tanh__real__neg__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( tanh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% tanh_real_neg_iff
thf(fact_5787_tanh__real__pos__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% tanh_real_pos_iff
thf(fact_5788_tanh__real__nonneg__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% tanh_real_nonneg_iff
thf(fact_5789_tanh__real__nonpos__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( tanh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% tanh_real_nonpos_iff
thf(fact_5790_flip__bit__integer_Oabs__eq,axiom,
! [Xa2: nat,X: int] :
( ( bit_se8732182000553998342ip_bit @ code_integer @ Xa2 @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se8732182000553998342ip_bit @ int @ Xa2 @ X ) ) ) ).
% flip_bit_integer.abs_eq
thf(fact_5791_unset__bit__integer_Oabs__eq,axiom,
! [Xa2: nat,X: int] :
( ( bit_se2638667681897837118et_bit @ code_integer @ Xa2 @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se2638667681897837118et_bit @ int @ Xa2 @ X ) ) ) ).
% unset_bit_integer.abs_eq
thf(fact_5792_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_5793_tanh__real__lt__1,axiom,
! [X: real] : ( ord_less @ real @ ( tanh @ real @ X ) @ ( one_one @ real ) ) ).
% tanh_real_lt_1
thf(fact_5794_tanh__real__gt__neg1,axiom,
! [X: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( tanh @ real @ X ) ) ).
% tanh_real_gt_neg1
thf(fact_5795_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_5796_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_5797_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_5798_tan__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( divide_divide @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( tan @ A @ X ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% tan_double
thf(fact_5799_complex__unimodular__polar,axiom,
! [Z: complex] :
( ( ( real_V7770717601297561774m_norm @ complex @ Z )
= ( one_one @ real ) )
=> ~ ! [T7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
=> ( ( ord_less @ real @ T7 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( Z
!= ( complex2 @ ( cos @ real @ T7 ) @ ( sin @ real @ T7 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_5800_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat,J: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( ord_less @ nat @ J
@ ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) )
=> ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I ) @ J )
= ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J ) @ I ) ) ) ) ) ).
% nth_nth_transpose_sorted
thf(fact_5801_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_5802_sorted__wrt__rev__linord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( sorted_wrt @ A
@ ^ [X4: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X4 )
@ L )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ L ) ) ) ) ).
% sorted_wrt_rev_linord
thf(fact_5803_sorted__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) ) ) ).
% sorted_transpose
thf(fact_5804_rev__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rev @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_5805_rev__update,axiom,
! [A: $tType,K: nat,Xs: list @ A,Y: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( rev @ A @ ( list_update @ A @ Xs @ K @ Y ) )
= ( list_update @ A @ ( rev @ A @ Xs ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) @ ( one_one @ nat ) ) @ Y ) ) ) ).
% rev_update
thf(fact_5806_sorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I3: nat] :
( ( ord_less @ nat @ ( suc @ I3 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ ( suc @ I3 ) ) @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_5807_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J ) @ ( nth @ A @ Xs @ I ) ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_5808_sorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J3 ) @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_5809_foldr__max__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Y: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
=> ( ( ( Xs
= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y )
= Y ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y )
= ( ord_max @ A @ ( nth @ A @ Xs @ ( zero_zero @ nat ) ) @ Y ) ) ) ) ) ) ).
% foldr_max_sorted
thf(fact_5810_lemma__tan__total,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ? [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ord_less @ real @ Y @ ( tan @ real @ X3 ) ) ) ) ).
% lemma_tan_total
thf(fact_5811_tan__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( tan @ real @ X ) ) ) ) ).
% tan_gt_zero
thf(fact_5812_lemma__tan__total1,axiom,
! [Y: real] :
? [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X3 )
= Y ) ) ).
% lemma_tan_total1
thf(fact_5813_tan__mono__lt__eq,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ) ) ).
% tan_mono_lt_eq
thf(fact_5814_tan__monotone_H,axiom,
! [Y: real,X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y @ X )
= ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X ) ) ) ) ) ) ) ).
% tan_monotone'
thf(fact_5815_tan__monotone,axiom,
! [Y: real,X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X ) ) ) ) ) ).
% tan_monotone
thf(fact_5816_tan__total,axiom,
! [Y: real] :
? [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X3 )
= Y )
& ! [Y4: real] :
( ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y4 )
& ( ord_less @ real @ Y4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ Y4 )
= Y ) )
=> ( Y4 = X3 ) ) ) ).
% tan_total
thf(fact_5817_add__tan__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) )
= ( divide_divide @ A @ ( sin @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_5818_tan__total__pos,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X3 )
= Y ) ) ) ).
% tan_total_pos
thf(fact_5819_tan__pos__pi2__le,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tan @ real @ X ) ) ) ) ).
% tan_pos_pi2_le
thf(fact_5820_tan__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( tan @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% tan_less_zero
thf(fact_5821_tan__mono__le__eq,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ) ) ).
% tan_mono_le_eq
thf(fact_5822_tan__mono__le,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y ) ) ) ) ) ).
% tan_mono_le
thf(fact_5823_tan__bound__pi2,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( abs_abs @ real @ ( tan @ real @ X ) ) @ ( one_one @ real ) ) ) ).
% tan_bound_pi2
thf(fact_5824_arctan,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) )
& ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ ( arctan @ Y ) )
= Y ) ) ).
% arctan
thf(fact_5825_arctan__tan,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arctan @ ( tan @ real @ X ) )
= X ) ) ) ).
% arctan_tan
thf(fact_5826_arctan__unique,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( tan @ real @ X )
= Y )
=> ( ( arctan @ Y )
= X ) ) ) ) ).
% arctan_unique
thf(fact_5827_lemma__tan__add1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) )
= ( divide_divide @ A @ ( cos @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_5828_tan__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( minus_minus @ A @ X @ Y ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( minus_minus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_5829_tan__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( plus_plus @ A @ X @ Y ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( plus_plus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_5830_tan__total__pi4,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ? [Z3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) @ Z3 )
& ( ord_less @ real @ Z3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
& ( ( tan @ real @ Z3 )
= X ) ) ) ).
% tan_total_pi4
thf(fact_5831_length__transpose__sorted,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ( Xs
= ( nil @ ( list @ A ) ) )
=> ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( zero_zero @ nat ) ) )
& ( ( Xs
!= ( nil @ ( list @ A ) ) )
=> ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% length_transpose_sorted
thf(fact_5832_transpose__column__length,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ) ).
% transpose_column_length
thf(fact_5833_arctan__def,axiom,
( arctan
= ( ^ [Y5: real] :
( the @ real
@ ^ [X4: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X4 )
& ( ord_less @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X4 )
= Y5 ) ) ) ) ) ).
% arctan_def
thf(fact_5834_tan__sec,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ A @ ( inverse_inverse @ A @ ( cos @ A @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% tan_sec
thf(fact_5835_transpose__column,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( map @ ( list @ A ) @ A
@ ^ [Ys3: list @ A] : ( nth @ A @ Ys3 @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ).
% transpose_column
thf(fact_5836_arcsin__def,axiom,
( arcsin
= ( ^ [Y5: real] :
( the @ real
@ ^ [X4: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X4 )
& ( ord_less_eq @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X4 )
= Y5 ) ) ) ) ) ).
% arcsin_def
thf(fact_5837_time__array__map__entry,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,F3: A > A,P3: array @ A,H2: heap_ext @ product_unit] :
( ( ( time_fails @ ( array @ A ) @ ( array_map_entry @ A @ I @ F3 @ P3 ) @ H2 )
=> ( ( time_time @ ( array @ A ) @ ( array_map_entry @ A @ I @ F3 @ P3 ) @ H2 )
= ( zero_zero @ nat ) ) )
& ( ~ ( time_fails @ ( array @ A ) @ ( array_map_entry @ A @ I @ F3 @ P3 ) @ H2 )
=> ( ( time_time @ ( array @ A ) @ ( array_map_entry @ A @ I @ F3 @ P3 ) @ H2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% time_array_map_entry
thf(fact_5838_sin__tan,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( sin @ real @ X )
= ( divide_divide @ real @ ( tan @ real @ X ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_tan
thf(fact_5839_real__sqrt__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% real_sqrt_less_iff
thf(fact_5840_real__sqrt__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% real_sqrt_le_iff
thf(fact_5841_real__sqrt__gt__0__iff,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ Y ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y ) ) ).
% real_sqrt_gt_0_iff
thf(fact_5842_real__sqrt__lt__0__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% real_sqrt_lt_0_iff
thf(fact_5843_real__sqrt__ge__0__iff,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ Y ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) ) ).
% real_sqrt_ge_0_iff
thf(fact_5844_real__sqrt__le__0__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% real_sqrt_le_0_iff
thf(fact_5845_real__sqrt__lt__1__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ).
% real_sqrt_lt_1_iff
thf(fact_5846_real__sqrt__gt__1__iff,axiom,
! [Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( sqrt @ Y ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y ) ) ).
% real_sqrt_gt_1_iff
thf(fact_5847_real__sqrt__ge__1__iff,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ Y ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y ) ) ).
% real_sqrt_ge_1_iff
thf(fact_5848_real__sqrt__le__1__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ).
% real_sqrt_le_1_iff
thf(fact_5849_sin__arcsin,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arcsin @ Y ) )
= Y ) ) ) ).
% sin_arcsin
thf(fact_5850_real__sqrt__pow2__iff,axiom,
! [X: real] :
( ( ( power_power @ real @ ( sqrt @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% real_sqrt_pow2_iff
thf(fact_5851_real__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( sqrt @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X ) ) ).
% real_sqrt_pow2
thf(fact_5852_real__sqrt__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_le_mono
thf(fact_5853_real__sqrt__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_less_mono
thf(fact_5854_real__sqrt__ge__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_ge_one
thf(fact_5855_real__sqrt__eq__zero__cancel,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( sqrt @ X )
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ).
% real_sqrt_eq_zero_cancel
thf(fact_5856_real__sqrt__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_ge_zero
thf(fact_5857_real__sqrt__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_gt_zero
thf(fact_5858_real__div__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( divide_divide @ real @ X @ ( sqrt @ X ) )
= ( sqrt @ X ) ) ) ).
% real_div_sqrt
thf(fact_5859_sqrt__add__le__add__sqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ X @ Y ) ) @ ( plus_plus @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).
% sqrt_add_le_add_sqrt
thf(fact_5860_le__real__sqrt__sumsq,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( times_times @ real @ X @ X ) @ ( times_times @ real @ Y @ Y ) ) ) ) ).
% le_real_sqrt_sumsq
thf(fact_5861_arcsin__le__arcsin,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_le_arcsin
thf(fact_5862_arcsin__minus,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( arcsin @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( arcsin @ X ) ) ) ) ) ).
% arcsin_minus
thf(fact_5863_arcsin__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ).
% arcsin_le_mono
thf(fact_5864_arcsin__eq__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ( arcsin @ X )
= ( arcsin @ Y ) )
= ( X = Y ) ) ) ) ).
% arcsin_eq_iff
thf(fact_5865_sqrt2__less__2,axiom,
ord_less @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% sqrt2_less_2
thf(fact_5866_sqrt__divide__self__eq,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( divide_divide @ real @ ( sqrt @ X ) @ X )
= ( inverse_inverse @ real @ ( sqrt @ X ) ) ) ) ).
% sqrt_divide_self_eq
thf(fact_5867_cos__arcsin,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_arcsin
thf(fact_5868_real__less__rsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ord_less @ real @ X @ ( sqrt @ Y ) ) ) ).
% real_less_rsqrt
thf(fact_5869_real__le__rsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ord_less_eq @ real @ X @ ( sqrt @ Y ) ) ) ).
% real_le_rsqrt
thf(fact_5870_sqrt__le__D,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ Y )
=> ( ord_less_eq @ real @ X @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% sqrt_le_D
thf(fact_5871_arcsin__less__arcsin,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_less_arcsin
thf(fact_5872_arcsin__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% arcsin_less_mono
thf(fact_5873_real__le__lsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ X @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% real_le_lsqrt
thf(fact_5874_real__sqrt__unique,axiom,
! [Y: real,X: real] :
( ( ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( sqrt @ X )
= Y ) ) ) ).
% real_sqrt_unique
thf(fact_5875_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_5876_real__sqrt__sum__squares__ge1,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge1
thf(fact_5877_real__sqrt__sum__squares__ge2,axiom,
! [Y: real,X: real] : ( ord_less_eq @ real @ Y @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge2
thf(fact_5878_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A2: real,C3: real,B2: real,D2: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ ( plus_plus @ real @ A2 @ C3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( plus_plus @ real @ B2 @ D2 ) @ ( 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 @ C3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_triangle_ineq
thf(fact_5879_sqrt__ge__absD,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( sqrt @ Y ) )
=> ( ord_less_eq @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) ).
% sqrt_ge_absD
thf(fact_5880_cos__arcsin__nonzero,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X ) )
!= ( zero_zero @ real ) ) ) ) ).
% cos_arcsin_nonzero
thf(fact_5881_real__less__lsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ X @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% real_less_lsqrt
thf(fact_5882_sqrt__sum__squares__le__sum,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ X @ Y ) ) ) ) ).
% sqrt_sum_squares_le_sum
thf(fact_5883_real__inv__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( inverse_inverse @ real @ X ) ) ) ).
% real_inv_sqrt_pow2
thf(fact_5884_real__sqrt__ge__abs1,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs1
thf(fact_5885_real__sqrt__ge__abs2,axiom,
! [Y: real,X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs2
thf(fact_5886_sqrt__sum__squares__le__sum__abs,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( abs_abs @ real @ X ) @ ( abs_abs @ real @ Y ) ) ) ).
% sqrt_sum_squares_le_sum_abs
thf(fact_5887_ln__sqrt,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( sqrt @ X ) )
= ( divide_divide @ real @ ( ln_ln @ real @ X ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% ln_sqrt
thf(fact_5888_real__sqrt__power__even,axiom,
! [N: nat,X: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( sqrt @ X ) @ N )
= ( power_power @ real @ X @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_power_even
thf(fact_5889_arsinh__real__aux,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% arsinh_real_aux
thf(fact_5890_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X: real,Y: real,Xa2: real,Ya2: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_ge_zero
thf(fact_5891_arith__geo__mean__sqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ ( times_times @ real @ X @ Y ) ) @ ( divide_divide @ real @ ( plus_plus @ real @ X @ Y ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arith_geo_mean_sqrt
thf(fact_5892_powr__half__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( sqrt @ X ) ) ) ).
% powr_half_sqrt
thf(fact_5893_cos__x__y__le__one,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( divide_divide @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( one_one @ real ) ) ).
% cos_x_y_le_one
thf(fact_5894_real__sqrt__sum__squares__less,axiom,
! [X: real,U: real,Y: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ).
% real_sqrt_sum_squares_less
thf(fact_5895_arcosh__real__def,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( arcosh @ real @ X )
= ( ln_ln @ real @ ( plus_plus @ real @ X @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arcosh_real_def
thf(fact_5896_arcsin__lt__bounded,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% arcsin_lt_bounded
thf(fact_5897_arcsin__bounded,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% arcsin_bounded
thf(fact_5898_arcsin__ubound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arcsin_ubound
thf(fact_5899_arcsin__lbound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).
% arcsin_lbound
thf(fact_5900_sqrt__sum__squares__half__less,axiom,
! [X: real,U: real,Y: real] :
( ( ord_less @ real @ X @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ) ) ).
% sqrt_sum_squares_half_less
thf(fact_5901_arcsin__sin,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arcsin @ ( sin @ real @ X ) )
= X ) ) ) ).
% arcsin_sin
thf(fact_5902_sin__cos__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) )
=> ( ( sin @ real @ X )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( cos @ real @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_cos_sqrt
thf(fact_5903_arcsin,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ ( arcsin @ Y ) )
= Y ) ) ) ) ).
% arcsin
thf(fact_5904_arcsin__pi,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq @ real @ ( arcsin @ Y ) @ pi )
& ( ( sin @ real @ ( arcsin @ Y ) )
= Y ) ) ) ) ).
% arcsin_pi
thf(fact_5905_arcsin__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( arcsin @ X ) @ Y )
= ( ord_less_eq @ real @ X @ ( sin @ real @ Y ) ) ) ) ) ) ) ).
% arcsin_le_iff
thf(fact_5906_le__arcsin__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ Y @ ( arcsin @ X ) )
= ( ord_less_eq @ real @ ( sin @ real @ Y ) @ X ) ) ) ) ) ) ).
% le_arcsin_iff
thf(fact_5907_cos__tan,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( cos @ real @ X )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_tan
thf(fact_5908_time__array__swap,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,F3: A > A,P3: array @ A,H2: heap_ext @ product_unit,X: A] :
( ( ( time_fails @ ( array @ A ) @ ( array_map_entry @ A @ I @ F3 @ P3 ) @ H2 )
=> ( ( time_time @ A @ ( array_swap @ A @ I @ X @ P3 ) @ H2 )
= ( zero_zero @ nat ) ) )
& ( ~ ( time_fails @ ( array @ A ) @ ( array_map_entry @ A @ I @ F3 @ P3 ) @ H2 )
=> ( ( time_time @ A @ ( array_swap @ A @ I @ X @ P3 ) @ H2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% time_array_swap
thf(fact_5909_sin__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ Y ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_arccos_abs
thf(fact_5910_sin__arccos,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_arccos
thf(fact_5911_cos__arccos,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arccos @ Y ) )
= Y ) ) ) ).
% cos_arccos
thf(fact_5912_arccos__le__arccos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_le_arccos
thf(fact_5913_arccos__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
& ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) ) )
=> ( ( ( arccos @ X )
= ( arccos @ Y ) )
= ( X = Y ) ) ) ).
% arccos_eq_iff
thf(fact_5914_arccos__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( arccos @ X ) @ ( arccos @ Y ) )
= ( ord_less_eq @ real @ Y @ X ) ) ) ) ).
% arccos_le_mono
thf(fact_5915_arccos__lbound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) ) ) ) ).
% arccos_lbound
thf(fact_5916_arccos__less__arccos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_less_arccos
thf(fact_5917_arccos__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arccos @ X ) @ ( arccos @ Y ) )
= ( ord_less @ real @ Y @ X ) ) ) ) ).
% arccos_less_mono
thf(fact_5918_arccos__ubound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y ) @ pi ) ) ) ).
% arccos_ubound
thf(fact_5919_arccos__cos,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( arccos @ ( cos @ real @ X ) )
= X ) ) ) ).
% arccos_cos
thf(fact_5920_cos__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arccos @ Y ) )
= Y ) ) ).
% cos_arccos_abs
thf(fact_5921_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_5922_arccos__lt__bounded,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) )
& ( ord_less @ real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% arccos_lt_bounded
thf(fact_5923_arccos__bounded,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) )
& ( ord_less_eq @ real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% arccos_bounded
thf(fact_5924_sin__arccos__nonzero,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X ) )
!= ( zero_zero @ real ) ) ) ) ).
% sin_arccos_nonzero
thf(fact_5925_arccos__cos2,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( arccos @ ( cos @ real @ X ) )
= ( uminus_uminus @ real @ X ) ) ) ) ).
% arccos_cos2
thf(fact_5926_arccos__minus,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( arccos @ ( uminus_uminus @ real @ X ) )
= ( minus_minus @ real @ pi @ ( arccos @ X ) ) ) ) ) ).
% arccos_minus
thf(fact_5927_arccos__def,axiom,
( arccos
= ( ^ [Y5: real] :
( the @ real
@ ^ [X4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less_eq @ real @ X4 @ pi )
& ( ( cos @ real @ X4 )
= Y5 ) ) ) ) ) ).
% arccos_def
thf(fact_5928_arccos,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) )
& ( ord_less_eq @ real @ ( arccos @ Y ) @ pi )
& ( ( cos @ real @ ( arccos @ Y ) )
= Y ) ) ) ) ).
% arccos
thf(fact_5929_arccos__minus__abs,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( arccos @ ( uminus_uminus @ real @ X ) )
= ( minus_minus @ real @ pi @ ( arccos @ X ) ) ) ) ).
% arccos_minus_abs
thf(fact_5930_arccos__le__pi2,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arccos_le_pi2
thf(fact_5931_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
@ ^ [N6: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N6 ) ) @ ( power_power @ A @ Z @ N6 ) )
@ ( 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_5932_remove__rev__alt__def,axiom,
! [A: $tType] :
( ( remove_rev @ A )
= ( ^ [X4: A,Xs2: list @ A] :
( filter2 @ A
@ ^ [Y5: A] : Y5 != X4
@ ( rev @ A @ Xs2 ) ) ) ) ).
% remove_rev_alt_def
thf(fact_5933_cot__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cot @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% cot_less_zero
thf(fact_5934_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_5935_sums__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( sums @ A
@ ^ [N6: nat] : ( zero_zero @ A )
@ ( zero_zero @ A ) ) ) ).
% sums_zero
thf(fact_5936_powser__sums__zero__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: nat > A,X: A] :
( ( sums @ A
@ ^ [N6: nat] : ( times_times @ A @ ( A2 @ N6 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N6 ) )
@ X )
= ( ( A2 @ ( zero_zero @ nat ) )
= X ) ) ) ).
% powser_sums_zero_iff
thf(fact_5937_sums__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I: nat,F3: nat > A] :
( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( R5 = I ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) )
@ ( F3 @ I ) ) ) ).
% sums_single
thf(fact_5938_sums__0,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A] :
( ! [N2: nat] :
( ( F3 @ N2 )
= ( zero_zero @ A ) )
=> ( sums @ A @ F3 @ ( zero_zero @ A ) ) ) ) ).
% sums_0
thf(fact_5939_sums__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,G3: nat > A,S: A,T2: A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F3 @ N2 ) @ ( G3 @ N2 ) )
=> ( ( sums @ A @ F3 @ S )
=> ( ( sums @ A @ G3 @ T2 )
=> ( ord_less_eq @ A @ S @ T2 ) ) ) ) ) ).
% sums_le
thf(fact_5940_sums__mult2__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C3: A,F3: nat > A,D2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N6: nat] : ( times_times @ A @ ( F3 @ N6 ) @ C3 )
@ ( times_times @ A @ D2 @ C3 ) )
= ( sums @ A @ F3 @ D2 ) ) ) ) ).
% sums_mult2_iff
thf(fact_5941_sums__mult__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C3: A,F3: nat > A,D2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N6: nat] : ( times_times @ A @ C3 @ ( F3 @ N6 ) )
@ ( times_times @ A @ C3 @ D2 ) )
= ( sums @ A @ F3 @ D2 ) ) ) ) ).
% sums_mult_iff
thf(fact_5942_sums__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F3: nat > A,A2: A] :
( ( sums @ A
@ ^ [N6: nat] : ( times_times @ A @ C3 @ ( F3 @ N6 ) )
@ A2 )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( sums @ A @ F3 @ ( divide_divide @ A @ A2 @ C3 ) ) ) ) ) ).
% sums_mult_D
thf(fact_5943_sums__Suc__imp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,S: A] :
( ( ( F3 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N6: nat] : ( F3 @ ( suc @ N6 ) )
@ S )
=> ( sums @ A @ F3 @ S ) ) ) ) ).
% sums_Suc_imp
thf(fact_5944_sums__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,S: A] :
( ( sums @ A
@ ^ [N6: nat] : ( F3 @ ( suc @ N6 ) )
@ S )
= ( sums @ A @ F3 @ ( plus_plus @ A @ S @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc_iff
thf(fact_5945_sums__Suc,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,L: A] :
( ( sums @ A
@ ^ [N6: nat] : ( F3 @ ( suc @ N6 ) )
@ L )
=> ( sums @ A @ F3 @ ( plus_plus @ A @ L @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc
thf(fact_5946_sums__zero__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [N: nat,F3: nat > A,S: A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ( F3 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [I3: nat] : ( F3 @ ( plus_plus @ nat @ I3 @ N ) )
@ S )
= ( sums @ A @ F3 @ S ) ) ) ) ).
% sums_zero_iff_shift
thf(fact_5947_sums__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N8: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ N8 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N8 )
=> ( ( F3 @ N2 )
= ( zero_zero @ A ) ) )
=> ( sums @ A @ F3 @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ N8 ) ) ) ) ) ).
% sums_finite
thf(fact_5948_sums__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P2: nat > $o,F3: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P2 ) )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( P2 @ R5 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( collect @ nat @ P2 ) ) ) ) ) ).
% sums_If_finite
thf(fact_5949_sums__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A5: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ A5 )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A5 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ A5 ) ) ) ) ).
% sums_If_finite_set
thf(fact_5950_powser__sums__if,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [M: nat,Z: A] :
( sums @ A
@ ^ [N6: nat] : ( times_times @ A @ ( if @ A @ ( N6 = M ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ ( power_power @ A @ Z @ N6 ) )
@ ( power_power @ A @ Z @ M ) ) ) ).
% powser_sums_if
thf(fact_5951_powser__sums__zero,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: nat > A] :
( sums @ A
@ ^ [N6: nat] : ( times_times @ A @ ( A2 @ N6 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N6 ) )
@ ( A2 @ ( zero_zero @ nat ) ) ) ) ).
% powser_sums_zero
thf(fact_5952_sums__If__finite__set_H,axiom,
! [A: $tType] :
( ( ( topolo1287966508704411220up_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G3: nat > A,S3: A,A5: set @ nat,S7: A,F3: nat > A] :
( ( sums @ A @ G3 @ S3 )
=> ( ( finite_finite2 @ nat @ A5 )
=> ( ( S7
= ( plus_plus @ A @ S3
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N6: nat] : ( minus_minus @ A @ ( F3 @ N6 ) @ ( G3 @ N6 ) )
@ A5 ) ) )
=> ( sums @ A
@ ^ [N6: nat] : ( if @ A @ ( member @ nat @ N6 @ A5 ) @ ( F3 @ N6 ) @ ( G3 @ N6 ) )
@ S7 ) ) ) ) ) ).
% sums_If_finite_set'
thf(fact_5953_sums__group,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A,S: A,K: nat] :
( ( sums @ A @ F3 @ S )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( sums @ A
@ ^ [N6: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ N6 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ N6 @ K ) @ K ) ) )
@ S ) ) ) ) ).
% sums_group
thf(fact_5954_geometric__sums,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C3 ) @ ( one_one @ real ) )
=> ( sums @ A @ ( power_power @ A @ C3 ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C3 ) ) ) ) ) ).
% geometric_sums
thf(fact_5955_power__half__series,axiom,
( sums @ real
@ ^ [N6: nat] : ( power_power @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( suc @ N6 ) )
@ ( one_one @ real ) ) ).
% power_half_series
thf(fact_5956_cot__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cot @ real @ X ) ) ) ) ).
% cot_gt_zero
thf(fact_5957_sin__x__sin__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P7: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N6: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P7 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) )
@ ( 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 @ P7 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P7 @ N6 ) ) ) ) @ ( semiring_char_0_fact @ real @ P7 ) ) ) @ ( power_power @ A @ X @ N6 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P7 @ N6 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P7 ) )
@ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ).
% sin_x_sin_y
thf(fact_5958_sums__cos__x__plus__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P7: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N6: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P7 ) @ ( 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 @ P7 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P7 @ N6 ) ) ) ) @ ( semiring_char_0_fact @ real @ P7 ) ) @ ( power_power @ A @ X @ N6 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P7 @ N6 ) ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P7 ) )
@ ( cos @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% sums_cos_x_plus_y
thf(fact_5959_scaleR__cancel__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X: A,B2: real] :
( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X )
= ( real_V8093663219630862766scaleR @ A @ B2 @ X ) )
= ( ( A2 = B2 )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_cancel_right
thf(fact_5960_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_5961_scaleR__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X: A] :
( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ real ) )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_eq_0_iff
thf(fact_5962_scaleR__zero__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( zero_zero @ real ) @ X )
= ( zero_zero @ A ) ) ) ).
% scaleR_zero_left
thf(fact_5963_scaleR__right__imp__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,A2: real,B2: real] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X )
= ( real_V8093663219630862766scaleR @ A @ B2 @ X ) )
=> ( A2 = B2 ) ) ) ) ).
% scaleR_right_imp_eq
thf(fact_5964_scaleR__right__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B2: real,A2: real,C3: A] :
( ( ord_less_eq @ real @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ C3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ C3 ) ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_5965_scaleR__right__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,X: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ) ).
% scaleR_right_mono
thf(fact_5966_scaleR__le__cancel__left,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B2 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% scaleR_le_cancel_left
thf(fact_5967_scaleR__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% scaleR_le_cancel_left_neg
thf(fact_5968_scaleR__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% scaleR_le_cancel_left_pos
thf(fact_5969_scaleR__left__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B2: A,A2: A,C3: real] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ real @ C3 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B2 ) ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_5970_scaleR__left__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X: A,Y: A,A2: real] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ) ).
% scaleR_left_mono
thf(fact_5971_vector__fraction__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U: real,V3: real,A2: A,X: A] :
( ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V3 ) @ A2 )
= X )
= ( ( ( V3
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ A ) ) )
& ( ( V3
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ U @ A2 )
= ( real_V8093663219630862766scaleR @ A @ V3 @ X ) ) ) ) ) ) ).
% vector_fraction_eq_iff
thf(fact_5972_eq__vector__fraction__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,U: real,V3: real,A2: A] :
( ( X
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V3 ) @ A2 ) )
= ( ( ( V3
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ A ) ) )
& ( ( V3
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ V3 @ X )
= ( real_V8093663219630862766scaleR @ A @ U @ A2 ) ) ) ) ) ) ).
% eq_vector_fraction_iff
thf(fact_5973_Real__Vector__Spaces_Ole__add__iff1,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,E3: A,C3: A,B2: real,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E3 ) @ D2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ E3 ) @ C3 ) @ D2 ) ) ) ).
% Real_Vector_Spaces.le_add_iff1
thf(fact_5974_Real__Vector__Spaces_Ole__add__iff2,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,E3: A,C3: A,B2: real,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E3 ) @ D2 ) )
= ( ord_less_eq @ A @ C3 @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ).
% Real_Vector_Spaces.le_add_iff2
thf(fact_5975_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_5976_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_5977_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_5978_scaleR__nonpos__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_5979_scaleR__nonneg__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_5980_scaleR__nonneg__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_5981_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_5982_split__scaleR__neg__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( zero_zero @ A ) ) ) ) ).
% split_scaleR_neg_le
thf(fact_5983_scaleR__mono_H,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,C3: A,D2: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ C3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_5984_scaleR__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,X: A,Y: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ Y ) ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_5985_scaleR__left__le__one__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X: A,A2: real] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ real @ A2 @ ( one_one @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ X ) ) ) ) ).
% scaleR_left_le_one_le
thf(fact_5986_scaleR__image__atLeastAtMost,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,X: A,Y: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( image @ A @ A @ ( real_V8093663219630862766scaleR @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ Y ) ) ) ) ) ).
% scaleR_image_atLeastAtMost
thf(fact_5987_neg__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) ) ) ) ) ).
% neg_le_divideR_eq
thf(fact_5988_neg__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) @ A2 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ B2 ) ) ) ) ).
% neg_divideR_le_eq
thf(fact_5989_pos__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ B2 ) ) ) ) ).
% pos_le_divideR_eq
thf(fact_5990_pos__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) ) ) ) ) ).
% pos_divideR_le_eq
thf(fact_5991_neg__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) )
= ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) ) ) ) ) ).
% neg_less_divideR_eq
thf(fact_5992_neg__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) @ A2 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ B2 ) ) ) ) ).
% neg_divideR_less_eq
thf(fact_5993_pos__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ B2 ) ) ) ) ).
% pos_less_divideR_eq
thf(fact_5994_pos__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) @ A2 )
= ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) ) ) ) ) ).
% pos_divideR_less_eq
thf(fact_5995_nonzero__inverse__scaleR__distrib,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [A2: real,X: A] :
( ( A2
!= ( zero_zero @ real ) )
=> ( ( X
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) )
= ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ A2 ) @ ( inverse_inverse @ A @ X ) ) ) ) ) ) ).
% nonzero_inverse_scaleR_distrib
thf(fact_5996_neg__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divideR_le_eq
thf(fact_5997_neg__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) ) ) ) ) ).
% neg_le_minus_divideR_eq
thf(fact_5998_pos__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) ) ) ) ) ).
% pos_minus_divideR_le_eq
thf(fact_5999_pos__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divideR_eq
thf(fact_6000_neg__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) @ A2 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divideR_less_eq
thf(fact_6001_neg__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) ) ) ) ) ).
% neg_less_minus_divideR_eq
thf(fact_6002_pos__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) @ A2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) ) ) ) ) ).
% pos_minus_divideR_less_eq
thf(fact_6003_pos__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divideR_eq
thf(fact_6004_exp__first__term,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X4: A] :
( plus_plus @ A @ ( one_one @ A )
@ ( suminf @ A
@ ^ [N6: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( suc @ N6 ) ) ) @ ( power_power @ A @ X4 @ ( suc @ N6 ) ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_6005_sinh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N6: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N6 ) ) @ ( power_power @ A @ X @ N6 ) ) )
@ ( sinh @ A @ X ) ) ) ).
% sinh_converges
thf(fact_6006_cos__x__cos__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P7: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N6: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P7 )
& ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) )
@ ( 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 @ P7 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P7 @ N6 ) ) ) ) @ ( semiring_char_0_fact @ real @ P7 ) ) @ ( power_power @ A @ X @ N6 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P7 @ N6 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P7 ) )
@ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ).
% cos_x_cos_y
thf(fact_6007_diffs__equiv,axiom,
! [A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( ring_1 @ A ) )
=> ! [C3: nat > A,X: A] :
( ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N6 ) @ ( power_power @ A @ X @ N6 ) ) )
=> ( sums @ A
@ ^ [N6: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N6 ) @ ( C3 @ N6 ) ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N6 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( suminf @ A
@ ^ [N6: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N6 ) @ ( power_power @ A @ X @ N6 ) ) ) ) ) ) ).
% diffs_equiv
thf(fact_6008_cosh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N6: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N6 ) ) @ ( power_power @ A @ X @ N6 ) ) @ ( zero_zero @ A ) )
@ ( cosh @ A @ X ) ) ) ).
% cosh_converges
thf(fact_6009_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_6010_cosh__real__ge__1,axiom,
! [X: real] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_ge_1
thf(fact_6011_cosh__real__nonneg,axiom,
! [X: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_nonneg
thf(fact_6012_cosh__real__nonneg__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ).
% cosh_real_nonneg_le_iff
thf(fact_6013_cosh__real__nonpos__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less_eq @ real @ Y @ X ) ) ) ) ).
% cosh_real_nonpos_le_iff
thf(fact_6014_cosh__real__pos,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_pos
thf(fact_6015_sinh__less__cosh__real,axiom,
! [X: real] : ( ord_less @ real @ ( sinh @ real @ X ) @ ( cosh @ real @ X ) ) ).
% sinh_less_cosh_real
thf(fact_6016_sinh__le__cosh__real,axiom,
! [X: real] : ( ord_less_eq @ real @ ( sinh @ real @ X ) @ ( cosh @ real @ X ) ) ).
% sinh_le_cosh_real
thf(fact_6017_cosh__real__strict__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) ) ) ) ).
% cosh_real_strict_mono
thf(fact_6018_cosh__real__nonneg__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% cosh_real_nonneg_less_iff
thf(fact_6019_cosh__real__nonpos__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less @ real @ Y @ X ) ) ) ) ).
% cosh_real_nonpos_less_iff
thf(fact_6020_arcosh__cosh__real,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( arcosh @ real @ ( cosh @ real @ X ) )
= X ) ) ).
% arcosh_cosh_real
thf(fact_6021_diffs__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( diffs @ A )
= ( ^ [C7: nat > A,N6: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N6 ) ) @ ( C7 @ ( suc @ N6 ) ) ) ) ) ) ).
% diffs_def
thf(fact_6022_tanh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cosh @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cosh @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( tanh @ A @ ( plus_plus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tanh @ A @ X ) @ ( tanh @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tanh @ A @ X ) @ ( tanh @ A @ Y ) ) ) ) ) ) ) ) ).
% tanh_add
thf(fact_6023_termdiff__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,K4: real,C3: nat > A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ K4 )
=> ( ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K4 )
=> ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( C3 @ N6 ) @ ( power_power @ A @ X3 @ N6 ) ) ) )
=> ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N6 ) @ ( power_power @ A @ X @ N6 ) ) ) ) ) ) ).
% termdiff_converges
thf(fact_6024_cosh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cosh @ A @ X )
= ( zero_zero @ A ) )
= ( ( power_power @ A @ ( exp @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% cosh_zero_iff
thf(fact_6025_cosh__ln__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( cosh @ real @ ( ln_ln @ real @ X ) )
= ( divide_divide @ real @ ( plus_plus @ real @ X @ ( inverse_inverse @ real @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% cosh_ln_real
thf(fact_6026_sint__range__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( ring_1_signed @ A @ int @ W2 ) )
& ( ord_less @ int @ ( ring_1_signed @ A @ int @ W2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% sint_range_size
thf(fact_6027_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_6028_signed__0,axiom,
! [B: $tType,A: $tType] :
( ( ( ring_1 @ A )
& ( type_len @ B ) )
=> ( ( ring_1_signed @ B @ A @ ( zero_zero @ ( word @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% signed_0
thf(fact_6029_signed__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_char_0 @ A ) )
=> ! [W2: word @ B] :
( ( ( ring_1_signed @ B @ A @ W2 )
= ( zero_zero @ A ) )
= ( W2
= ( zero_zero @ ( word @ B ) ) ) ) ) ).
% signed_eq_0_iff
thf(fact_6030_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_6031_sint__above__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,X: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W2 ) @ ( one_one @ nat ) ) ) @ X )
=> ( ord_less @ int @ ( ring_1_signed @ A @ int @ W2 ) @ X ) ) ) ).
% sint_above_size
thf(fact_6032_sint__below__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int,W2: word @ A] :
( ( ord_less_eq @ int @ X @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W2 ) @ ( one_one @ nat ) ) ) ) )
=> ( ord_less_eq @ int @ X @ ( ring_1_signed @ A @ int @ W2 ) ) ) ) ).
% sint_below_size
thf(fact_6033_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D6: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z9: int,Z2: int] :
( ( ord_less_eq @ int @ D6 @ Z2 )
& ( ord_less @ int @ Z9 @ Z2 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_6034_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D6: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z9: int,Z2: int] :
( ( ord_less_eq @ int @ D6 @ Z9 )
& ( ord_less @ int @ Z9 @ Z2 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_6035_set__decode__0,axiom,
! [X: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ ( nat_set_decode @ X ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% set_decode_0
thf(fact_6036_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_6037_of__bool__eq__0__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P2: $o] :
( ( ( zero_neq_one_of_bool @ A @ P2 )
= ( zero_zero @ A ) )
= ~ P2 ) ) ).
% of_bool_eq_0_iff
thf(fact_6038_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_6039_of__bool__less__eq__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P2: $o,Q2: $o] :
( ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P2 ) @ ( zero_neq_one_of_bool @ A @ Q2 ) )
= ( P2
=> Q2 ) ) ) ).
% of_bool_less_eq_iff
thf(fact_6040_of__bool__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P2: $o,Q2: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P2 ) @ ( zero_neq_one_of_bool @ A @ Q2 ) )
= ( ~ P2
& Q2 ) ) ) ).
% of_bool_less_iff
thf(fact_6041_of__nat__of__bool,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P2: $o] :
( ( semiring_1_of_nat @ A @ ( zero_neq_one_of_bool @ nat @ P2 ) )
= ( zero_neq_one_of_bool @ A @ P2 ) ) ) ).
% of_nat_of_bool
thf(fact_6042_zero__less__of__bool__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P2: $o] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P2 ) )
= P2 ) ) ).
% zero_less_of_bool_iff
thf(fact_6043_of__bool__less__one__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P2: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P2 ) @ ( one_one @ A ) )
= ~ P2 ) ) ).
% of_bool_less_one_iff
thf(fact_6044_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_6045_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_6046_set__encode__inverse,axiom,
! [A5: set @ nat] :
( ( finite_finite2 @ nat @ A5 )
=> ( ( nat_set_decode @ ( nat_set_encode @ A5 ) )
= A5 ) ) ).
% set_encode_inverse
thf(fact_6047_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_6048_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_6049_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_6050_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_6051_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_6052_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A5: set @ B,P2: B > $o,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P2 @ X4 ) ) @ ( F3 @ X4 ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P2 ) ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_6053_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A5: set @ B,F3: B > A,P2: B > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ ( zero_neq_one_of_bool @ A @ ( P2 @ X4 ) ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P2 ) ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_6054_mod__word__by__minus__1__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A] :
( ( modulo_modulo @ ( word @ A ) @ W2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( times_times @ ( word @ A ) @ W2 @ ( zero_neq_one_of_bool @ ( word @ A ) @ ( ord_less @ ( word @ A ) @ W2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ) ).
% mod_word_by_minus_1_eq
thf(fact_6055_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_6056_set__decode__Suc,axiom,
! [N: nat,X: nat] :
( ( member @ nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
= ( member @ nat @ N @ ( nat_set_decode @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_6057_sum__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A5: set @ B,P2: B > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( zero_neq_one_of_bool @ A @ ( P2 @ X4 ) )
@ A5 )
= ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P2 ) ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_6058_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_6059_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_6060_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_6061_finite__set__decode,axiom,
! [N: nat] : ( finite_finite2 @ nat @ ( nat_set_decode @ N ) ) ).
% finite_set_decode
thf(fact_6062_zero__less__eq__of__bool,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P2: $o] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P2 ) ) ) ).
% zero_less_eq_of_bool
thf(fact_6063_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P7: $o] : ( if @ A @ P7 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_6064_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P2: A > $o,P3: $o] :
( ( P2 @ ( zero_neq_one_of_bool @ A @ P3 ) )
= ( ( P3
=> ( P2 @ ( one_one @ A ) ) )
& ( ~ P3
=> ( P2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_6065_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P2: A > $o,P3: $o] :
( ( P2 @ ( zero_neq_one_of_bool @ A @ P3 ) )
= ( ~ ( ( P3
& ~ ( P2 @ ( one_one @ A ) ) )
| ( ~ P3
& ~ ( P2 @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_6066_of__bool__less__eq__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P2: $o] : ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P2 ) @ ( one_one @ A ) ) ) ).
% of_bool_less_eq_one
thf(fact_6067_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( product_case_prod @ int @ int @ int
@ ^ [Q8: int,R5: int] :
( plus_plus @ int @ Q8
@ ( zero_neq_one_of_bool @ int
@ ( R5
!= ( zero_zero @ int ) ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_6068_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_6069_horner__sum__of__bool__2__less,axiom,
! [Bs2: list @ $o] : ( ord_less @ int @ ( groups4207007520872428315er_sum @ $o @ int @ ( zero_neq_one_of_bool @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Bs2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ $o ) @ Bs2 ) ) ) ).
% horner_sum_of_bool_2_less
thf(fact_6070_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_6071_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_6072_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_6073_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_6074_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_6075_sort__key__by__quicksort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ B @ A )
= ( ^ [F4: B > A,Xs2: list @ B] :
( append @ B
@ ( linorder_sort_key @ B @ A @ F4
@ ( filter2 @ B
@ ^ [X4: B] : ( ord_less @ A @ ( F4 @ X4 ) @ ( F4 @ ( nth @ B @ Xs2 @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Xs2 ) )
@ ( append @ B
@ ( filter2 @ B
@ ^ [X4: B] :
( ( F4 @ X4 )
= ( F4 @ ( nth @ B @ Xs2 @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Xs2 )
@ ( linorder_sort_key @ B @ A @ F4
@ ( filter2 @ B
@ ^ [X4: B] : ( ord_less @ A @ ( F4 @ ( nth @ B @ Xs2 @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( F4 @ X4 ) )
@ Xs2 ) ) ) ) ) ) ) ).
% sort_key_by_quicksort
thf(fact_6076_sort__by__quicksort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ Xs )
= ( append @ A
@ ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ ( filter2 @ A
@ ^ [X4: A] : ( ord_less @ A @ X4 @ ( nth @ A @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ Xs ) )
@ ( append @ A
@ ( filter2 @ A
@ ^ [X4: A] :
( X4
= ( nth @ A @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ Xs )
@ ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ ( filter2 @ A @ ( ord_less @ A @ ( nth @ A @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Xs ) ) ) ) ) ) ).
% sort_by_quicksort
thf(fact_6077_False__map2__or,axiom,
! [Xs: list @ $o,Ys: list @ $o] :
( ( ord_less_eq @ ( set @ $o ) @ ( set2 @ $o @ Xs ) @ ( insert @ $o @ $false @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ( ( size_size @ ( list @ $o ) @ Ys )
= ( size_size @ ( list @ $o ) @ Xs ) )
=> ( ( map @ ( product_prod @ $o @ $o ) @ $o @ ( product_case_prod @ $o @ $o @ $o @ (|) ) @ ( zip @ $o @ $o @ Xs @ Ys ) )
= Ys ) ) ) ).
% False_map2_or
thf(fact_6078_False__map2__and,axiom,
! [Xs: list @ $o,Ys: list @ $o] :
( ( ord_less_eq @ ( set @ $o ) @ ( set2 @ $o @ Xs ) @ ( insert @ $o @ $false @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ( ( size_size @ ( list @ $o ) @ Ys )
= ( size_size @ ( list @ $o ) @ Xs ) )
=> ( ( map @ ( product_prod @ $o @ $o ) @ $o @ ( product_case_prod @ $o @ $o @ $o @ (&) ) @ ( zip @ $o @ $o @ Xs @ Ys ) )
= Xs ) ) ) ).
% False_map2_and
thf(fact_6079_sorted__sort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ Xs ) ) ) ).
% sorted_sort
thf(fact_6080_sorted__sort__id,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ Xs )
= Xs ) ) ) ).
% sorted_sort_id
thf(fact_6081_sorted__sort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( linorder_sort_key @ B @ A @ F3 @ Xs ) ) ) ) ).
% sorted_sort_key
thf(fact_6082_word__2p__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W2: word @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W2 ) )
=> ( ( ord_less @ ( word @ A ) @ W2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% word_2p_lem
thf(fact_6083_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N6: nat,A8: A] :
( if @ A
@ ( N6
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N6 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_6084_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_6085_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_6086_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_6087_unsigned__0,axiom,
! [B: $tType,A: $tType] :
( ( ( semiring_1 @ A )
& ( type_len @ B ) )
=> ( ( semiring_1_unsigned @ B @ A @ ( zero_zero @ ( word @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% unsigned_0
thf(fact_6088_uint__nonnegative,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_unsigned @ A @ int @ W2 ) ) ) ).
% uint_nonnegative
thf(fact_6089_uint__le__0__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( zero_zero @ int ) )
= ( ( semiring_1_unsigned @ A @ int @ X )
= ( zero_zero @ int ) ) ) ) ).
% uint_le_0_iff
thf(fact_6090_uint__ge__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_unsigned @ A @ int @ X ) ) ) ).
% uint_ge_0
thf(fact_6091_uint__lt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( zero_zero @ int ) ) ) ).
% uint_lt_0
thf(fact_6092_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_6093_word__le__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less_eq @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) ) ) ) ).
% word_le_no
thf(fact_6094_word__less__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) ) ) ) ).
% word_less_no
thf(fact_6095_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_6096_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_6097_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_6098_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_6099_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_6100_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_6101_word__less__iff__unsigned,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( linordered_semidom @ A ) )
=> ( ( ord_less @ ( word @ B ) )
= ( ^ [A8: word @ B,B8: word @ B] : ( ord_less @ A @ ( semiring_1_unsigned @ B @ A @ A8 ) @ ( semiring_1_unsigned @ B @ A @ B8 ) ) ) ) ) ).
% word_less_iff_unsigned
thf(fact_6102_unsigned__greater__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( unique1627219031080169319umeral @ A ) )
=> ! [W2: word @ B] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_unsigned @ B @ A @ W2 ) ) ) ).
% unsigned_greater_eq
thf(fact_6103_UNIV__bool,axiom,
( ( top_top @ ( set @ $o ) )
= ( insert @ $o @ $false @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% UNIV_bool
thf(fact_6104_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_6105_word__less__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less @ ( word @ A ) )
= ( ^ [A8: word @ A,B8: word @ A] : ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ A8 ) @ ( semiring_1_unsigned @ A @ int @ B8 ) ) ) ) ) ).
% word_less_def
thf(fact_6106_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_6107_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_6108_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_6109_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_6110_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_6111_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_6112_word__le__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ ( word @ A ) )
= ( ^ [A8: word @ A,B8: word @ A] : ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ A8 ) @ ( semiring_1_unsigned @ A @ int @ B8 ) ) ) ) ) ).
% word_le_def
thf(fact_6113_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_6114_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_6115_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_6116_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_6117_take__bit__integer_Oabs__eq,axiom,
! [Xa2: nat,X: int] :
( ( bit_se2584673776208193580ke_bit @ code_integer @ Xa2 @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se2584673776208193580ke_bit @ int @ Xa2 @ X ) ) ) ).
% take_bit_integer.abs_eq
thf(fact_6118_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_6119_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_6120_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_6121_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_6122_unsigned__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( semiring_char_0 @ A ) )
=> ! [W2: word @ B] :
( ( ( semiring_1_unsigned @ B @ A @ W2 )
= ( zero_zero @ A ) )
= ( W2
= ( zero_zero @ ( word @ B ) ) ) ) ) ).
% unsigned_eq_0_iff
thf(fact_6123_word__less__eq__iff__unsigned,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( linordered_semidom @ A ) )
=> ( ( ord_less_eq @ ( word @ B ) )
= ( ^ [A8: word @ B,B8: word @ B] : ( ord_less_eq @ A @ ( semiring_1_unsigned @ B @ A @ A8 ) @ ( semiring_1_unsigned @ B @ A @ B8 ) ) ) ) ) ).
% word_less_eq_iff_unsigned
thf(fact_6124_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_6125_uint__add__ge0,axiom,
! [Aa: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ Aa ) )
=> ! [Xa2: word @ Aa,X: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ Aa @ int @ Xa2 ) @ ( semiring_1_unsigned @ A @ int @ X ) ) ) ) ).
% uint_add_ge0
thf(fact_6126_uint__mult__ge0,axiom,
! [Aa: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ Aa ) )
=> ! [Xa2: word @ Aa,X: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( semiring_1_unsigned @ Aa @ int @ Xa2 ) @ ( semiring_1_unsigned @ A @ int @ X ) ) ) ) ).
% uint_mult_ge0
thf(fact_6127_nat__uint__less__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,Z: nat,X: word @ A] :
( ( ( nat2 @ ( semiring_1_unsigned @ A @ int @ Y ) )
= Z )
=> ( ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ord_less @ nat @ ( nat2 @ ( semiring_1_unsigned @ A @ int @ X ) ) @ Z ) ) ) ) ).
% nat_uint_less_helper
thf(fact_6128_uint__add__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] : ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ).
% uint_add_le
thf(fact_6129_uint__plus__simple__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ).
% uint_plus_simple_iff
thf(fact_6130_uint__plus__simple,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ).
% uint_plus_simple
thf(fact_6131_no__ulen__sub,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ X )
= ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y ) @ ( semiring_1_unsigned @ A @ int @ X ) ) ) ) ).
% no_ulen_sub
thf(fact_6132_uint__sub__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y ) @ ( semiring_1_unsigned @ A @ int @ X ) )
= ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ).
% uint_sub_lem
thf(fact_6133_uint__sub__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X @ Y ) ) ) ) ).
% uint_sub_ge
thf(fact_6134_uint__minus__simple__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ X )
= ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ).
% uint_minus_simple_iff
thf(fact_6135_uint__minus__simple__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ ( word @ A ) )
= ( ^ [Y5: word @ A,X4: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X4 @ Y5 ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X4 ) @ ( semiring_1_unsigned @ A @ int @ Y5 ) ) ) ) ) ) ).
% uint_minus_simple_alt
thf(fact_6136_udvd__incr__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Up: int,Uq: int,Ua: int,N: int,K4: word @ A,N3: int] :
( ( ord_less @ int @ Up @ Uq )
=> ( ( Up
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K4 ) ) ) )
=> ( ( Uq
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N3 @ ( semiring_1_unsigned @ A @ int @ K4 ) ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ Up @ ( semiring_1_unsigned @ A @ int @ K4 ) ) @ Uq ) ) ) ) ) ).
% udvd_incr_lem
thf(fact_6137_udvd__incr__lem0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Up: int,Uq: int,N: int,K4: word @ A,N3: int] :
( ( ord_less @ int @ Up @ Uq )
=> ( ( Up
= ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K4 ) ) )
=> ( ( Uq
= ( times_times @ int @ N3 @ ( semiring_1_unsigned @ A @ int @ K4 ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ Up @ ( semiring_1_unsigned @ A @ int @ K4 ) ) @ Uq ) ) ) ) ) ).
% udvd_incr_lem0
thf(fact_6138_udvd__incr0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P3: word @ A,Q3: word @ A,N: int,K4: word @ A,N3: int] :
( ( ord_less @ ( word @ A ) @ P3 @ Q3 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P3 )
= ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K4 ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( times_times @ int @ N3 @ ( semiring_1_unsigned @ A @ int @ K4 ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P3 @ K4 ) @ Q3 ) ) ) ) ) ).
% udvd_incr0
thf(fact_6139_udvd__decr0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P3: word @ A,Q3: word @ A,N: int,K4: word @ A,N3: int] :
( ( ord_less @ ( word @ A ) @ P3 @ Q3 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P3 )
= ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K4 ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( times_times @ int @ N3 @ ( semiring_1_unsigned @ A @ int @ K4 ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( times_times @ int @ N3 @ ( semiring_1_unsigned @ A @ int @ K4 ) ) )
=> ( ord_less_eq @ ( word @ A ) @ P3 @ ( minus_minus @ ( word @ A ) @ Q3 @ K4 ) ) ) ) ) ) ) ).
% udvd_decr0
thf(fact_6140_udvd__unfold__int,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( udvd @ A )
= ( ^ [A8: word @ A,B8: word @ A] :
? [N6: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N6 )
& ( ( semiring_1_unsigned @ A @ int @ B8 )
= ( times_times @ int @ N6 @ ( semiring_1_unsigned @ A @ int @ A8 ) ) ) ) ) ) ) ).
% udvd_unfold_int
thf(fact_6141_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_6142_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_6143_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_6144_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_6145_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_6146_udvd__incr_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P3: word @ A,Q3: word @ A,Ua: int,N: int,K4: word @ A,N3: int] :
( ( ord_less @ ( word @ A ) @ P3 @ Q3 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P3 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K4 ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N3 @ ( semiring_1_unsigned @ A @ int @ K4 ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P3 @ K4 ) @ Q3 ) ) ) ) ) ).
% udvd_incr'
thf(fact_6147_udvd__decr_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P3: word @ A,Q3: word @ A,Ua: int,N: int,K4: word @ A,N3: int] :
( ( ord_less @ ( word @ A ) @ P3 @ Q3 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P3 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K4 ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N3 @ ( semiring_1_unsigned @ A @ int @ K4 ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N3 @ ( semiring_1_unsigned @ A @ int @ K4 ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ P3 @ ( minus_minus @ ( word @ A ) @ Q3 @ K4 ) ) ) ) ) ) ) ).
% udvd_decr'
thf(fact_6148_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_6149_bin__last__bintrunc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [L: nat,N: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ L @ N ) ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% bin_last_bintrunc
thf(fact_6150_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_6151_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_6152_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_6153_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_6154_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_6155_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_6156_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_6157_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_6158_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_6159_uint__range__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_unsigned @ A @ int @ W2 ) )
& ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ W2 ) ) ) ) ) ).
% uint_range_size
thf(fact_6160_uint__2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% uint_2p
thf(fact_6161_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_6162_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_6163_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_6164_signed__take__bit__eq__take__bit__shift,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N6: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N6 ) @ ( plus_plus @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N6 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N6 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_6165_no__plus__overflow__uint__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) ) ) ).
% no_plus_overflow_uint_size
thf(fact_6166_uint__plus__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) )
& ( ~ ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) ) ) ) ) ).
% uint_plus_if_size
thf(fact_6167_uint__sub__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y ) @ ( semiring_1_unsigned @ A @ int @ X ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y ) @ ( semiring_1_unsigned @ A @ int @ X ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) ) ) ) ) ).
% uint_sub_if_size
thf(fact_6168_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_6169_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_6170_transpose__rectangle,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N: nat] :
( ( ( Xs
= ( nil @ ( list @ A ) ) )
=> ( N
= ( zero_zero @ nat ) ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I2 ) )
= N ) )
=> ( ( transpose @ A @ Xs )
= ( map @ nat @ ( list @ A )
@ ^ [I3: nat] :
( map @ nat @ A
@ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J3 ) @ I3 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_6171_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_6172_upt__0__eq__Nil__conv,axiom,
! [J: nat] :
( ( ( upt @ ( zero_zero @ nat ) @ J )
= ( nil @ nat ) )
= ( J
= ( zero_zero @ nat ) ) ) ).
% upt_0_eq_Nil_conv
thf(fact_6173_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( upt @ I @ J )
= ( nil @ nat ) ) ) ).
% upt_conv_Nil
thf(fact_6174_upt__merge,axiom,
! [I: nat,J: nat,K: nat] :
( ( ( ord_less_eq @ nat @ I @ J )
& ( ord_less_eq @ nat @ J @ K ) )
=> ( ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ K ) )
= ( upt @ I @ K ) ) ) ).
% upt_merge
thf(fact_6175_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_6176_Suc__unat__minus__one,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( suc @ ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) ) )
= ( semiring_1_unsigned @ A @ nat @ X ) ) ) ) ).
% Suc_unat_minus_one
thf(fact_6177_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_6178_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_6179_sum__list__upt,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups8242544230860333062m_list @ nat @ ( upt @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum_list_upt
thf(fact_6180_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_6181_upt__eq__append__conv,axiom,
! [I: nat,J: nat,Xs: list @ nat,Ys: list @ nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( upt @ I @ J )
= ( append @ nat @ Xs @ Ys ) )
= ( ? [K3: nat] :
( ( ord_less_eq @ nat @ I @ K3 )
& ( ord_less_eq @ nat @ K3 @ J )
& ( ( upt @ I @ K3 )
= Xs )
& ( ( upt @ K3 @ J )
= Ys ) ) ) ) ) ).
% upt_eq_append_conv
thf(fact_6182_sorted__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( upt @ M @ N ) ) ).
% sorted_upt
thf(fact_6183_unat__of__nat__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,C3: nat] :
( ( ord_less @ ( word @ A ) @ A2 @ B2 )
=> ( ( ( semiring_1_unsigned @ A @ nat @ B2 )
= C3 )
=> ( ord_less @ ( word @ A ) @ A2 @ ( semiring_1_of_nat @ ( word @ A ) @ C3 ) ) ) ) ) ).
% unat_of_nat_less
thf(fact_6184_unat__eq__zero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( semiring_1_unsigned @ A @ nat @ X )
= ( zero_zero @ nat ) )
= ( X
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% unat_eq_zero
thf(fact_6185_unat__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ nat @ ( zero_zero @ ( word @ A ) ) )
= ( zero_zero @ nat ) ) ) ).
% unat_0
thf(fact_6186_word__less__nat__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less @ ( word @ A ) )
= ( ^ [A8: word @ A,B8: word @ A] : ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ A8 ) @ ( semiring_1_unsigned @ A @ nat @ B8 ) ) ) ) ) ).
% word_less_nat_alt
thf(fact_6187_unat__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( ord_less @ ( word @ A ) @ A2 @ B2 )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) ) ) ).
% unat_mono
thf(fact_6188_word__le__nat__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ ( word @ A ) )
= ( ^ [A8: word @ A,B8: word @ A] : ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ A8 ) @ ( semiring_1_unsigned @ A @ nat @ B8 ) ) ) ) ) ).
% word_le_nat_alt
thf(fact_6189_le__ucast__ucast__le,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ A,Y: word @ B] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) )
=> ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ Y ) ) ) ).
% le_ucast_ucast_le
thf(fact_6190_le__unat__uoi,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: nat,Z: word @ A] :
( ( ord_less_eq @ nat @ Y @ ( semiring_1_unsigned @ A @ nat @ Z ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ Y ) )
= Y ) ) ) ).
% le_unat_uoi
thf(fact_6191_map__add__upt_H,axiom,
! [Ofs: nat,A2: nat,B2: nat] :
( ( map @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ I3 @ Ofs )
@ ( upt @ A2 @ B2 ) )
= ( upt @ ( plus_plus @ nat @ A2 @ Ofs ) @ ( plus_plus @ nat @ B2 @ Ofs ) ) ) ).
% map_add_upt'
thf(fact_6192_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map @ nat @ nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_6193_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ I3 @ N )
@ ( upt @ ( zero_zero @ nat ) @ M ) )
= ( upt @ N @ ( plus_plus @ nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_6194_sorted__wrt__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less @ nat ) @ ( upt @ M @ N ) ) ).
% sorted_wrt_upt
thf(fact_6195_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% upt_0
thf(fact_6196_butlast__upt,axiom,
! [M: nat,N: nat] :
( ( butlast @ nat @ ( upt @ M @ N ) )
= ( upt @ M @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% butlast_upt
thf(fact_6197_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list @ nat,Q3: nat] :
( ( ( cons @ nat @ M @ ( cons @ nat @ N @ Ns ) )
= ( upt @ M @ Q3 ) )
= ( ( cons @ nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q3 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_6198_filter__upt__take__conv,axiom,
! [A: $tType,P2: A > $o,M: nat,L: list @ A,N: nat] :
( ( filter2 @ nat
@ ^ [I3: nat] : ( P2 @ ( nth @ A @ ( take @ A @ M @ L ) @ I3 ) )
@ ( upt @ N @ M ) )
= ( filter2 @ nat
@ ^ [I3: nat] : ( P2 @ ( nth @ A @ L @ I3 ) )
@ ( upt @ N @ M ) ) ) ).
% filter_upt_take_conv
thf(fact_6199_unat__eq__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( semiring_1_unsigned @ A @ nat @ X )
= ( suc @ ( zero_zero @ nat ) ) )
= ( X
= ( one_one @ ( word @ A ) ) ) ) ) ).
% unat_eq_1
thf(fact_6200_unat__gt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
= ( X
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% unat_gt_0
thf(fact_6201_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_6202_un__ui__le,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [A2: word @ A,B2: word @ B] :
( ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ B @ nat @ B2 ) )
= ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ B @ int @ B2 ) ) ) ) ).
% un_ui_le
thf(fact_6203_unat__plus__simple,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ) ).
% unat_plus_simple
thf(fact_6204_unat__sub,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B2: word @ A,A2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ B2 @ A2 )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) )
= ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) ) ) ) ).
% unat_sub
thf(fact_6205_unat__less__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( semiring_1_of_nat @ ( word @ A ) @ N ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ N ) ) ) ).
% unat_less_helper
thf(fact_6206_word__of__nat__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less @ nat @ N @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) @ X ) ) ) ).
% word_of_nat_less
thf(fact_6207_upt__filter__extend,axiom,
! [U: nat,U4: nat,P2: nat > $o] :
( ( ord_less_eq @ nat @ U @ U4 )
=> ( ! [I2: nat] :
( ( ( ord_less_eq @ nat @ U @ I2 )
& ( ord_less @ nat @ I2 @ U4 ) )
=> ~ ( P2 @ I2 ) )
=> ( ( filter2 @ nat @ P2 @ ( upt @ ( zero_zero @ nat ) @ U ) )
= ( filter2 @ nat @ P2 @ ( upt @ ( zero_zero @ nat ) @ U4 ) ) ) ) ) ).
% upt_filter_extend
thf(fact_6208_word__of__nat__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less_eq @ nat @ N @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) @ X ) ) ) ).
% word_of_nat_le
thf(fact_6209_word__unat__less__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: nat] :
( ( ord_less_eq @ ( word @ A ) @ A2 @ ( semiring_1_of_nat @ ( word @ A ) @ B2 ) )
=> ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ B2 ) ) ) ).
% word_unat_less_le
thf(fact_6210_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map @ nat @ nat
@ ^ [N6: nat] : ( minus_minus @ nat @ N6 @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_6211_upt__append,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( append @ nat @ ( upt @ ( zero_zero @ nat ) @ I ) @ ( upt @ I @ J ) )
= ( upt @ ( zero_zero @ nat ) @ J ) ) ) ).
% upt_append
thf(fact_6212_atLeastAtMost__upt,axiom,
( ( set_or1337092689740270186AtMost @ nat )
= ( ^ [N6: nat,M5: nat] : ( set2 @ nat @ ( upt @ N6 @ ( suc @ M5 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_6213_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_6214_atLeast__upt,axiom,
( ( set_ord_lessThan @ nat )
= ( ^ [N6: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N6 ) ) ) ) ).
% atLeast_upt
thf(fact_6215_map__replicate__trivial,axiom,
! [A: $tType,X: A,I: nat] :
( ( map @ nat @ A
@ ^ [I3: nat] : X
@ ( upt @ ( zero_zero @ nat ) @ I ) )
= ( replicate @ A @ I @ X ) ) ).
% map_replicate_trivial
thf(fact_6216_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list @ nat] :
( ( ( upt @ I @ J )
= ( cons @ nat @ X @ Xs ) )
= ( ( ord_less @ nat @ I @ J )
& ( I = X )
& ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_6217_unat__1__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiring_1_unsigned @ A @ nat @ X ) ) ) ) ).
% unat_1_0
thf(fact_6218_unat__max__word__pos,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiring_1_unsigned @ A @ nat @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% unat_max_word_pos
thf(fact_6219_unatSuc2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A] :
( ( ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) )
= ( suc @ ( semiring_1_unsigned @ A @ nat @ N ) ) ) ) ) ).
% unatSuc2
thf(fact_6220_unatSuc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A] :
( ( ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N ) )
= ( suc @ ( semiring_1_unsigned @ A @ nat @ N ) ) ) ) ) ).
% unatSuc
thf(fact_6221_Suc__unat__diff__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X )
=> ( ( suc @ ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) ) )
= ( semiring_1_unsigned @ A @ nat @ X ) ) ) ) ).
% Suc_unat_diff_1
thf(fact_6222_unat__Suc2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A] :
( ( N
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) )
= ( suc @ ( semiring_1_unsigned @ A @ nat @ N ) ) ) ) ) ).
% unat_Suc2
thf(fact_6223_uno__simps_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Z: word @ A,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiring_1_unsigned @ A @ nat @ Z ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ ( modulo_modulo @ nat @ M @ ( semiring_1_unsigned @ A @ nat @ Z ) ) ) )
= ( modulo_modulo @ nat @ M @ ( semiring_1_unsigned @ A @ nat @ Z ) ) ) ) ) ).
% uno_simps(1)
thf(fact_6224_measure__unat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P3: word @ A] :
( ( P3
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ P3 @ ( one_one @ ( word @ A ) ) ) ) @ ( semiring_1_unsigned @ A @ nat @ P3 ) ) ) ) ).
% measure_unat
thf(fact_6225_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N6: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N6 ) ) ) ) ) ).
% atMost_upto
thf(fact_6226_upt__rec,axiom,
( upt
= ( ^ [I3: nat,J3: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I3 @ J3 ) @ ( cons @ nat @ I3 @ ( upt @ ( suc @ I3 ) @ J3 ) ) @ ( nil @ nat ) ) ) ) ).
% upt_rec
thf(fact_6227_map__upt__Suc,axiom,
! [A: $tType,F3: nat > A,N: nat] :
( ( map @ nat @ A @ F3 @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( cons @ A @ ( F3 @ ( zero_zero @ nat ) )
@ ( map @ nat @ A
@ ^ [I3: nat] : ( F3 @ ( suc @ I3 ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_6228_map__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( map @ nat @ A @ ( nth @ A @ Xs ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= Xs ) ).
% map_nth
thf(fact_6229_nth__map__upt,axiom,
! [A: $tType,I: nat,N: nat,M: nat,F3: nat > A] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N @ M ) )
=> ( ( nth @ A @ ( map @ nat @ A @ F3 @ ( upt @ M @ N ) ) @ I )
= ( F3 @ ( plus_plus @ nat @ M @ I ) ) ) ) ).
% nth_map_upt
thf(fact_6230_upt__eq__lel__conv,axiom,
! [L: nat,H2: nat,Is1: list @ nat,I: nat,Is2: list @ nat] :
( ( ( upt @ L @ H2 )
= ( append @ nat @ Is1 @ ( cons @ nat @ I @ Is2 ) ) )
= ( ( Is1
= ( upt @ L @ I ) )
& ( Is2
= ( upt @ ( suc @ I ) @ H2 ) )
& ( ord_less_eq @ nat @ L @ I )
& ( ord_less @ nat @ I @ H2 ) ) ) ).
% upt_eq_lel_conv
thf(fact_6231_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_6232_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_6233_map__upt__eqI,axiom,
! [A: $tType,Xs: list @ A,N: nat,M: nat,F3: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( minus_minus @ nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I2 )
= ( F3 @ ( plus_plus @ nat @ M @ I2 ) ) ) )
=> ( ( map @ nat @ A @ F3 @ ( upt @ M @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_6234_map__nth__upt__drop__take__conv,axiom,
! [A: $tType,N8: nat,L: list @ A,M7: nat] :
( ( ord_less_eq @ nat @ N8 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( map @ nat @ A @ ( nth @ A @ L ) @ ( upt @ M7 @ N8 ) )
= ( drop @ A @ M7 @ ( take @ A @ N8 @ L ) ) ) ) ).
% map_nth_upt_drop_take_conv
thf(fact_6235_lt__plus__1__le__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,MaxBound: word @ A,X: word @ A] :
( ( ord_less @ nat @ N @ ( semiring_1_unsigned @ A @ nat @ MaxBound ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) )
= ( ord_less_eq @ ( word @ A ) @ X @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% lt_plus_1_le_word
thf(fact_6236_filter__upt__last,axiom,
! [A: $tType,P2: A > $o,L: list @ A,Js2: list @ nat,J: nat,I: nat] :
( ( ( filter2 @ nat
@ ^ [K3: nat] : ( P2 @ ( nth @ A @ L @ K3 ) )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
= ( append @ nat @ Js2 @ ( cons @ nat @ J @ ( nil @ nat ) ) ) )
=> ( ( ord_less @ nat @ J @ I )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ~ ( P2 @ ( nth @ A @ L @ I ) ) ) ) ) ).
% filter_upt_last
thf(fact_6237_word__div__eq__1__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,M: word @ A] :
( ( ( divide_divide @ ( word @ A ) @ N @ M )
= ( one_one @ ( word @ A ) ) )
= ( ( ord_less_eq @ ( word @ A ) @ M @ N )
& ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ N ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ M ) ) ) ) ) ) ).
% word_div_eq_1_iff
thf(fact_6238_unat__plus__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) )
& ( ~ ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) ) ) ) ) ).
% unat_plus_if_size
thf(fact_6239_unat__sub__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) )
& ( ~ ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ) ) ).
% unat_sub_if_size
thf(fact_6240_no__plus__overflow__unat__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) ) ) ).
% no_plus_overflow_unat_size
thf(fact_6241_make__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,F3: nat > A] :
( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_make @ A @ N @ F3 )
@ ^ [R5: array @ A] : ( snga_assn @ A @ R5 @ ( map @ nat @ A @ F3 @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% make_rule
thf(fact_6242_signed__take__bit__eq__take__bit__minus,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N6: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N6 ) @ K3 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N6 ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N6 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_6243_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_6244_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_6245_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_6246_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_6247_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_6248_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ N ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_6249_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W2 ) ) ) @ ( suc @ N ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W2 ) @ N ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_6250_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_6251_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_6252_bin__nth__minus__Bit0,axiom,
! [N: nat,W2: num] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) @ N )
= ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W2 ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% bin_nth_minus_Bit0
thf(fact_6253_bin__nth__minus__Bit1,axiom,
! [N: nat,W2: num] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ ( bit1 @ W2 ) ) @ N )
= ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W2 ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% bin_nth_minus_Bit1
thf(fact_6254_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_6255_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_6256_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_6257_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_6258_map__bit__range__eq__if__take__bit__eq,axiom,
! [N: nat,K: int,L: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2584673776208193580ke_bit @ int @ N @ L ) )
=> ( ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ int @ K ) @ ( upt @ ( zero_zero @ nat ) @ N ) )
= ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ int @ L ) @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% map_bit_range_eq_if_take_bit_eq
thf(fact_6259_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_6260_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_6261_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_6262_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_6263_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_6264_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_6265_bit__horner__sum__bit__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Bs2: list @ $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs2 ) @ N )
= ( ( ord_less @ nat @ N @ ( size_size @ ( list @ $o ) @ Bs2 ) )
& ( nth @ $o @ Bs2 @ N ) ) ) ) ).
% bit_horner_sum_bit_iff
thf(fact_6266_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_6267_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A8: A,N6: nat] :
( ( ( N6
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A8 ) )
& ( ( N6
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N6 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_6268_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_6269_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_6270_num_Osize__gen_I3_J,axiom,
! [X33: num] :
( ( size_num @ ( bit1 @ X33 ) )
= ( plus_plus @ nat @ ( size_num @ X33 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(3)
thf(fact_6271_hash__code__prod__simps,axiom,
! [A: $tType,B: $tType,H_a: A > uint32,H_b: B > uint32,X: A,Xa2: B] :
( ( hash_hash_code_prod @ A @ B @ H_a @ H_b @ ( product_Pair @ A @ B @ X @ Xa2 ) )
= ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_a @ X ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_b @ Xa2 ) @ ( numeral_numeral @ uint32 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% hash_code_prod_simps
thf(fact_6272_test__bit__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% test_bit_1
thf(fact_6273_test__bit__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ N )
=> ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W2 ) ) ) ) ).
% test_bit_size
thf(fact_6274_word__eqI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U: word @ A,V3: word @ A] :
( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( word @ A ) @ U ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ U @ N2 )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V3 @ N2 ) ) )
=> ( U = V3 ) ) ) ).
% word_eqI
thf(fact_6275_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_6276_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_6277_finite__bit__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A] : ( finite_finite2 @ nat @ ( collect @ nat @ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 ) ) ) ) ).
% finite_bit_word
thf(fact_6278_bit__integer_Oabs__eq,axiom,
! [X: int] :
( ( bit_se5641148757651400278ts_bit @ code_integer @ ( code_integer_of_int @ X ) )
= ( bit_se5641148757651400278ts_bit @ int @ X ) ) ).
% bit_integer.abs_eq
thf(fact_6279_test__bit__over,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ X ) @ N )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N ) ) ) ).
% test_bit_over
thf(fact_6280_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_6281_lsb__this__or__next,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) ) ) ) ).
% lsb_this_or_next
thf(fact_6282_word__leI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U: word @ A,V3: word @ A] :
( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( word @ A ) @ U ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ U @ N2 )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V3 @ N2 ) ) )
=> ( ord_less_eq @ ( word @ A ) @ U @ V3 ) ) ) ).
% word_leI
thf(fact_6283_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_6284_overflow__imp__lsb,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) ) ) ) ).
% overflow_imp_lsb
thf(fact_6285_test__bit__bin_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [W3: word @ A,N6: nat] :
( ( ord_less @ nat @ N6 @ ( size_size @ ( word @ A ) @ W3 ) )
& ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W3 ) @ N6 ) ) ) ) ) ).
% test_bit_bin'
thf(fact_6286_bang__is__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ M )
=> ( ord_less_eq @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) @ X ) ) ) ).
% bang_is_le
thf(fact_6287_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_6288_odd__iff__lsb,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ X ) ) )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) ) ) ) ).
% odd_iff_lsb
thf(fact_6289_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_6290_mlex__eq,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F4: A > nat,R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y5: A] :
( ( ord_less @ nat @ ( F4 @ X4 ) @ ( F4 @ Y5 ) )
| ( ( ord_less_eq @ nat @ ( F4 @ X4 ) @ ( F4 @ Y5 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R4 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_6291_sort__key__by__quicksort__code,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ B @ A )
= ( ^ [F4: B > A,Xs2: list @ B] :
( case_list @ ( list @ B ) @ B @ ( nil @ B )
@ ^ [X4: B] :
( case_list @ ( list @ B ) @ B @ Xs2
@ ^ [Y5: B] :
( case_list @ ( list @ B ) @ B @ ( if @ ( list @ B ) @ ( ord_less_eq @ A @ ( F4 @ X4 ) @ ( F4 @ Y5 ) ) @ Xs2 @ ( cons @ B @ Y5 @ ( cons @ B @ X4 @ ( nil @ B ) ) ) )
@ ^ [Ab2: B,List2: list @ B] :
( product_case_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( list @ B )
@ ^ [Lts: list @ B] :
( product_case_prod @ ( list @ B ) @ ( list @ B ) @ ( list @ B )
@ ^ [Eqs: list @ B,Gts: list @ B] : ( append @ B @ ( linorder_sort_key @ B @ A @ F4 @ Lts ) @ ( append @ B @ Eqs @ ( linorder_sort_key @ B @ A @ F4 @ Gts ) ) ) )
@ ( linorder_part @ B @ A @ F4 @ ( F4 @ ( nth @ B @ Xs2 @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Xs2 ) ) ) )
@ Xs2 ) ) ) ) ).
% sort_key_by_quicksort_code
thf(fact_6292_mlex__less,axiom,
! [A: $tType,F3: A > nat,X: A,Y: A,R3: set @ ( product_prod @ A @ A )] :
( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F3 @ R3 ) ) ) ).
% mlex_less
thf(fact_6293_mlex__iff,axiom,
! [A: $tType,X: A,Y: A,F3: A > nat,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F3 @ R3 ) )
= ( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
| ( ( ( F3 @ X )
= ( F3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ).
% mlex_iff
thf(fact_6294_mlex__leq,axiom,
! [A: $tType,F3: A > nat,X: A,Y: A,R3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F3 @ R3 ) ) ) ) ).
% mlex_leq
thf(fact_6295_zip__Cons,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys: list @ B] :
( ( zip @ A @ B @ Xs @ ( cons @ B @ Y @ Ys ) )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ A @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Z2: A,Zs3: list @ A] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Z2 @ Y ) @ ( zip @ A @ B @ Zs3 @ Ys ) )
@ Xs ) ) ).
% zip_Cons
thf(fact_6296_zip__Cons1,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ B @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Y5: B,Ys3: list @ B] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y5 ) @ ( zip @ A @ B @ Xs @ Ys3 ) )
@ Ys ) ) ).
% zip_Cons1
thf(fact_6297_part__code_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Pivot: A,X: B,Xs: list @ B] :
( ( linorder_part @ B @ A @ F3 @ Pivot @ ( cons @ B @ X @ Xs ) )
= ( product_case_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
@ ^ [Lts: list @ B] :
( product_case_prod @ ( list @ B ) @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
@ ^ [Eqs: list @ B,Gts: list @ B] : ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ ( F3 @ X ) @ Pivot ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( cons @ B @ X @ Lts ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ Gts ) ) @ ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ Pivot @ ( F3 @ X ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ ( cons @ B @ X @ Gts ) ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( cons @ B @ X @ Eqs ) @ Gts ) ) ) ) )
@ ( linorder_part @ B @ A @ F3 @ Pivot @ Xs ) ) ) ) ).
% part_code(2)
thf(fact_6298_part__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_part @ B @ A )
= ( ^ [F4: B > A,Pivot2: A,Xs2: list @ B] :
( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) )
@ ( filter2 @ B
@ ^ [X4: B] : ( ord_less @ A @ ( F4 @ X4 ) @ Pivot2 )
@ Xs2 )
@ ( product_Pair @ ( list @ B ) @ ( list @ B )
@ ( filter2 @ B
@ ^ [X4: B] :
( ( F4 @ X4 )
= Pivot2 )
@ Xs2 )
@ ( filter2 @ B
@ ^ [X4: B] : ( ord_less @ A @ Pivot2 @ ( F4 @ X4 ) )
@ Xs2 ) ) ) ) ) ) ).
% part_def
thf(fact_6299_transpose__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( transpose @ A @ ( transpose @ A @ Xs ) )
= ( takeWhile @ ( list @ A )
@ ^ [X4: list @ A] :
( X4
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_transpose
thf(fact_6300_Code__Numeral_Opositive__def,axiom,
( code_positive
= ( numeral_numeral @ code_integer ) ) ).
% Code_Numeral.positive_def
thf(fact_6301_length__takeWhile__le,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_takeWhile_le
thf(fact_6302_sorted__takeWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P2: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) ) ) ).
% sorted_takeWhile
thf(fact_6303_takeWhile__nth,axiom,
! [A: $tType,J: nat,P2: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) )
=> ( ( nth @ A @ ( takeWhile @ A @ P2 @ Xs ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ).
% takeWhile_nth
thf(fact_6304_nth__length__takeWhile,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P2 @ ( nth @ A @ Xs @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) ) ) ) ).
% nth_length_takeWhile
thf(fact_6305_drop__takeWhile,axiom,
! [A: $tType,I: nat,P2: A > $o,L: list @ A] :
( ( ord_less_eq @ nat @ I @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ L ) ) )
=> ( ( drop @ A @ I @ ( takeWhile @ A @ P2 @ L ) )
= ( takeWhile @ A @ P2 @ ( drop @ A @ I @ L ) ) ) ) ).
% drop_takeWhile
thf(fact_6306_length__takeWhile__less__P__nth,axiom,
! [A: $tType,J: nat,P2: A > $o,Xs: list @ A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( P2 @ ( nth @ A @ Xs @ I2 ) ) )
=> ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) ) ) ) ).
% length_takeWhile_less_P_nth
thf(fact_6307_eq__len__takeWhile__conv,axiom,
! [A: $tType,I: nat,P2: A > $o,L: list @ A] :
( ( I
= ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ L ) ) )
= ( ( ord_less_eq @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I )
=> ( P2 @ ( nth @ A @ L @ J3 ) ) )
& ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ~ ( P2 @ ( nth @ A @ L @ I ) ) ) ) ) ).
% eq_len_takeWhile_conv
thf(fact_6308_less__length__takeWhile__conv,axiom,
! [A: $tType,I: nat,P2: A > $o,L: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ L ) ) )
= ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
& ! [J3: nat] :
( ( ord_less_eq @ nat @ J3 @ I )
=> ( P2 @ ( nth @ A @ L @ J3 ) ) ) ) ) ).
% less_length_takeWhile_conv
thf(fact_6309_takeWhile__eq__take__P__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P2: A > $o] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P2 @ ( nth @ A @ Xs @ I2 ) ) ) )
=> ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P2 @ ( nth @ A @ Xs @ N ) ) )
=> ( ( takeWhile @ A @ P2 @ Xs )
= ( take @ A @ N @ Xs ) ) ) ) ).
% takeWhile_eq_take_P_nth
thf(fact_6310_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,T2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F3 @ Xs ) ) )
=> ( ( filter2 @ B
@ ^ [X4: B] : ( ord_less @ A @ T2 @ ( F3 @ X4 ) )
@ Xs )
= ( takeWhile @ B
@ ^ [X4: B] : ( ord_less @ A @ T2 @ ( F3 @ X4 ) )
@ Xs ) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
thf(fact_6311_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_6312_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P3 )
= P3 ) ).
% case_prod_Pair_iden
thf(fact_6313_concat__bit__0,axiom,
! [K: int,L: int] :
( ( bit_concat_bit @ ( zero_zero @ nat ) @ K @ L )
= L ) ).
% concat_bit_0
thf(fact_6314_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_6315_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_6316_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Q2: A > B > $o,R3: $o,X: A,Y: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q2 )
& R3 )
=> ( R3
& ( ( P2 @ X @ Y )
=> ( Q2 @ X @ Y ) ) ) ) ).
% predicate2D_conj
thf(fact_6317_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_6318_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F3: ( A > B ) > C,G3: C] :
( ( F3
= ( ^ [X4: A > B] : G3 ) )
=> ( ( F3
@ ^ [X4: A] : ( zero_zero @ B ) )
= G3 ) ) ) ).
% fun_cong_unused_0
thf(fact_6319_eq__subset,axiom,
! [A: $tType,P2: A > A > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y6: A,Z4: A] : Y6 = Z4
@ ^ [A8: A,B8: A] :
( ( P2 @ A8 @ B8 )
| ( A8 = B8 ) ) ) ).
% eq_subset
thf(fact_6320_quicksort_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A,Y: list @ A] :
( ( ( linorder_quicksort @ A @ X )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( nil @ A ) ) )
=> ~ ! [X3: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs3 ) )
=> ( Y
!= ( append @ A
@ ( linorder_quicksort @ A
@ ( filter2 @ A
@ ^ [Y5: A] :
~ ( ord_less_eq @ A @ X3 @ Y5 )
@ Xs3 ) )
@ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter2 @ A @ ( ord_less_eq @ A @ X3 ) @ Xs3 ) ) ) ) ) ) ) ) ) ).
% quicksort.elims
thf(fact_6321_quicksort_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( linorder_quicksort @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A
@ ( linorder_quicksort @ A
@ ( filter2 @ A
@ ^ [Y5: A] :
~ ( ord_less_eq @ A @ X @ Y5 )
@ Xs ) )
@ ( append @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter2 @ A @ ( ord_less_eq @ A @ X ) @ Xs ) ) ) ) ) ) ).
% quicksort.simps(2)
thf(fact_6322_sorted__quicksort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linorder_quicksort @ A @ Xs ) ) ) ).
% sorted_quicksort
thf(fact_6323_quicksort_Opelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A,Y: list @ A] :
( ( ( linorder_quicksort @ A @ X )
= Y )
=> ( ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( nil @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ ( nil @ A ) ) ) )
=> ~ ! [X3: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs3 ) )
=> ( ( Y
= ( append @ A
@ ( linorder_quicksort @ A
@ ( filter2 @ A
@ ^ [Y5: A] :
~ ( ord_less_eq @ A @ X3 @ Y5 )
@ Xs3 ) )
@ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter2 @ A @ ( ord_less_eq @ A @ X3 ) @ Xs3 ) ) ) ) )
=> ~ ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ ( cons @ A @ X3 @ Xs3 ) ) ) ) ) ) ) ) ).
% quicksort.pelims
thf(fact_6324_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_6325_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_6326_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_6327_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_6328_bit_Oxor__self,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ X )
= ( zero_zero @ A ) ) ) ).
% bit.xor_self
thf(fact_6329_xor__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ).
% xor_nat_numerals(1)
thf(fact_6330_xor__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ Y ) ) ) ).
% xor_nat_numerals(2)
thf(fact_6331_xor__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% xor_nat_numerals(3)
thf(fact_6332_xor__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ X ) ) ) ).
% xor_nat_numerals(4)
thf(fact_6333_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M5: nat,N6: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ N6
@ ( if @ nat
@ ( N6
= ( zero_zero @ nat ) )
@ M5
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_6334_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_6335_sdiv__word__min,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ A2 ) @ ( one_one @ nat ) ) ) ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) ) ) ).
% sdiv_word_min
thf(fact_6336_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M5: nat,N6: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ N6
@ ( if @ nat
@ ( N6
= ( zero_zero @ nat ) )
@ M5
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_6337_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_6338_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_6339_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_6340_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_6341_or__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(2)
thf(fact_6342_or__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(4)
thf(fact_6343_or__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(3)
thf(fact_6344_or__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(1)
thf(fact_6345_xor__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( bit_se5824344971392196577ns_xor @ code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se5824344971392196577ns_xor @ int @ Xa2 @ X ) ) ) ).
% xor_integer.abs_eq
thf(fact_6346_XOR__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y ) ) ) ) ).
% XOR_lower
thf(fact_6347_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_6348_bit_Odisj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( zero_zero @ A ) )
= X ) ) ).
% bit.disj_zero_right
thf(fact_6349_XOR__upper,axiom,
! [X: int,N: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_6350_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_6351_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_6352_sdiv__word__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] : ( ord_less_eq @ int @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ A2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sdiv_word_max
thf(fact_6353_test__bit__split,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ C )
& ( type_len @ A ) )
=> ! [C3: word @ C,A2: word @ A,B2: word @ B] :
( ( ( word_split @ C @ A @ B @ C3 )
= ( product_Pair @ ( word @ A ) @ ( word @ B ) @ A2 @ B2 ) )
=> ( ! [N9: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B2 @ N9 )
= ( ( ord_less @ nat @ N9 @ ( size_size @ ( word @ B ) @ B2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C3 @ N9 ) ) )
& ! [M2: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A2 @ M2 )
= ( ( ord_less @ nat @ M2 @ ( size_size @ ( word @ A ) @ A2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C3 @ ( plus_plus @ nat @ M2 @ ( size_size @ ( word @ B ) @ B2 ) ) ) ) ) ) ) ) ).
% test_bit_split
thf(fact_6354_test__bit__split_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ C )
& ( type_len @ A ) )
=> ! [C3: word @ C,A2: word @ A,B2: word @ B] :
( ( ( word_split @ C @ A @ B @ C3 )
= ( product_Pair @ ( word @ A ) @ ( word @ B ) @ A2 @ B2 ) )
=> ! [N9: nat,M2: nat] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B2 @ N9 )
= ( ( ord_less @ nat @ N9 @ ( size_size @ ( word @ B ) @ B2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C3 @ N9 ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A2 @ M2 )
= ( ( ord_less @ nat @ M2 @ ( size_size @ ( word @ A ) @ A2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C3 @ ( plus_plus @ nat @ M2 @ ( size_size @ ( word @ B ) @ B2 ) ) ) ) ) ) ) ) ).
% test_bit_split'
thf(fact_6355_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_6356_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_6357_le__word__or2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] : ( ord_less_eq @ ( word @ A ) @ X @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) ) ) ).
% le_word_or2
thf(fact_6358_le__word__or1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [C3: word @ A,Y: word @ A] : ( ord_less_eq @ ( word @ A ) @ C3 @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ Y @ C3 ) ) ) ).
% le_word_or1
thf(fact_6359_OR__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) ) ) ) ).
% OR_lower
thf(fact_6360_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_6361_or__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( bit_se1065995026697491101ons_or @ code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se1065995026697491101ons_or @ int @ Xa2 @ X ) ) ) ).
% or_integer.abs_eq
thf(fact_6362_OR__upper,axiom,
! [X: int,N: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_6363_test__bit__split__eq,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ C )
& ( type_len @ B ) )
=> ! [C3: word @ C,A2: word @ A,B2: word @ B] :
( ( ( word_split @ C @ A @ B @ C3 )
= ( product_Pair @ ( word @ A ) @ ( word @ B ) @ A2 @ B2 ) )
= ( ! [N6: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B2 @ N6 )
= ( ( ord_less @ nat @ N6 @ ( size_size @ ( word @ B ) @ B2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C3 @ N6 ) ) )
& ! [M5: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A2 @ M5 )
= ( ( ord_less @ nat @ M5 @ ( size_size @ ( word @ A ) @ A2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C3 @ ( plus_plus @ nat @ M5 @ ( size_size @ ( word @ B ) @ B2 ) ) ) ) ) ) ) ) ).
% test_bit_split_eq
thf(fact_6364_word__cat__split__size,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( type_len @ C )
& ( type_len @ A )
& ( type_len @ B ) )
=> ! [T2: word @ A,U: word @ B,V3: word @ C] :
( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ T2 ) @ ( plus_plus @ nat @ ( size_size @ ( word @ B ) @ U ) @ ( size_size @ ( word @ C ) @ V3 ) ) )
=> ( ( ( product_Pair @ ( word @ B ) @ ( word @ C ) @ U @ V3 )
= ( word_split @ A @ B @ C @ T2 ) )
=> ( T2
= ( word_cat @ B @ C @ A @ U @ V3 ) ) ) ) ) ).
% word_cat_split_size
thf(fact_6365_word__split__cat__alt,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( type_len @ C )
& ( type_len @ B )
& ( type_len @ A ) )
=> ! [W2: word @ A,U: word @ B,V3: word @ C] :
( ( W2
= ( word_cat @ B @ C @ A @ U @ V3 ) )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( size_size @ ( word @ B ) @ U ) @ ( size_size @ ( word @ C ) @ V3 ) ) @ ( size_size @ ( word @ A ) @ W2 ) )
=> ( ( word_split @ A @ B @ C @ W2 )
= ( product_Pair @ ( word @ B ) @ ( word @ C ) @ U @ V3 ) ) ) ) ) ).
% word_split_cat_alt
thf(fact_6366_test__bit__cat,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( type_len @ C )
& ( type_len @ B )
& ( type_len @ A ) )
=> ! [A2: word @ B,B2: word @ C,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_cat @ B @ C @ A @ A2 @ B2 ) @ N )
= ( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ ( word_cat @ B @ C @ A @ A2 @ B2 ) ) )
& ( ( ord_less @ nat @ N @ ( size_size @ ( word @ C ) @ B2 ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ B2 @ N ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( word @ C ) @ B2 ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ A2 @ ( minus_minus @ nat @ N @ ( size_size @ ( word @ C ) @ B2 ) ) ) ) ) ) ) ).
% test_bit_cat
thf(fact_6367_word__cat__split__alt,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B )
& ( type_len @ C ) )
=> ! [W2: word @ A,U: word @ B,V3: word @ C] :
( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ W2 ) @ ( plus_plus @ nat @ ( size_size @ ( word @ B ) @ U ) @ ( size_size @ ( word @ C ) @ V3 ) ) )
=> ( ( ( word_split @ A @ B @ C @ W2 )
= ( product_Pair @ ( word @ B ) @ ( word @ C ) @ U @ V3 ) )
=> ( ( word_cat @ B @ C @ A @ U @ V3 )
= W2 ) ) ) ) ).
% word_cat_split_alt
thf(fact_6368_cat__slices,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [A2: word @ A,N: nat,C3: word @ B,B2: word @ C] :
( ( A2
= ( slice2 @ B @ A @ N @ C3 ) )
=> ( ( B2
= ( slice2 @ B @ C @ ( zero_zero @ nat ) @ C3 ) )
=> ( ( N
= ( size_size @ ( word @ C ) @ B2 ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( word @ B ) @ C3 ) @ ( plus_plus @ nat @ ( size_size @ ( word @ A ) @ A2 ) @ ( size_size @ ( word @ C ) @ B2 ) ) )
=> ( ( word_cat @ A @ C @ B @ A2 @ B2 )
= C3 ) ) ) ) ) ) ).
% cat_slices
thf(fact_6369_DERIV__even__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N ) ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_6370_at__within__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: A] :
( ( topolo174197925503356063within @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ) ).
% at_within_empty
thf(fact_6371_has__real__derivative__pos__inc__right,axiom,
! [F3: real > real,L: real,X: real,S3: set @ real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ S3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( member @ real @ ( plus_plus @ real @ X @ H7 ) @ S3 )
=> ( ( ord_less @ real @ H7 @ D3 )
=> ( ord_less @ real @ ( F3 @ X ) @ ( F3 @ ( plus_plus @ real @ X @ H7 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_6372_has__real__derivative__neg__dec__right,axiom,
! [F3: real > real,L: real,X: real,S3: set @ real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ S3 ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( member @ real @ ( plus_plus @ real @ X @ H7 ) @ S3 )
=> ( ( ord_less @ real @ H7 @ D3 )
=> ( ord_less @ real @ ( F3 @ ( plus_plus @ real @ X @ H7 ) ) @ ( F3 @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_6373_DERIV__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,X: A,S: set @ A,G3: A > A,E6: A] :
( ( has_field_derivative @ A @ F3 @ D5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G3 @ E6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D5 @ ( G3 @ X ) ) @ ( times_times @ A @ ( F3 @ X ) @ E6 ) ) @ ( times_times @ A @ ( G3 @ X ) @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% DERIV_divide
thf(fact_6374_DERIV__inverse_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F3 @ D5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( inverse_inverse @ A @ ( F3 @ X4 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F3 @ X ) ) @ D5 ) @ ( inverse_inverse @ A @ ( F3 @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_inverse'
thf(fact_6375_MVT2,axiom,
! [A2: real,B2: real,F3: real > real,F8: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ( has_field_derivative @ real @ F3 @ ( F8 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
& ( ord_less @ real @ Z3 @ B2 )
& ( ( minus_minus @ real @ ( F3 @ B2 ) @ ( F3 @ A2 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( F8 @ Z3 ) ) ) ) ) ) ).
% MVT2
thf(fact_6376_DERIV__neg__dec__right,axiom,
! [F3: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( ord_less @ real @ H7 @ D3 )
=> ( ord_less @ real @ ( F3 @ ( plus_plus @ real @ X @ H7 ) ) @ ( F3 @ X ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_6377_DERIV__pos__inc__right,axiom,
! [F3: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( ord_less @ real @ H7 @ D3 )
=> ( ord_less @ real @ ( F3 @ X ) @ ( F3 @ ( plus_plus @ real @ X @ H7 ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_6378_DERIV__ln,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( inverse_inverse @ real @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln
thf(fact_6379_DERIV__pos__imp__increasing,axiom,
! [A2: real,B2: real,F3: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F3 @ Y4 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y4 ) ) ) )
=> ( ord_less @ real @ ( F3 @ A2 ) @ ( F3 @ B2 ) ) ) ) ).
% DERIV_pos_imp_increasing
thf(fact_6380_DERIV__neg__imp__decreasing,axiom,
! [A2: real,B2: real,F3: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F3 @ Y4 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y4 @ ( zero_zero @ real ) ) ) ) )
=> ( ord_less @ real @ ( F3 @ B2 ) @ ( F3 @ A2 ) ) ) ) ).
% DERIV_neg_imp_decreasing
thf(fact_6381_deriv__nonneg__imp__mono,axiom,
! [A2: real,B2: real,G3: real > real,G7: real > real] :
( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
=> ( has_field_derivative @ real @ G3 @ ( G7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( G7 @ X3 ) ) )
=> ( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ord_less_eq @ real @ ( G3 @ A2 ) @ ( G3 @ B2 ) ) ) ) ) ).
% deriv_nonneg_imp_mono
thf(fact_6382_DERIV__nonpos__imp__nonincreasing,axiom,
! [A2: real,B2: real,F3: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F3 @ Y4 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less_eq @ real @ Y4 @ ( zero_zero @ real ) ) ) ) )
=> ( ord_less_eq @ real @ ( F3 @ B2 ) @ ( F3 @ A2 ) ) ) ) ).
% DERIV_nonpos_imp_nonincreasing
thf(fact_6383_DERIV__nonneg__imp__nondecreasing,axiom,
! [A2: real,B2: real,F3: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F3 @ Y4 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y4 ) ) ) )
=> ( ord_less_eq @ real @ ( F3 @ A2 ) @ ( F3 @ B2 ) ) ) ) ).
% DERIV_nonneg_imp_nondecreasing
thf(fact_6384_DERIV__neg__dec__left,axiom,
! [F3: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( ord_less @ real @ H7 @ D3 )
=> ( ord_less @ real @ ( F3 @ X ) @ ( F3 @ ( minus_minus @ real @ X @ H7 ) ) ) ) ) ) ) ) ).
% DERIV_neg_dec_left
thf(fact_6385_DERIV__pos__inc__left,axiom,
! [F3: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( ord_less @ real @ H7 @ D3 )
=> ( ord_less @ real @ ( F3 @ ( minus_minus @ real @ X @ H7 ) ) @ ( F3 @ X ) ) ) ) ) ) ) ).
% DERIV_pos_inc_left
thf(fact_6386_slice__id,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [T2: word @ A] :
( ( slice2 @ A @ A @ ( zero_zero @ nat ) @ T2 )
= T2 ) ) ).
% slice_id
thf(fact_6387_DERIV__const,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [K: A,F5: filter @ A] :
( has_field_derivative @ A
@ ^ [X4: A] : K
@ ( zero_zero @ A )
@ F5 ) ) ).
% DERIV_const
thf(fact_6388_has__real__derivative__pos__inc__left,axiom,
! [F3: real > real,L: real,X: real,S3: set @ real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ S3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( member @ real @ ( minus_minus @ real @ X @ H7 ) @ S3 )
=> ( ( ord_less @ real @ H7 @ D3 )
=> ( ord_less @ real @ ( F3 @ ( minus_minus @ real @ X @ H7 ) ) @ ( F3 @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
thf(fact_6389_has__real__derivative__neg__dec__left,axiom,
! [F3: real > real,L: real,X: real,S3: set @ real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ S3 ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H7 )
=> ( ( member @ real @ ( minus_minus @ real @ X @ H7 ) @ S3 )
=> ( ( ord_less @ real @ H7 @ D3 )
=> ( ord_less @ real @ ( F3 @ X ) @ ( F3 @ ( minus_minus @ real @ X @ H7 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
thf(fact_6390_at__le,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,T2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
=> ( ord_less_eq @ ( filter @ A ) @ ( topolo174197925503356063within @ A @ X @ S ) @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ).
% at_le
thf(fact_6391_has__field__derivative__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Y: A,X: A,S: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F3 @ Y @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_field_derivative @ A @ F3 @ Y @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% has_field_derivative_subset
thf(fact_6392_DERIV__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,F8: A,X: A,S: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_field_derivative @ A @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% DERIV_subset
thf(fact_6393_DERIV__local__const,axiom,
! [F3: real > real,L: real,X: real,D2: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y3 ) ) @ D2 )
=> ( ( F3 @ X )
= ( F3 @ Y3 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_const
thf(fact_6394_DERIV__power__Suc,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,X: A,S: set @ A,N: nat] :
( ( has_field_derivative @ A @ F3 @ D5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( power_power @ A @ ( F3 @ X4 ) @ ( suc @ N ) )
@ ( times_times @ A @ ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) ) @ ( times_times @ A @ D5 @ ( power_power @ A @ ( F3 @ X ) @ N ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_power_Suc
thf(fact_6395_DERIV__power,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,X: A,S: set @ A,N: nat] :
( ( has_field_derivative @ A @ F3 @ D5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( power_power @ A @ ( F3 @ X4 ) @ N )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( times_times @ A @ D5 @ ( power_power @ A @ ( F3 @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_power
thf(fact_6396_DERIV__inverse,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,S: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( inverse_inverse @ A ) @ ( uminus_uminus @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_inverse
thf(fact_6397_DERIV__local__max,axiom,
! [F3: real > real,L: real,X: real,D2: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y3 ) ) @ D2 )
=> ( ord_less_eq @ real @ ( F3 @ Y3 ) @ ( F3 @ X ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_max
thf(fact_6398_DERIV__local__min,axiom,
! [F3: real > real,L: real,X: real,D2: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y3 ) ) @ D2 )
=> ( ord_less_eq @ real @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_min
thf(fact_6399_DERIV__ln__divide,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln_divide
thf(fact_6400_DERIV__pow,axiom,
! [N: nat,X: real,S: set @ real] :
( has_field_derivative @ real
@ ^ [X4: real] : ( power_power @ real @ X4 @ N )
@ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ X @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ S ) ) ).
% DERIV_pow
thf(fact_6401_at__within__Icc__at,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less @ A @ A2 @ X )
=> ( ( ord_less @ A @ X @ B2 )
=> ( ( topolo174197925503356063within @ A @ X @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% at_within_Icc_at
thf(fact_6402_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_6403_DERIV__quotient,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D2: A,X: A,S: set @ A,G3: A > A,E3: A] :
( ( has_field_derivative @ A @ F3 @ D2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G3 @ E3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [Y5: A] : ( divide_divide @ A @ ( F3 @ Y5 ) @ ( G3 @ Y5 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D2 @ ( G3 @ X ) ) @ ( times_times @ A @ E3 @ ( F3 @ X ) ) ) @ ( power_power @ A @ ( G3 @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% DERIV_quotient
thf(fact_6404_trivial__limit__at__left__bot,axiom,
! [A: $tType] :
( ( ( order_bot @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ( ( topolo174197925503356063within @ A @ ( bot_bot @ A ) @ ( set_ord_lessThan @ A @ ( bot_bot @ A ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_left_bot
thf(fact_6405_DERIV__inverse__fun,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D2: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F3 @ D2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( inverse_inverse @ A @ ( F3 @ X4 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ D2 @ ( inverse_inverse @ A @ ( power_power @ A @ ( F3 @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_inverse_fun
thf(fact_6406_termdiffs__sums__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K4: real,C3: nat > A,F3: A > A,F8: A,Z: A] :
( ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K4 )
=> ( sums @ A
@ ^ [N6: nat] : ( times_times @ A @ ( C3 @ N6 ) @ ( power_power @ A @ Z3 @ N6 ) )
@ ( F3 @ Z3 ) ) )
=> ( ( has_field_derivative @ A @ F3 @ F8 @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K4 )
=> ( sums @ A
@ ^ [N6: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N6 ) @ ( power_power @ A @ Z @ N6 ) )
@ F8 ) ) ) ) ) ).
% termdiffs_sums_strong
thf(fact_6407_ucast__slice,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) )
= ( slice2 @ B @ A @ ( zero_zero @ nat ) ) ) ) ).
% ucast_slice
thf(fact_6408_has__real__derivative__powr,axiom,
! [Z: real,R2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Z )
=> ( has_field_derivative @ real
@ ^ [Z2: real] : ( powr @ real @ Z2 @ 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_6409_slice__cat2,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [A2: word @ B,T2: word @ A] :
( ( slice2 @ A @ A @ ( zero_zero @ nat ) @ ( word_cat @ B @ A @ A @ A2 @ T2 ) )
= T2 ) ) ).
% slice_cat2
thf(fact_6410_termdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K4: A,X: A] :
( ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( C3 @ N6 ) @ ( power_power @ A @ K4 @ N6 ) ) )
=> ( ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N6 ) @ ( power_power @ A @ K4 @ N6 ) ) )
=> ( ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C3 ) @ N6 ) @ ( power_power @ A @ K4 @ N6 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K4 ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] :
( suminf @ A
@ ^ [N6: nat] : ( times_times @ A @ ( C3 @ N6 ) @ ( power_power @ A @ X4 @ N6 ) ) )
@ ( suminf @ A
@ ^ [N6: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N6 ) @ ( power_power @ A @ X @ N6 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% termdiffs
thf(fact_6411_termdiffs__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K4: A,X: A] :
( ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( C3 @ N6 ) @ ( power_power @ A @ K4 @ N6 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K4 ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] :
( suminf @ A
@ ^ [N6: nat] : ( times_times @ A @ ( C3 @ N6 ) @ ( power_power @ A @ X4 @ N6 ) ) )
@ ( suminf @ A
@ ^ [N6: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N6 ) @ ( power_power @ A @ X @ N6 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong
thf(fact_6412_termdiffs__strong_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K4: real,C3: nat > A,Z: A] :
( ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K4 )
=> ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( C3 @ N6 ) @ ( power_power @ A @ Z3 @ N6 ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K4 )
=> ( has_field_derivative @ A
@ ^ [Z2: A] :
( suminf @ A
@ ^ [N6: nat] : ( times_times @ A @ ( C3 @ N6 ) @ ( power_power @ A @ Z2 @ N6 ) ) )
@ ( suminf @ A
@ ^ [N6: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N6 ) @ ( power_power @ A @ Z @ N6 ) ) )
@ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong'
thf(fact_6413_DERIV__fun__powr,axiom,
! [G3: real > real,M: real,X: real,R2: real] :
( ( has_field_derivative @ real @ G3 @ M @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X ) )
=> ( has_field_derivative @ real
@ ^ [X4: real] : ( powr @ real @ ( G3 @ X4 ) @ R2 )
@ ( times_times @ real @ ( times_times @ real @ R2 @ ( powr @ real @ ( G3 @ X ) @ ( minus_minus @ real @ R2 @ ( semiring_1_of_nat @ real @ ( one_one @ nat ) ) ) ) ) @ M )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_fun_powr
thf(fact_6414_DERIV__log,axiom,
! [X: real,B2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( log @ B2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( times_times @ real @ ( ln_ln @ real @ B2 ) @ X ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_log
thf(fact_6415_DERIV__powr,axiom,
! [G3: real > real,M: real,X: real,F3: real > real,R2: real] :
( ( has_field_derivative @ real @ G3 @ M @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X ) )
=> ( ( has_field_derivative @ real @ F3 @ R2 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X4: real] : ( powr @ real @ ( G3 @ X4 ) @ ( F3 @ X4 ) )
@ ( times_times @ real @ ( powr @ real @ ( G3 @ X ) @ ( F3 @ X ) ) @ ( plus_plus @ real @ ( times_times @ real @ R2 @ ( ln_ln @ real @ ( G3 @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ M @ ( F3 @ X ) ) @ ( G3 @ X ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_powr
thf(fact_6416_DERIV__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( tan @ A ) @ ( inverse_inverse @ A @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_tan
thf(fact_6417_DERIV__real__sqrt,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ sqrt @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_real_sqrt
thf(fact_6418_DERIV__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sin @ A @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( cot @ A ) @ ( uminus_uminus @ A @ ( inverse_inverse @ A @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_cot
thf(fact_6419_has__field__derivative__tanh,axiom,
! [A17: $tType] :
( ( ( real_Vector_banach @ A17 )
& ( real_V3459762299906320749_field @ A17 ) )
=> ! [G3: A17 > A17,X: A17,Db: A17,S: set @ A17] :
( ( ( cosh @ A17 @ ( G3 @ X ) )
!= ( zero_zero @ A17 ) )
=> ( ( has_field_derivative @ A17 @ G3 @ Db @ ( topolo174197925503356063within @ A17 @ X @ S ) )
=> ( has_field_derivative @ A17
@ ^ [X4: A17] : ( tanh @ A17 @ ( G3 @ X4 ) )
@ ( times_times @ A17 @ ( minus_minus @ A17 @ ( one_one @ A17 ) @ ( power_power @ A17 @ ( tanh @ A17 @ ( G3 @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ Db )
@ ( topolo174197925503356063within @ A17 @ X @ S ) ) ) ) ) ).
% has_field_derivative_tanh
thf(fact_6420_DERIV__real__sqrt__generic,axiom,
! [X: real,D5: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( D5
= ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( D5
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( has_field_derivative @ real @ sqrt @ D5 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_6421_arcosh__real__has__field__derivative,axiom,
! [X: real,A5: set @ real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( has_field_derivative @ real @ ( arcosh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ A5 ) ) ) ).
% arcosh_real_has_field_derivative
thf(fact_6422_artanh__real__has__field__derivative,axiom,
! [X: real,A5: set @ real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ ( artanh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ A5 ) ) ) ).
% artanh_real_has_field_derivative
thf(fact_6423_DERIV__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_real_root
thf(fact_6424_DERIV__arccos,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arccos @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arccos
thf(fact_6425_DERIV__arcsin,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arcsin @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arcsin
thf(fact_6426_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F3: real > real,X: real,N: nat] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M3: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_6427_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F3: real > real,X: real,N: nat] :
( ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
& ! [M3: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_6428_DERIV__odd__real__root,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_6429_Maclaurin,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F3: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less @ real @ T7 @ H2 )
& ( ( F3 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_6430_Maclaurin2,axiom,
! [H2: real,Diff: nat > real > real,F3: real > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 )
& ( ( F3 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_6431_Maclaurin__minus,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F3: real > real] :
( ( ord_less @ real @ H2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ H2 @ T7 )
& ( ord_less_eq @ real @ T7 @ ( zero_zero @ real ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ H2 @ T7 )
& ( ord_less @ real @ T7 @ ( zero_zero @ real ) )
& ( ( F3 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_6432_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F3: real > real,N: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ! [M3: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T7 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_6433_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F3: real > real,N: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_6434_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A2: real,B2: real,C3: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ A2 @ T7 )
& ( ord_less_eq @ real @ T7 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ C3 )
=> ( ( ord_less_eq @ real @ C3 @ B2 )
=> ( ( ord_less_eq @ real @ A2 @ X )
=> ( ( ord_less_eq @ real @ X @ B2 )
=> ( ( X != C3 )
=> ? [T7: real] :
( ( ( ord_less @ real @ X @ C3 )
=> ( ( ord_less @ real @ X @ T7 )
& ( ord_less @ real @ T7 @ C3 ) ) )
& ( ~ ( ord_less @ real @ X @ C3 )
=> ( ( ord_less @ real @ C3 @ T7 )
& ( ord_less @ real @ T7 @ X ) ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ C3 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C3 ) @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C3 ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_6435_Taylor__up,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A2: real,B2: real,C3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ A2 @ T7 )
& ( ord_less_eq @ real @ T7 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ C3 )
=> ( ( ord_less @ real @ C3 @ B2 )
=> ? [T7: real] :
( ( ord_less @ real @ C3 @ T7 )
& ( ord_less @ real @ T7 @ B2 )
& ( ( F3 @ B2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ C3 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C3 ) @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C3 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_6436_Taylor__down,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A2: real,B2: real,C3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ A2 @ T7 )
& ( ord_less_eq @ real @ T7 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less @ real @ A2 @ C3 )
=> ( ( ord_less_eq @ real @ C3 @ B2 )
=> ? [T7: real] :
( ( ord_less @ real @ A2 @ T7 )
& ( ord_less @ real @ T7 @ C3 )
& ( ( F3 @ A2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ C3 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C3 ) @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C3 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_6437_slice__cat1,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [A2: word @ B,B2: word @ C] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( size_size @ ( word @ B ) @ A2 ) @ ( size_size @ ( word @ C ) @ B2 ) ) @ ( size_size @ ( word @ A ) @ ( word_cat @ B @ C @ A @ A2 @ B2 ) ) )
=> ( ( slice2 @ A @ B @ ( size_size @ ( word @ C ) @ B2 ) @ ( word_cat @ B @ C @ A @ A2 @ B2 ) )
= A2 ) ) ) ).
% slice_cat1
thf(fact_6438_Maclaurin__lemma2,axiom,
! [N: nat,H2: real,Diff: nat > real > real,K: nat,B5: real] :
( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( N
= ( suc @ K ) )
=> ! [M2: nat,T9: real] :
( ( ( ord_less @ nat @ M2 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T9 )
& ( ord_less_eq @ real @ T9 @ H2 ) )
=> ( has_field_derivative @ real
@ ^ [U3: real] :
( minus_minus @ real @ ( Diff @ M2 @ U3 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P7: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ M2 @ P7 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P7 ) ) @ ( power_power @ real @ U3 @ P7 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ M2 ) ) )
@ ( times_times @ real @ B5 @ ( divide_divide @ real @ ( power_power @ real @ U3 @ ( minus_minus @ nat @ N @ M2 ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) )
@ ( minus_minus @ real @ ( Diff @ ( suc @ M2 ) @ T9 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P7: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ ( suc @ M2 ) @ P7 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P7 ) ) @ ( power_power @ real @ T9 @ P7 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ M2 ) ) ) )
@ ( times_times @ real @ B5 @ ( divide_divide @ real @ ( power_power @ real @ T9 @ ( minus_minus @ nat @ N @ ( suc @ M2 ) ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ T9 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_6439_split__slices,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( type_len @ C )
& ( type_len @ B )
& ( type_len @ A ) )
=> ! [W2: word @ C,U: word @ A,V3: word @ B] :
( ( ( word_split @ C @ A @ B @ W2 )
= ( product_Pair @ ( word @ A ) @ ( word @ B ) @ U @ V3 ) )
=> ( ( U
= ( slice2 @ C @ A @ ( size_size @ ( word @ B ) @ V3 ) @ W2 ) )
& ( V3
= ( slice2 @ C @ B @ ( zero_zero @ nat ) @ W2 ) ) ) ) ) ).
% split_slices
thf(fact_6440_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real
@ ^ [X17: 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 @ X17 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) )
@ ( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( power_power @ real @ X @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_arctan_series
thf(fact_6441_DERIV__real__root__generic,axiom,
! [N: nat,X: real,D5: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( D5
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( D5
= ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( D5
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ D5 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_6442_DERIV__power__series_H,axiom,
! [R3: real,F3: nat > real,X0: real] :
( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R3 ) @ R3 ) )
=> ( summable @ real
@ ^ [N6: nat] : ( times_times @ real @ ( times_times @ real @ ( F3 @ N6 ) @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) @ ( power_power @ real @ X3 @ N6 ) ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R3 ) @ R3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ( has_field_derivative @ real
@ ^ [X4: real] :
( suminf @ real
@ ^ [N6: nat] : ( times_times @ real @ ( F3 @ N6 ) @ ( power_power @ real @ X4 @ ( suc @ N6 ) ) ) )
@ ( suminf @ real
@ ^ [N6: nat] : ( times_times @ real @ ( times_times @ real @ ( F3 @ N6 ) @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) @ ( power_power @ real @ X0 @ N6 ) ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_6443_has__derivative__arcsin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,X: A,G7: A > real,S: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G3 @ X ) )
=> ( ( ord_less @ real @ ( G3 @ X ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G3 @ G7 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( arcsin @ ( G3 @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G7 @ X4 ) @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G3 @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% has_derivative_arcsin
thf(fact_6444_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_6445_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_6446_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_6447_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_6448_infinite__Ioo__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ioo_iff
thf(fact_6449_Sup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanLessThan
thf(fact_6450_cSup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_greaterThanLessThan
thf(fact_6451_Inf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_greaterThanLessThan
thf(fact_6452_cInf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_greaterThanLessThan
thf(fact_6453_has__derivative__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F8: A > B,X: A,S: set @ A,T2: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_derivative @ A @ B @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% has_derivative_subset
thf(fact_6454_infinite__Ioo,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ioo
thf(fact_6455_has__derivative__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [C3: B,F5: filter @ A] :
( has_derivative @ A @ B
@ ^ [X4: A] : C3
@ ^ [X4: A] : ( zero_zero @ B )
@ F5 ) ) ).
% has_derivative_const
thf(fact_6456_has__derivative__zero__unique,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F5: A > B,X: A] :
( ( has_derivative @ A @ B
@ ^ [X4: A] : ( zero_zero @ B )
@ F5
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( F5
= ( ^ [H4: A] : ( zero_zero @ B ) ) ) ) ) ).
% has_derivative_zero_unique
thf(fact_6457_has__derivative__in__compose2,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [T2: set @ A,G3: A > B,G7: A > A > B,F3: C > A,S: set @ C,X: C,F8: C > A] :
( ! [X3: A] :
( ( member @ A @ X3 @ T2 )
=> ( has_derivative @ A @ B @ G3 @ ( G7 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F3 @ S ) @ T2 )
=> ( ( member @ C @ X @ S )
=> ( ( has_derivative @ C @ A @ F3 @ F8 @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( has_derivative @ C @ B
@ ^ [X4: C] : ( G3 @ ( F3 @ X4 ) )
@ ^ [Y5: C] : ( G7 @ ( F3 @ X ) @ ( F8 @ Y5 ) )
@ ( topolo174197925503356063within @ C @ X @ S ) ) ) ) ) ) ) ).
% has_derivative_in_compose2
thf(fact_6458_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_6459_ivl__disj__int__two_I4_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(4)
thf(fact_6460_ivl__disj__int__two_I5_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(5)
thf(fact_6461_ivl__disj__int__two_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(1)
thf(fact_6462_ivl__disj__int__one_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(1)
thf(fact_6463_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_6464_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_6465_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_6466_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) ).
% atLeastAtMost_diff_ends
thf(fact_6467_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_6468_has__derivative__divide_H,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,F8: C > A,X: C,S3: set @ C,G3: C > A,G7: C > A] :
( ( has_derivative @ C @ A @ F3 @ F8 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( has_derivative @ C @ A @ G3 @ G7 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( ( G3 @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ^ [H4: C] : ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ ( F8 @ H4 ) @ ( G3 @ X ) ) @ ( times_times @ A @ ( F3 @ X ) @ ( G7 @ H4 ) ) ) @ ( times_times @ A @ ( G3 @ X ) @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ) ).
% has_derivative_divide'
thf(fact_6469_has__derivative__inverse_H,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,S3: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A @ ( inverse_inverse @ A )
@ ^ [H4: A] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ X ) @ H4 ) @ ( inverse_inverse @ A @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_derivative_inverse'
thf(fact_6470_has__derivative__inverse,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: C > A,X: C,F8: C > A,S3: set @ C] :
( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F3 @ F8 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( inverse_inverse @ A @ ( F3 @ X4 ) )
@ ^ [H4: C] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F3 @ X ) ) @ ( F8 @ H4 ) ) @ ( inverse_inverse @ A @ ( F3 @ X ) ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ).
% has_derivative_inverse
thf(fact_6471_DERIV__isconst3,axiom,
! [A2: real,B2: real,X: real,Y: real,F3: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( member @ real @ X @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( member @ real @ Y @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( has_field_derivative @ real @ F3 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( F3 @ X )
= ( F3 @ Y ) ) ) ) ) ) ).
% DERIV_isconst3
thf(fact_6472_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_6473_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_6474_has__derivative__ln,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,X: A,G7: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X ) )
=> ( ( has_derivative @ A @ real @ G3 @ G7 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( ln_ln @ real @ ( G3 @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G7 @ X4 ) @ ( inverse_inverse @ real @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_ln
thf(fact_6475_has__derivative__divide,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: C > A,F8: C > A,X: C,S3: set @ C,G3: C > A,G7: C > A] :
( ( has_derivative @ C @ A @ F3 @ F8 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( has_derivative @ C @ A @ G3 @ G7 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( ( G3 @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ^ [H4: C] : ( plus_plus @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( F3 @ X ) ) @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( G3 @ X ) ) @ ( G7 @ H4 ) ) @ ( inverse_inverse @ A @ ( G3 @ X ) ) ) ) @ ( divide_divide @ A @ ( F8 @ H4 ) @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ) ).
% has_derivative_divide
thf(fact_6476_has__derivative__prod,axiom,
! [B: $tType,I5: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [I6: set @ I5,F3: I5 > A > B,F8: I5 > A > B,X: A,S3: set @ A] :
( ! [I2: I5] :
( ( member @ I5 @ I2 @ I6 )
=> ( has_derivative @ A @ B @ ( F3 @ I2 ) @ ( F8 @ I2 ) @ ( topolo174197925503356063within @ A @ X @ S3 ) ) )
=> ( has_derivative @ A @ B
@ ^ [X4: A] :
( groups7121269368397514597t_prod @ I5 @ B
@ ^ [I3: I5] : ( F3 @ I3 @ X4 )
@ I6 )
@ ^ [Y5: A] :
( groups7311177749621191930dd_sum @ I5 @ B
@ ^ [I3: I5] :
( times_times @ B @ ( F8 @ I3 @ Y5 )
@ ( groups7121269368397514597t_prod @ I5 @ B
@ ^ [J3: I5] : ( F3 @ J3 @ X )
@ ( minus_minus @ ( set @ I5 ) @ I6 @ ( insert @ I5 @ I3 @ ( bot_bot @ ( set @ I5 ) ) ) ) ) )
@ I6 )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_derivative_prod
thf(fact_6477_has__derivative__powr,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,G7: A > real,X: A,X6: set @ A,F3: A > real,F8: A > real] :
( ( has_derivative @ A @ real @ G3 @ G7 @ ( topolo174197925503356063within @ A @ X @ X6 ) )
=> ( ( has_derivative @ A @ real @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X @ X6 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X ) )
=> ( ( member @ A @ X @ X6 )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( powr @ real @ ( G3 @ X4 ) @ ( F3 @ X4 ) )
@ ^ [H4: A] : ( times_times @ real @ ( powr @ real @ ( G3 @ X ) @ ( F3 @ X ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( F8 @ H4 ) @ ( ln_ln @ real @ ( G3 @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ ( G7 @ H4 ) @ ( F3 @ X ) ) @ ( G3 @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ X6 ) ) ) ) ) ) ) ).
% has_derivative_powr
thf(fact_6478_has__derivative__real__sqrt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,X: A,G7: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X ) )
=> ( ( has_derivative @ A @ real @ G3 @ G7 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( sqrt @ ( G3 @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G7 @ X4 ) @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ ( G3 @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_real_sqrt
thf(fact_6479_DERIV__series_H,axiom,
! [F3: real > nat > real,F8: real > nat > real,X0: real,A2: real,B2: real,L5: nat > real] :
( ! [N2: nat] :
( has_field_derivative @ real
@ ^ [X4: real] : ( F3 @ X4 @ N2 )
@ ( F8 @ X0 @ N2 )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( summable @ real @ ( F3 @ X3 ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( summable @ real @ ( F8 @ X0 ) )
=> ( ( summable @ real @ L5 )
=> ( ! [N2: nat,X3: real,Y3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( member @ real @ Y3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( F3 @ X3 @ N2 ) @ ( F3 @ Y3 @ N2 ) ) ) @ ( times_times @ real @ ( L5 @ N2 ) @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y3 ) ) ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X4: real] : ( suminf @ real @ ( F3 @ X4 ) )
@ ( suminf @ real @ ( F8 @ X0 ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_series'
thf(fact_6480_has__derivative__arccos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,X: A,G7: A > real,S: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G3 @ X ) )
=> ( ( ord_less @ real @ ( G3 @ X ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G3 @ G7 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( arccos @ ( G3 @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G7 @ X4 ) @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G3 @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% has_derivative_arccos
thf(fact_6481_termdiffs__aux,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K4: A,X: A] :
( ( summable @ A
@ ^ [N6: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C3 ) @ N6 ) @ ( power_power @ A @ K4 @ N6 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K4 ) )
=> ( filterlim @ A @ A
@ ^ [H4: A] :
( suminf @ A
@ ^ [N6: nat] : ( times_times @ A @ ( C3 @ N6 ) @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ X @ H4 ) @ N6 ) @ ( power_power @ A @ X @ N6 ) ) @ H4 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N6 ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N6 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_aux
thf(fact_6482_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
@ ^ [L4: code_integer,J3: code_integer] :
( if @ int
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L4 ) )
@ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L4 ) ) @ ( one_one @ int ) ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_6483_int__of__integer__max,axiom,
! [K: code_integer,L: code_integer] :
( ( code_int_of_integer @ ( ord_max @ code_integer @ K @ L ) )
= ( ord_max @ int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).
% int_of_integer_max
thf(fact_6484_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% finite_greaterThanLessThan
thf(fact_6485_finite__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or5935395276787703475ssThan @ code_integer @ L @ U ) ) ).
% finite_greaterThanLessThan_integer
thf(fact_6486_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_6487_integer__of__int__int__of__integer,axiom,
! [K: code_integer] :
( ( code_integer_of_int @ ( code_int_of_integer @ K ) )
= K ) ).
% integer_of_int_int_of_integer
thf(fact_6488_int__of__integer__integer__of__int,axiom,
! [K: int] :
( ( code_int_of_integer @ ( code_integer_of_int @ K ) )
= K ) ).
% int_of_integer_integer_of_int
thf(fact_6489_int__of__integer__inverse,axiom,
! [X: code_integer] :
( ( code_integer_of_int @ ( code_int_of_integer @ X ) )
= X ) ).
% int_of_integer_inverse
thf(fact_6490_of__int__integer__of,axiom,
! [K: code_integer] :
( ( ring_1_of_int @ code_integer @ ( code_int_of_integer @ K ) )
= K ) ).
% of_int_integer_of
thf(fact_6491_int__of__integer__of__int,axiom,
! [K: int] :
( ( code_int_of_integer @ ( ring_1_of_int @ code_integer @ K ) )
= K ) ).
% int_of_integer_of_int
thf(fact_6492_zero__integer_Orep__eq,axiom,
( ( code_int_of_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ int ) ) ).
% zero_integer.rep_eq
thf(fact_6493_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ ( suc @ L ) ) ) ).
% card_greaterThanLessThan
thf(fact_6494_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_6495_one__integer_Orep__eq,axiom,
( ( code_int_of_integer @ ( one_one @ code_integer ) )
= ( one_one @ int ) ) ).
% one_integer.rep_eq
thf(fact_6496_uminus__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( uminus_uminus @ code_integer @ X ) )
= ( uminus_uminus @ int @ ( code_int_of_integer @ X ) ) ) ).
% uminus_integer.rep_eq
thf(fact_6497_plus__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( plus_plus @ code_integer @ X @ Xa2 ) )
= ( plus_plus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% plus_integer.rep_eq
thf(fact_6498_minus__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( minus_minus @ code_integer @ X @ Xa2 ) )
= ( minus_minus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% minus_integer.rep_eq
thf(fact_6499_times__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( times_times @ code_integer @ X @ Xa2 ) )
= ( times_times @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% times_integer.rep_eq
thf(fact_6500_abs__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( abs_abs @ code_integer @ X ) )
= ( abs_abs @ int @ ( code_int_of_integer @ X ) ) ) ).
% abs_integer.rep_eq
thf(fact_6501_divide__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( divide_divide @ code_integer @ X @ Xa2 ) )
= ( divide_divide @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% divide_integer.rep_eq
thf(fact_6502_modulo__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( modulo_modulo @ code_integer @ X @ Xa2 ) )
= ( modulo_modulo @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% modulo_integer.rep_eq
thf(fact_6503_sgn__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( sgn_sgn @ code_integer @ X ) )
= ( sgn_sgn @ int @ ( code_int_of_integer @ X ) ) ) ).
% sgn_integer.rep_eq
thf(fact_6504_tendsto__mult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,F3: B > A,L: A,F5: filter @ B] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ C3 @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C3 @ L ) )
@ F5 )
= ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% tendsto_mult_left_iff
thf(fact_6505_tendsto__mult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,F3: B > A,L: A,F5: filter @ B] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C3 ) )
@ F5 )
= ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% tendsto_mult_right_iff
thf(fact_6506_power__tendsto__0__iff,axiom,
! [A: $tType,N: nat,F3: A > real,F5: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real
@ ^ [X4: A] : ( power_power @ real @ ( F3 @ X4 ) @ N )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 )
= ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) ) ) ).
% power_tendsto_0_iff
thf(fact_6507_LIM__offset__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,A2: A,L5: B] :
( ( filterlim @ A @ B
@ ^ [H4: A] : ( F3 @ ( plus_plus @ A @ A2 @ H4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero_cancel
thf(fact_6508_LIM__offset__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,L5: B,A2: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H4: A] : ( F3 @ ( plus_plus @ A @ A2 @ H4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero
thf(fact_6509_LIM__isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,A2: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [H4: A] : ( F3 @ ( plus_plus @ A @ A2 @ H4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A2 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_isCont_iff
thf(fact_6510_real__LIM__sandwich__zero,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: A > real,A2: A,G3: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ! [X3: A] :
( ( X3 != A2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( G3 @ X3 ) ) )
=> ( ! [X3: A] :
( ( X3 != A2 )
=> ( ord_less_eq @ real @ ( G3 @ X3 ) @ ( F3 @ X3 ) ) )
=> ( filterlim @ A @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% real_LIM_sandwich_zero
thf(fact_6511_LIM__imp__LIM,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,L: B,A2: A,G3: A > C,M: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ! [X3: A] :
( ( X3 != A2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( minus_minus @ C @ ( G3 @ X3 ) @ M ) ) @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F3 @ X3 ) @ L ) ) ) )
=> ( filterlim @ A @ C @ G3 @ ( topolo7230453075368039082e_nhds @ C @ M ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_imp_LIM
thf(fact_6512_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
@ ^ [X4: A] : K
@ ( topolo7230453075368039082e_nhds @ Aa @ ( zero_zero @ Aa ) )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_not_zero
thf(fact_6513_tendsto__within__subset,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: A > B,L: filter @ B,X: A,S3: set @ A,T4: set @ A] :
( ( filterlim @ A @ B @ F3 @ L @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T4 @ S3 )
=> ( filterlim @ A @ B @ F3 @ L @ ( topolo174197925503356063within @ A @ X @ T4 ) ) ) ) ) ).
% tendsto_within_subset
thf(fact_6514_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L ) @ U )
= ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_6515_or__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( bit_se1065995026697491101ons_or @ code_integer @ X @ Xa2 ) )
= ( bit_se1065995026697491101ons_or @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% or_integer.rep_eq
thf(fact_6516_bit__integer_Orep__eq,axiom,
( ( bit_se5641148757651400278ts_bit @ code_integer )
= ( ^ [X4: code_integer] : ( bit_se5641148757651400278ts_bit @ int @ ( code_int_of_integer @ X4 ) ) ) ) ).
% bit_integer.rep_eq
thf(fact_6517_tendsto__add__zero,axiom,
! [B: $tType,D: $tType] :
( ( topolo6943815403480290642id_add @ B )
=> ! [F3: D > B,F5: filter @ D,G3: D > B] :
( ( filterlim @ D @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( ( filterlim @ D @ B @ G3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( filterlim @ D @ B
@ ^ [X4: D] : ( plus_plus @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ) ).
% tendsto_add_zero
thf(fact_6518_tendsto__mult__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F5: filter @ D,G3: D > A] :
( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( ( filterlim @ D @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ A
@ ^ [X4: D] : ( times_times @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ) ).
% tendsto_mult_zero
thf(fact_6519_tendsto__mult__left__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F5: filter @ D,C3: A] :
( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ A
@ ^ [X4: D] : ( times_times @ A @ ( F3 @ X4 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ).
% tendsto_mult_left_zero
thf(fact_6520_tendsto__mult__right__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F5: filter @ D,C3: A] :
( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ A
@ ^ [X4: D] : ( times_times @ A @ C3 @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ).
% tendsto_mult_right_zero
thf(fact_6521_int__of__integer__induct,axiom,
! [Y: int,P2: int > $o] :
( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
=> ( ! [X3: code_integer] : ( P2 @ ( code_int_of_integer @ X3 ) )
=> ( P2 @ Y ) ) ) ).
% int_of_integer_induct
thf(fact_6522_int__of__integer__cases,axiom,
! [Y: int] :
( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
=> ~ ! [X3: code_integer] :
( Y
!= ( code_int_of_integer @ X3 ) ) ) ).
% int_of_integer_cases
thf(fact_6523_int__of__integer,axiom,
! [X: code_integer] : ( member @ int @ ( code_int_of_integer @ X ) @ ( top_top @ ( set @ int ) ) ) ).
% int_of_integer
thf(fact_6524_type__definition__integer,axiom,
type_definition @ code_integer @ int @ code_int_of_integer @ code_integer_of_int @ ( top_top @ ( set @ int ) ) ).
% type_definition_integer
thf(fact_6525_integer__of__int__inverse,axiom,
! [Y: int] :
( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
=> ( ( code_int_of_integer @ ( code_integer_of_int @ Y ) )
= Y ) ) ).
% integer_of_int_inverse
thf(fact_6526_tendsto__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: A > A,A2: A,F5: filter @ A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( ( cos @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X4: A] : ( tan @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tan @ A @ A2 ) )
@ F5 ) ) ) ) ).
% tendsto_tan
thf(fact_6527_tendsto__divide,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: B > A,A2: A,F5: filter @ B,G3: B > A,B2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F5 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( divide_divide @ A @ A2 @ B2 ) )
@ F5 ) ) ) ) ) ).
% tendsto_divide
thf(fact_6528_tendsto__divide__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: B > A,F5: filter @ B,C3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( divide_divide @ A @ ( F3 @ X4 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ).
% tendsto_divide_zero
thf(fact_6529_int__of__integer__less__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_less @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Y ) )
= ( ord_less @ code_integer @ X @ Y ) ) ).
% int_of_integer_less_iff
thf(fact_6530_less__integer_Orep__eq,axiom,
( ( ord_less @ code_integer )
= ( ^ [X4: code_integer,Xa5: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa5 ) ) ) ) ).
% less_integer.rep_eq
thf(fact_6531_integer__less__iff,axiom,
( ( ord_less @ code_integer )
= ( ^ [K3: code_integer,L4: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L4 ) ) ) ) ).
% integer_less_iff
thf(fact_6532_tendsto__null__sum,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I6: set @ B,F3: A > B > C,F5: filter @ A] :
( ! [I2: B] :
( ( member @ B @ I2 @ I6 )
=> ( filterlim @ A @ C
@ ^ [X4: A] : ( F3 @ X4 @ I2 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) )
=> ( filterlim @ A @ C
@ ^ [I3: A] : ( groups7311177749621191930dd_sum @ B @ C @ ( F3 @ I3 ) @ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) ) ) ).
% tendsto_null_sum
thf(fact_6533_tendsto__log,axiom,
! [A: $tType,F3: A > real,A2: real,F5: filter @ A,G3: A > real,B2: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 )
=> ( ( filterlim @ A @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( log @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( log @ A2 @ B2 ) )
@ F5 ) ) ) ) ) ) ).
% tendsto_log
thf(fact_6534_tendsto__null__power,axiom,
! [B: $tType,A: $tType] :
( ( real_V2822296259951069270ebra_1 @ B )
=> ! [F3: A > B,F5: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ) ).
% tendsto_null_power
thf(fact_6535_Lim__transform__eq,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,G3: B > A,F5: filter @ B,A2: A] :
( ( filterlim @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
= ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) ) ) ) ).
% Lim_transform_eq
thf(fact_6536_LIM__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X4: A] : ( minus_minus @ B @ ( F3 @ X4 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) ) ) ).
% LIM_zero_cancel
thf(fact_6537_Lim__transform2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,A2: A,F5: filter @ B,G3: B > A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( filterlim @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 )
=> ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) ) ) ) ).
% Lim_transform2
thf(fact_6538_Lim__transform,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: B > A,A2: A,F5: filter @ B,F3: B > A] :
( ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( filterlim @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) ) ) ) ).
% Lim_transform
thf(fact_6539_LIM__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X4: A] : ( minus_minus @ B @ ( F3 @ X4 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 )
= ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) ) ) ).
% LIM_zero_iff
thf(fact_6540_LIM__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( minus_minus @ B @ ( F3 @ X4 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ).
% LIM_zero
thf(fact_6541_tendsto__norm__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 ) ) ) ).
% tendsto_norm_zero
thf(fact_6542_tendsto__norm__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 )
= ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) ) ) ).
% tendsto_norm_zero_iff
thf(fact_6543_tendsto__norm__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) ) ) ).
% tendsto_norm_zero_cancel
thf(fact_6544_tendsto__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: B > A,A2: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( inverse_inverse @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( inverse_inverse @ A @ A2 ) )
@ F5 ) ) ) ) ).
% tendsto_inverse
thf(fact_6545_int__of__integer__inject,axiom,
! [X: code_integer,Y: code_integer] :
( ( ( code_int_of_integer @ X )
= ( code_int_of_integer @ Y ) )
= ( X = Y ) ) ).
% int_of_integer_inject
thf(fact_6546_integer__eqI,axiom,
! [K: code_integer,L: code_integer] :
( ( ( code_int_of_integer @ K )
= ( code_int_of_integer @ L ) )
=> ( K = L ) ) ).
% integer_eqI
thf(fact_6547_integer__eq__iff,axiom,
( ( ^ [Y6: code_integer,Z4: code_integer] : Y6 = Z4 )
= ( ^ [K3: code_integer,L4: code_integer] :
( ( code_int_of_integer @ K3 )
= ( code_int_of_integer @ L4 ) ) ) ) ).
% integer_eq_iff
thf(fact_6548_tendsto__sgn,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,L: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 )
=> ( ( L
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( sgn_sgn @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sgn_sgn @ A @ L ) )
@ F5 ) ) ) ) ).
% tendsto_sgn
thf(fact_6549_tendsto__artanh,axiom,
! [A: $tType,F3: A > real,A2: real,F5: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ A2 )
=> ( ( ord_less @ real @ A2 @ ( one_one @ real ) )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( artanh @ real @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( artanh @ real @ A2 ) )
@ F5 ) ) ) ) ).
% tendsto_artanh
thf(fact_6550_tendsto__arcosh,axiom,
! [B: $tType,F3: B > real,A2: real,F5: filter @ B] :
( ( filterlim @ B @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( filterlim @ B @ real
@ ^ [X4: B] : ( arcosh @ real @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A2 ) )
@ F5 ) ) ) ).
% tendsto_arcosh
thf(fact_6551_tendsto__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: A > A,A2: A,F5: filter @ A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( ( sin @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X4: A] : ( cot @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cot @ A @ A2 ) )
@ F5 ) ) ) ) ).
% tendsto_cot
thf(fact_6552_tendsto__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,A2: A,F5: filter @ C] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( ( cosh @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( filterlim @ C @ A
@ ^ [X4: C] : ( tanh @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tanh @ A @ A2 ) )
@ F5 ) ) ) ) ).
% tendsto_tanh
thf(fact_6553_tendsto__mono,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F5: filter @ B,F7: filter @ B,F3: B > A,L: A] :
( ( ord_less_eq @ ( filter @ B ) @ F5 @ F7 )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F7 )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% tendsto_mono
thf(fact_6554_filterlim__mono,axiom,
! [B: $tType,A: $tType,F3: A > B,F24: filter @ B,F14: filter @ A,F25: filter @ B,F16: filter @ A] :
( ( filterlim @ A @ B @ F3 @ F24 @ F14 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F24 @ F25 )
=> ( ( ord_less_eq @ ( filter @ A ) @ F16 @ F14 )
=> ( filterlim @ A @ B @ F3 @ F25 @ F16 ) ) ) ) ).
% filterlim_mono
thf(fact_6555_less__eq__integer_Orep__eq,axiom,
( ( ord_less_eq @ code_integer )
= ( ^ [X4: code_integer,Xa5: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa5 ) ) ) ) ).
% less_eq_integer.rep_eq
thf(fact_6556_integer__less__eq__iff,axiom,
( ( ord_less_eq @ code_integer )
= ( ^ [K3: code_integer,L4: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L4 ) ) ) ) ).
% integer_less_eq_iff
thf(fact_6557_tendsto__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,A2: B,F5: filter @ A,G3: A > C,B2: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F5 )
=> ( ( filterlim @ A @ C @ G3 @ ( topolo7230453075368039082e_nhds @ C @ B2 ) @ F5 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X4: A] : ( product_Pair @ B @ C @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A2 @ B2 ) )
@ F5 ) ) ) ) ).
% tendsto_Pair
thf(fact_6558_LIM__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A2: A,F3: A > D,L5: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 )
=> ( ( filterlim @ A @ D @ F3 @ ( topolo7230453075368039082e_nhds @ D @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ D
@ ^ [H4: A] : ( F3 @ ( plus_plus @ A @ A2 @ H4 ) )
@ ( topolo7230453075368039082e_nhds @ D @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_offset_zero_iff
thf(fact_6559_xor__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( bit_se5824344971392196577ns_xor @ code_integer @ X @ Xa2 ) )
= ( bit_se5824344971392196577ns_xor @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% xor_integer.rep_eq
thf(fact_6560_take__bit__integer_Orep__eq,axiom,
! [X: nat,Xa2: code_integer] :
( ( code_int_of_integer @ ( bit_se2584673776208193580ke_bit @ code_integer @ X @ Xa2 ) )
= ( bit_se2584673776208193580ke_bit @ int @ X @ ( code_int_of_integer @ Xa2 ) ) ) ).
% take_bit_integer.rep_eq
thf(fact_6561_LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [R3: real,A2: A,F3: A > B,G3: A > B,L: B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ( ! [X3: A] :
( ( X3 != A2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ R3 )
=> ( ( F3 @ X3 )
= ( G3 @ X3 ) ) ) )
=> ( ( filterlim @ A @ B @ G3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_equal2
thf(fact_6562_LIM__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,L5: B,A2: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S8 )
& ! [X4: A] :
( ( ( X4 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X4 @ A2 ) ) @ S8 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F3 @ X4 ) @ L5 ) ) @ R5 ) ) ) ) ) ) ) ).
% LIM_eq
thf(fact_6563_LIM__I,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F3: A > B,L5: B] :
( ! [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [S9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S9 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ S9 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F3 @ X3 ) @ L5 ) ) @ R ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_I
thf(fact_6564_LIM__D,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,L5: B,A2: A,R2: real] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [S4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S4 )
& ! [X5: A] :
( ( ( X5 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X5 @ A2 ) ) @ S4 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F3 @ X5 ) @ L5 ) ) @ R2 ) ) ) ) ) ) ).
% LIM_D
thf(fact_6565_DERIV__LIM__iff,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > A,A2: A,D5: A] :
( ( filterlim @ A @ A
@ ^ [H4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ A2 @ H4 ) ) @ ( F3 @ A2 ) ) @ H4 )
@ ( topolo7230453075368039082e_nhds @ A @ D5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [X4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ X4 ) @ ( F3 @ A2 ) ) @ ( minus_minus @ A @ X4 @ A2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ D5 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_LIM_iff
thf(fact_6566_unset__bit__integer_Orep__eq,axiom,
! [X: nat,Xa2: code_integer] :
( ( code_int_of_integer @ ( bit_se2638667681897837118et_bit @ code_integer @ X @ Xa2 ) )
= ( bit_se2638667681897837118et_bit @ int @ X @ ( code_int_of_integer @ Xa2 ) ) ) ).
% unset_bit_integer.rep_eq
thf(fact_6567_LIM__fun__less__zero,axiom,
! [F3: real > real,L: real,C3: real] :
( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
& ! [X5: real] :
( ( ( X5 != C3 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C3 @ X5 ) ) @ R ) )
=> ( ord_less @ real @ ( F3 @ X5 ) @ ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_less_zero
thf(fact_6568_LIM__fun__not__zero,axiom,
! [F3: real > real,L: real,C3: real] :
( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( L
!= ( zero_zero @ real ) )
=> ? [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
& ! [X5: real] :
( ( ( X5 != C3 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C3 @ X5 ) ) @ R ) )
=> ( ( F3 @ X5 )
!= ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_not_zero
thf(fact_6569_LIM__fun__gt__zero,axiom,
! [F3: real > real,L: real,C3: real] :
( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
& ! [X5: real] :
( ( ( X5 != C3 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C3 @ X5 ) ) @ R ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X5 ) ) ) ) ) ) ).
% LIM_fun_gt_zero
thf(fact_6570_Code__Numeral_Oset__bit__integer_Orep__eq,axiom,
! [X: nat,Xa2: code_integer] :
( ( code_int_of_integer @ ( bit_se5668285175392031749et_bit @ code_integer @ X @ Xa2 ) )
= ( bit_se5668285175392031749et_bit @ int @ X @ ( code_int_of_integer @ Xa2 ) ) ) ).
% Code_Numeral.set_bit_integer.rep_eq
thf(fact_6571_LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,B2: B,A2: A,G3: B > C,C3: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G3 @ ( topolo7230453075368039082e_nhds @ C @ C3 ) @ ( topolo174197925503356063within @ B @ B2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ D4 ) )
=> ( ( F3 @ X3 )
!= B2 ) ) )
=> ( filterlim @ A @ C
@ ^ [X4: A] : ( G3 @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C3 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_compose2
thf(fact_6572_flip__bit__integer_Orep__eq,axiom,
! [X: nat,Xa2: code_integer] :
( ( code_int_of_integer @ ( bit_se8732182000553998342ip_bit @ code_integer @ X @ Xa2 ) )
= ( bit_se8732182000553998342ip_bit @ int @ X @ ( code_int_of_integer @ Xa2 ) ) ) ).
% flip_bit_integer.rep_eq
thf(fact_6573_DERIV__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,X: A] :
( ( has_field_derivative @ A @ F3 @ D5 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X @ H4 ) ) @ ( F3 @ X ) ) @ H4 )
@ ( topolo7230453075368039082e_nhds @ A @ D5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_def
thf(fact_6574_DERIV__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,X: A] :
( ( has_field_derivative @ A @ F3 @ D5 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A
@ ^ [H4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X @ H4 ) ) @ ( F3 @ X ) ) @ H4 )
@ ( topolo7230453075368039082e_nhds @ A @ D5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_D
thf(fact_6575_lim__exp__minus__1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( filterlim @ A @ A
@ ^ [Z2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z2 ) @ ( one_one @ A ) ) @ Z2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% lim_exp_minus_1
thf(fact_6576_greaterThanLessThan__upt,axiom,
( ( set_or5935395276787703475ssThan @ nat )
= ( ^ [N6: nat,M5: nat] : ( set2 @ nat @ ( upt @ ( suc @ N6 ) @ M5 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_6577_lemma__termdiff4,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [K: real,F3: A > B,K4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ! [H: A] :
( ( H
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ H ) ) @ ( times_times @ real @ K4 @ ( real_V7770717601297561774m_norm @ A @ H ) ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% lemma_termdiff4
thf(fact_6578_field__has__derivative__at,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,X: A] :
( ( has_derivative @ A @ A @ F3 @ ( times_times @ A @ D5 ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X @ H4 ) ) @ ( F3 @ X ) ) @ H4 )
@ ( topolo7230453075368039082e_nhds @ A @ D5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% field_has_derivative_at
thf(fact_6579_filterlim__at__to__0,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: A > B,F5: filter @ B,A2: A] :
( ( filterlim @ A @ B @ F3 @ F5 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [X4: A] : ( F3 @ ( plus_plus @ A @ X4 @ A2 ) )
@ F5
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_at_to_0
thf(fact_6580_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or7035219750837199246ssThan @ code_integer @ ( plus_plus @ code_integer @ L @ ( one_one @ code_integer ) ) @ U )
= ( set_or5935395276787703475ssThan @ code_integer @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_integer
thf(fact_6581_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_6582_powser__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A2: nat > A,F3: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ S )
=> ( sums @ A
@ ^ [N6: nat] : ( times_times @ A @ ( A2 @ N6 ) @ ( power_power @ A @ X3 @ N6 ) )
@ ( F3 @ X3 ) ) )
=> ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( A2 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0
thf(fact_6583_powser__limit__0__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A2: nat > A,F3: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X3: A] :
( ( X3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ S )
=> ( sums @ A
@ ^ [N6: nat] : ( times_times @ A @ ( A2 @ N6 ) @ ( power_power @ A @ X3 @ N6 ) )
@ ( F3 @ X3 ) ) ) )
=> ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( A2 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0_strong
thf(fact_6584_lemma__termdiff5,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_Vector_banach @ B ) )
=> ! [K: real,F3: nat > real,G3: A > nat > B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ( summable @ real @ F3 )
=> ( ! [H: A,N2: nat] :
( ( H
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( G3 @ H @ N2 ) ) @ ( times_times @ real @ ( F3 @ N2 ) @ ( real_V7770717601297561774m_norm @ A @ H ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H4: A] : ( suminf @ B @ ( G3 @ H4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% lemma_termdiff5
thf(fact_6585_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_6586_divmod__integer__def,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L4: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ K3 @ L4 ) @ ( modulo_modulo @ code_integer @ K3 @ L4 ) ) ) ) ).
% divmod_integer_def
thf(fact_6587_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
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_6588_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
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) @ ( one_one @ nat ) ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_6589_tendsto__zero__mult__left__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,A2: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N6: nat] : ( times_times @ A @ C3 @ ( A2 @ N6 ) )
@ ( 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_6590_tendsto__zero__mult__right__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,A2: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N6: nat] : ( times_times @ A @ ( A2 @ N6 ) @ C3 )
@ ( 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_6591_tendsto__zero__divide__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,A2: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N6: nat] : ( divide_divide @ A @ ( A2 @ N6 ) @ C3 )
@ ( 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_6592_approx__from__above__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [U5: nat > A] :
( ! [N9: nat] : ( ord_less @ A @ X @ ( U5 @ N9 ) )
& ( filterlim @ nat @ A @ U5 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_above_dense_linorder
thf(fact_6593_approx__from__below__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ? [U5: nat > A] :
( ! [N9: nat] : ( ord_less @ A @ ( U5 @ N9 ) @ X )
& ( filterlim @ nat @ A @ U5 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_below_dense_linorder
thf(fact_6594_LIMSEQ__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,L: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N6: nat] : ( F3 @ ( suc @ N6 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_Suc
thf(fact_6595_LIMSEQ__imp__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,L: A] :
( ( filterlim @ nat @ A
@ ^ [N6: nat] : ( F3 @ ( suc @ N6 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_imp_Suc
thf(fact_6596_filterlim__Suc,axiom,
filterlim @ nat @ nat @ suc @ ( at_top @ nat ) @ ( at_top @ nat ) ).
% filterlim_Suc
thf(fact_6597_filterlim__sequentially__Suc,axiom,
! [A: $tType,F3: nat > A,F5: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X4: nat] : ( F3 @ ( suc @ X4 ) )
@ F5
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F3 @ F5 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_6598_LIMSEQ__le__const2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X6: nat > A,X: A,A2: A] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ? [N10: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N10 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ N2 ) @ A2 ) )
=> ( ord_less_eq @ A @ X @ A2 ) ) ) ) ).
% LIMSEQ_le_const2
thf(fact_6599_LIMSEQ__le__const,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X6: nat > A,X: A,A2: A] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ? [N10: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N10 @ N2 )
=> ( ord_less_eq @ A @ A2 @ ( X6 @ N2 ) ) )
=> ( ord_less_eq @ A @ A2 @ X ) ) ) ) ).
% LIMSEQ_le_const
thf(fact_6600_Lim__bounded2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: nat > A,L: A,N8: nat,C4: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N8 @ N2 )
=> ( ord_less_eq @ A @ C4 @ ( F3 @ N2 ) ) )
=> ( ord_less_eq @ A @ C4 @ L ) ) ) ) ).
% Lim_bounded2
thf(fact_6601_Lim__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: nat > A,L: A,M7: nat,C4: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ M7 @ N2 )
=> ( ord_less_eq @ A @ ( F3 @ N2 ) @ C4 ) )
=> ( ord_less_eq @ A @ L @ C4 ) ) ) ) ).
% Lim_bounded
thf(fact_6602_LIMSEQ__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X6: nat > A,X: A,Y7: nat > A,Y: A] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y7 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ? [N10: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N10 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( Y7 @ N2 ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ).
% LIMSEQ_le
thf(fact_6603_lim__mono,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [N8: nat,X6: nat > A,Y7: nat > A,X: A,Y: A] :
( ! [N2: nat] :
( ( ord_less_eq @ nat @ N8 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( Y7 @ N2 ) ) )
=> ( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y7 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ).
% lim_mono
thf(fact_6604_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_6605_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_6606_Inf__as__limit,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ? [U5: nat > A] :
( ! [N9: nat] : ( member @ A @ ( U5 @ N9 ) @ A5 )
& ( filterlim @ nat @ A @ U5 @ ( topolo7230453075368039082e_nhds @ A @ ( complete_Inf_Inf @ A @ A5 ) ) @ ( at_top @ nat ) ) ) ) ) ).
% Inf_as_limit
thf(fact_6607_summable__LIMSEQ__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ_zero
thf(fact_6608_mult__nat__left__at__top,axiom,
! [C3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C3 )
=> ( filterlim @ nat @ nat @ ( times_times @ nat @ C3 ) @ ( at_top @ nat ) @ ( at_top @ nat ) ) ) ).
% mult_nat_left_at_top
thf(fact_6609_mult__nat__right__at__top,axiom,
! [C3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C3 )
=> ( filterlim @ nat @ nat
@ ^ [X4: nat] : ( times_times @ nat @ X4 @ C3 )
@ ( at_top @ nat )
@ ( at_top @ nat ) ) ) ).
% mult_nat_right_at_top
thf(fact_6610_monoseq__convergent,axiom,
! [X6: nat > real,B5: real] :
( ( topological_monoseq @ real @ X6 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( X6 @ I2 ) ) @ B5 )
=> ~ ! [L8: real] :
~ ( filterlim @ nat @ real @ X6 @ ( topolo7230453075368039082e_nhds @ real @ L8 ) @ ( at_top @ nat ) ) ) ) ).
% monoseq_convergent
thf(fact_6611_monoseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: nat > A,X: A] :
( ( topological_monoseq @ A @ A2 )
=> ( ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ! [N9: nat] : ( ord_less_eq @ A @ ( A2 @ N9 ) @ X )
& ! [M2: nat,N9: nat] :
( ( ord_less_eq @ nat @ M2 @ N9 )
=> ( ord_less_eq @ A @ ( A2 @ M2 ) @ ( A2 @ N9 ) ) ) )
| ( ! [N9: nat] : ( ord_less_eq @ A @ X @ ( A2 @ N9 ) )
& ! [M2: nat,N9: nat] :
( ( ord_less_eq @ nat @ M2 @ N9 )
=> ( ord_less_eq @ A @ ( A2 @ N9 ) @ ( A2 @ M2 ) ) ) ) ) ) ) ) ).
% monoseq_le
thf(fact_6612_lim__const__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: A] :
( filterlim @ nat @ A
@ ^ [N6: nat] : ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ N6 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_const_over_n
thf(fact_6613_LIMSEQ__linear,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X6: nat > A,X: A,L: nat] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L )
=> ( filterlim @ nat @ A
@ ^ [N6: nat] : ( X6 @ ( times_times @ nat @ N6 @ L ) )
@ ( topolo7230453075368039082e_nhds @ A @ X )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_linear
thf(fact_6614_lim__inverse__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N6: nat] : ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N6 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_inverse_n
thf(fact_6615_telescope__summable_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C3: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N6: nat] : ( minus_minus @ A @ ( F3 @ N6 ) @ ( F3 @ ( suc @ N6 ) ) ) ) ) ) ).
% telescope_summable'
thf(fact_6616_telescope__summable,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C3: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N6: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N6 ) ) @ ( F3 @ N6 ) ) ) ) ) ).
% telescope_summable
thf(fact_6617_nested__sequence__unique,axiom,
! [F3: nat > real,G3: nat > real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( G3 @ ( suc @ N2 ) ) @ ( G3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( F3 @ N2 ) @ ( G3 @ N2 ) )
=> ( ( filterlim @ nat @ real
@ ^ [N6: nat] : ( minus_minus @ real @ ( F3 @ N6 ) @ ( G3 @ N6 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) )
=> ? [L2: real] :
( ! [N9: nat] : ( ord_less_eq @ real @ ( F3 @ N9 ) @ L2 )
& ( filterlim @ nat @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L2 ) @ ( at_top @ nat ) )
& ! [N9: nat] : ( ord_less_eq @ real @ L2 @ ( G3 @ N9 ) )
& ( filterlim @ nat @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ L2 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_6618_LIMSEQ__inverse__zero,axiom,
! [X6: nat > real] :
( ! [R: real] :
? [N10: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N10 @ N2 )
=> ( ord_less @ real @ R @ ( X6 @ N2 ) ) )
=> ( filterlim @ nat @ real
@ ^ [N6: nat] : ( inverse_inverse @ real @ ( X6 @ N6 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_zero
thf(fact_6619_LIMSEQ__inverse__real__of__nat,axiom,
( filterlim @ nat @ real
@ ^ [N6: nat] : ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat
thf(fact_6620_LIMSEQ__root__const,axiom,
! [C3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( filterlim @ nat @ real
@ ^ [N6: nat] : ( root @ N6 @ C3 )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_root_const
thf(fact_6621_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N6: nat] : ( plus_plus @ real @ R2 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_6622_increasing__LIMSEQ,axiom,
! [F3: nat > real,L: real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( F3 @ N2 ) @ L )
=> ( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [N9: nat] : ( ord_less_eq @ real @ L @ ( plus_plus @ real @ ( F3 @ N9 ) @ E2 ) ) )
=> ( filterlim @ nat @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( at_top @ nat ) ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_6623_lim__1__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N6: nat] : ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N6 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_1_over_n
thf(fact_6624_LIMSEQ__n__over__Suc__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N6: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N6 ) @ ( semiring_1_of_nat @ A @ ( suc @ N6 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_n_over_Suc_n
thf(fact_6625_LIMSEQ__Suc__n__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N6: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ N6 ) ) @ ( semiring_1_of_nat @ A @ N6 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_Suc_n_over_n
thf(fact_6626_telescope__sums,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C3: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N6: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N6 ) ) @ ( F3 @ N6 ) )
@ ( minus_minus @ A @ C3 @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% telescope_sums
thf(fact_6627_telescope__sums_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C3: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N6: nat] : ( minus_minus @ A @ ( F3 @ N6 ) @ ( F3 @ ( suc @ N6 ) ) )
@ ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ C3 ) ) ) ) ).
% telescope_sums'
thf(fact_6628_LIMSEQ__realpow__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ X ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_realpow_zero
thf(fact_6629_LIMSEQ__divide__realpow__zero,axiom,
! [X: real,A2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( filterlim @ nat @ real
@ ^ [N6: nat] : ( divide_divide @ real @ A2 @ ( power_power @ real @ X @ N6 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_6630_LIMSEQ__abs__realpow__zero,axiom,
! [C3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ C3 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ ( abs_abs @ real @ C3 ) ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ).
% LIMSEQ_abs_realpow_zero
thf(fact_6631_LIMSEQ__abs__realpow__zero2,axiom,
! [C3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ C3 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ C3 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ).
% LIMSEQ_abs_realpow_zero2
thf(fact_6632_LIMSEQ__inverse__realpow__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( filterlim @ nat @ real
@ ^ [N6: nat] : ( inverse_inverse @ real @ ( power_power @ real @ X @ N6 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_6633_sums__def_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F4: nat > A,S8: A] :
( filterlim @ nat @ A
@ ^ [N6: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N6 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S8 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def'
thf(fact_6634_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N6: nat] : ( plus_plus @ real @ R2 @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_6635_root__test__convergence,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,X: real] :
( ( filterlim @ nat @ real
@ ^ [N6: nat] : ( root @ N6 @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N6 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ nat ) )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% root_test_convergence
thf(fact_6636_LIMSEQ__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No2: nat] :
! [N6: nat] :
( ( ord_less_eq @ nat @ No2 @ N6 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ N6 ) @ L5 ) ) @ R5 ) ) ) ) ) ) ).
% LIMSEQ_iff
thf(fact_6637_LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A,L5: A] :
( ! [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [No3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No3 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ N2 ) @ L5 ) ) @ R ) ) )
=> ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_I
thf(fact_6638_LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A,L5: A,R2: real] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ No @ N9 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ N9 ) @ L5 ) ) @ R2 ) ) ) ) ) ).
% LIMSEQ_D
thf(fact_6639_LIMSEQ__power__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X ) @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_power_zero
thf(fact_6640_tendsto__power__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F3: B > nat,F5: filter @ B,X: A] :
( ( filterlim @ B @ nat @ F3 @ ( at_top @ nat ) @ F5 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ B @ A
@ ^ [Y5: B] : ( power_power @ A @ X @ ( F3 @ Y5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ) ).
% tendsto_power_zero
thf(fact_6641_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N6: nat] : ( times_times @ real @ R2 @ ( plus_plus @ real @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_6642_LIMSEQ__norm__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ! [N2: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_norm_0
thf(fact_6643_field__derivative__lim__unique,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Df: A,Z: A,S: nat > A,A2: A] :
( ( has_field_derivative @ A @ F3 @ 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
@ ^ [N6: nat] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ Z @ ( S @ N6 ) ) ) @ ( F3 @ Z ) ) @ ( S @ N6 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) )
=> ( Df = A2 ) ) ) ) ) ) ).
% field_derivative_lim_unique
thf(fact_6644_powser__times__n__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A
@ ^ [N6: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ N6 ) @ ( power_power @ A @ X @ N6 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% powser_times_n_limit_0
thf(fact_6645_lim__n__over__pown,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( filterlim @ nat @ A
@ ^ [N6: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N6 ) @ ( power_power @ A @ X @ N6 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% lim_n_over_pown
thf(fact_6646_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
@ ^ [N6: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N6 ) @ ( A2 @ N6 ) ) ) ) ) ) ).
% summable
thf(fact_6647_zeroseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real
@ ^ [N6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% zeroseq_arctan_series
thf(fact_6648_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
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_6649_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
@ ^ [N6: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(3)
thf(fact_6650_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 ) )
=> ? [L2: real] :
( ! [N9: nat] :
( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) ) )
@ L2 )
& ( filterlim @ nat @ real
@ ^ [N6: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L2 )
@ ( at_top @ nat ) )
& ! [N9: nat] :
( ord_less_eq @ real @ L2
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) @ ( one_one @ nat ) ) ) ) )
& ( filterlim @ nat @ real
@ ^ [N6: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L2 )
@ ( at_top @ nat ) ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_6651_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
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_6652_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
@ ^ [N6: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_6653_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
@ ^ [L4: code_integer,J3: code_integer] :
( if @ num
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ num @ ( code_num_of_integer @ L4 ) @ ( code_num_of_integer @ L4 ) )
@ ( plus_plus @ num @ ( plus_plus @ num @ ( code_num_of_integer @ L4 ) @ ( code_num_of_integer @ L4 ) ) @ one2 ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_6654_has__derivative__at2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F8: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ( filterlim @ A @ B
@ ^ [Y5: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y5 @ X ) ) ) @ ( minus_minus @ B @ ( F3 @ Y5 ) @ ( plus_plus @ B @ ( F3 @ X ) @ ( F8 @ ( minus_minus @ A @ Y5 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at2
thf(fact_6655_bounded__linear__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X4: A] : ( zero_zero @ B ) ) ) ).
% bounded_linear_zero
thf(fact_6656_bounded__linear_Obounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ? [K8: real] :
! [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K8 ) ) ) ) ).
% bounded_linear.bounded
thf(fact_6657_bounded__linear_Otendsto__zero,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G3: C > A,F5: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( filterlim @ C @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ C @ B
@ ^ [X4: C] : ( F3 @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ) ).
% bounded_linear.tendsto_zero
thf(fact_6658_bounded__linear_Ononneg__bounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ? [K8: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ K8 )
& ! [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K8 ) ) ) ) ) ).
% bounded_linear.nonneg_bounded
thf(fact_6659_has__derivative__within__singleton__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G3: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ G3 @ ( topolo174197925503356063within @ A @ X @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( real_V3181309239436604168linear @ A @ B @ G3 ) ) ) ).
% has_derivative_within_singleton_iff
thf(fact_6660_bounded__linear_Opos__bounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ? [K8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K8 )
& ! [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K8 ) ) ) ) ) ).
% bounded_linear.pos_bounded
thf(fact_6661_bounded__linear__intro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,K4: real] :
( ! [X3: A,Y3: A] :
( ( F3 @ ( plus_plus @ A @ X3 @ Y3 ) )
= ( plus_plus @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ! [R: real,X3: A] :
( ( F3 @ ( real_V8093663219630862766scaleR @ A @ R @ X3 ) )
= ( real_V8093663219630862766scaleR @ B @ R @ ( F3 @ X3 ) ) )
=> ( ! [X3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X3 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K4 ) )
=> ( real_V3181309239436604168linear @ A @ B @ F3 ) ) ) ) ) ).
% bounded_linear_intro
thf(fact_6662_has__derivativeI,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F8: A > B,X: A,F3: A > B,S: set @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F8 )
=> ( ( filterlim @ A @ B
@ ^ [Y5: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y5 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y5 ) @ ( F3 @ X ) ) @ ( F8 @ ( minus_minus @ A @ Y5 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ B @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivativeI
thf(fact_6663_has__derivative__at__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F8: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ( filterlim @ A @ B
@ ^ [Y5: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y5 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y5 ) @ ( F3 @ X ) ) @ ( F8 @ ( minus_minus @ A @ Y5 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_at_within
thf(fact_6664_has__derivative__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F8: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ? [E4: A > B] :
( ! [H4: A] :
( ( F3 @ ( plus_plus @ A @ X @ H4 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F3 @ X ) @ ( F8 @ H4 ) ) @ ( E4 @ H4 ) ) )
& ( filterlim @ A @ real
@ ^ [H4: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E4 @ H4 ) ) @ ( real_V7770717601297561774m_norm @ A @ H4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% has_derivative_iff_Ex
thf(fact_6665_has__derivative__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F8: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ( filterlim @ A @ B
@ ^ [Y5: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y5 @ X ) ) ) @ ( minus_minus @ B @ ( F3 @ Y5 ) @ ( plus_plus @ B @ ( F3 @ X ) @ ( F8 @ ( minus_minus @ A @ Y5 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_within
thf(fact_6666_has__derivative__at,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,D5: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ D5 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ D5 )
& ( filterlim @ A @ real
@ ^ [H4: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ ( plus_plus @ A @ X @ H4 ) ) @ ( F3 @ X ) ) @ ( D5 @ H4 ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at
thf(fact_6667_has__derivative__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( has_derivative @ A @ B )
= ( ^ [F4: A > B,F17: A > B,F11: filter @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F17 )
& ( filterlim @ A @ B
@ ^ [Y5: A] :
( real_V8093663219630862766scaleR @ B
@ ( inverse_inverse @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( minus_minus @ A @ Y5
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F11
@ ^ [X4: A] : X4 ) ) ) )
@ ( minus_minus @ B
@ ( minus_minus @ B @ ( F4 @ Y5 )
@ ( F4
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F11
@ ^ [X4: A] : X4 ) ) )
@ ( F17
@ ( minus_minus @ A @ Y5
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F11
@ ^ [X4: A] : X4 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F11 ) ) ) ) ) ).
% has_derivative_def
thf(fact_6668_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
@ ^ [L4: code_integer,J3: code_integer] :
( if @ nat
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ nat @ ( code_nat_of_integer @ L4 ) @ ( code_nat_of_integer @ L4 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( code_nat_of_integer @ L4 ) @ ( code_nat_of_integer @ L4 ) ) @ ( one_one @ nat ) ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_6669_nat__of__integer__of__nat,axiom,
! [N: nat] :
( ( code_nat_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
= N ) ).
% nat_of_integer_of_nat
thf(fact_6670_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_6671_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_6672_nat__of__integer_Orep__eq,axiom,
( code_nat_of_integer
= ( ^ [X4: code_integer] : ( nat2 @ ( code_int_of_integer @ X4 ) ) ) ) ).
% nat_of_integer.rep_eq
thf(fact_6673_nat__of__integer_Oabs__eq,axiom,
! [X: int] :
( ( code_nat_of_integer @ ( code_integer_of_int @ X ) )
= ( nat2 @ X ) ) ).
% nat_of_integer.abs_eq
thf(fact_6674_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_6675_nat__of__integer__less__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ X )
=> ( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ Y )
=> ( ( ord_less @ nat @ ( code_nat_of_integer @ X ) @ ( code_nat_of_integer @ Y ) )
= ( ord_less @ code_integer @ X @ Y ) ) ) ) ).
% nat_of_integer_less_iff
thf(fact_6676_image__atLeastZeroLessThan__integer,axiom,
! [U: code_integer] :
( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ U )
=> ( ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ U )
= ( image @ nat @ code_integer @ ( semiring_1_of_nat @ code_integer ) @ ( set_ord_lessThan @ nat @ ( code_nat_of_integer @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_integer
thf(fact_6677_has__derivative__at__within__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [X: A,S3: set @ A,F3: A > B,F8: A > B] :
( ( member @ A @ X @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( has_derivative @ A @ B @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ? [E4: A > B] :
( ! [H4: A] :
( ( member @ A @ ( plus_plus @ A @ X @ H4 ) @ S3 )
=> ( ( F3 @ ( plus_plus @ A @ X @ H4 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F3 @ X ) @ ( F8 @ H4 ) ) @ ( E4 @ H4 ) ) ) )
& ( filterlim @ A @ real
@ ^ [H4: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E4 @ H4 ) ) @ ( real_V7770717601297561774m_norm @ A @ H4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% has_derivative_at_within_iff_Ex
thf(fact_6678_has__derivativeI__sandwich,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [E3: real,F8: A > B,S: set @ A,X: A,F3: A > B,H8: A > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ( ( real_V3181309239436604168linear @ A @ B @ F8 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ S )
=> ( ( Y3 != X )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y3 @ X ) @ E3 )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y3 ) @ ( F3 @ X ) ) @ ( F8 @ ( minus_minus @ A @ Y3 @ X ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y3 @ X ) ) ) @ ( H8 @ Y3 ) ) ) ) )
=> ( ( filterlim @ A @ real @ H8 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ B @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ) ).
% has_derivativeI_sandwich
thf(fact_6679_open__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo1002775350975398744n_open @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% open_empty
thf(fact_6680_dist__0__norm,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( real_V557655796197034286t_dist @ A @ ( zero_zero @ A ) @ X )
= ( real_V7770717601297561774m_norm @ A @ X ) ) ) ).
% dist_0_norm
thf(fact_6681_zero__less__dist__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X @ Y ) )
= ( X != Y ) ) ) ).
% zero_less_dist_iff
thf(fact_6682_dist__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( zero_zero @ real ) )
= ( X = Y ) ) ) ).
% dist_le_zero_iff
thf(fact_6683_open__INT,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A5: set @ B,B5: B > ( set @ A )] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( topolo1002775350975398744n_open @ A @ ( B5 @ X3 ) ) )
=> ( topolo1002775350975398744n_open @ A @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A5 ) ) ) ) ) ) ).
% open_INT
thf(fact_6684_not__open__singleton,axiom,
! [A: $tType] :
( ( topolo8386298272705272623_space @ A )
=> ! [X: A] :
~ ( topolo1002775350975398744n_open @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% not_open_singleton
thf(fact_6685_hausdorff,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ? [U6: set @ A,V5: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U6 )
& ( topolo1002775350975398744n_open @ A @ V5 )
& ( member @ A @ X @ U6 )
& ( member @ A @ Y @ V5 )
& ( ( inf_inf @ ( set @ A ) @ U6 @ V5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% hausdorff
thf(fact_6686_separation__t2,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ? [U7: set @ A,V6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U7 )
& ( topolo1002775350975398744n_open @ A @ V6 )
& ( member @ A @ X @ U7 )
& ( member @ A @ Y @ V6 )
& ( ( inf_inf @ ( set @ A ) @ U7 @ V6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% separation_t2
thf(fact_6687_dist__triangle,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Z: A,Y: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z ) ) ) ) ).
% dist_triangle
thf(fact_6688_dist__triangle2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A,Z: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z ) ) ) ) ).
% dist_triangle2
thf(fact_6689_dist__triangle3,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A,A2: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ A2 @ X ) @ ( real_V557655796197034286t_dist @ A @ A2 @ Y ) ) ) ) ).
% dist_triangle3
thf(fact_6690_dist__triangle__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Z: A,Y: A,E3: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z ) ) @ E3 )
=> ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E3 ) ) ) ).
% dist_triangle_le
thf(fact_6691_dist__triangle__less__add,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X1: A,Y: A,E1: real,X2: A,E22: real] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ Y ) @ E1 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y ) @ E22 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X2 ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% dist_triangle_less_add
thf(fact_6692_dist__triangle__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Z: A,Y: A,E3: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z ) ) @ E3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E3 ) ) ) ).
% dist_triangle_lt
thf(fact_6693_Sup__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A5: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less @ A @ X3 @ X ) )
=> ~ ( member @ A @ ( complete_Sup_Sup @ A @ A5 ) @ A5 ) ) ) ) ).
% Sup_notin_open
thf(fact_6694_Inf__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A5: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less @ A @ X @ X3 ) )
=> ~ ( member @ A @ ( complete_Inf_Inf @ A @ A5 ) @ A5 ) ) ) ) ).
% Inf_notin_open
thf(fact_6695_norm__conv__dist,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A )
= ( ^ [X4: A] : ( real_V557655796197034286t_dist @ A @ X4 @ ( zero_zero @ A ) ) ) ) ) ).
% norm_conv_dist
thf(fact_6696_dist__commute__lessI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y: A,X: A,E3: real] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X ) @ E3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E3 ) ) ) ).
% dist_commute_lessI
thf(fact_6697_open__ball,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,D2: real] :
( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [Y5: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y5 ) @ D2 ) ) ) ) ).
% open_ball
thf(fact_6698_open__dist,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ S6 )
=> ? [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
& ! [Y5: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y5 @ X4 ) @ E4 )
=> ( member @ A @ Y5 @ S6 ) ) ) ) ) ) ) ).
% open_dist
thf(fact_6699_dist__not__less__zero,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( zero_zero @ real ) ) ) ).
% dist_not_less_zero
thf(fact_6700_dist__pos__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X @ Y ) ) ) ) ).
% dist_pos_lt
thf(fact_6701_zero__le__dist,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X @ Y ) ) ) ).
% zero_le_dist
thf(fact_6702_openI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ? [T10: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T10 )
& ( member @ A @ X3 @ T10 )
& ( ord_less_eq @ ( set @ A ) @ T10 @ S3 ) ) )
=> ( topolo1002775350975398744n_open @ A @ S3 ) ) ) ).
% openI
thf(fact_6703_open__subopen,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ S6 )
=> ? [T11: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T11 )
& ( member @ A @ X4 @ T11 )
& ( ord_less_eq @ ( set @ A ) @ T11 @ S6 ) ) ) ) ) ) ).
% open_subopen
thf(fact_6704_first__countable__basis,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X: A] :
? [A7: nat > ( set @ A )] :
( ! [I4: nat] :
( ( member @ A @ X @ ( A7 @ I4 ) )
& ( topolo1002775350975398744n_open @ A @ ( A7 @ I4 ) ) )
& ! [S10: set @ A] :
( ( ( topolo1002775350975398744n_open @ A @ S10 )
& ( member @ A @ X @ S10 ) )
=> ? [I2: nat] : ( ord_less_eq @ ( set @ A ) @ ( A7 @ I2 ) @ S10 ) ) ) ) ).
% first_countable_basis
thf(fact_6705_at__within__open__subset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: A,S3: set @ A,T4: set @ A] :
( ( member @ A @ A2 @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ T4 )
=> ( ( topolo174197925503356063within @ A @ A2 @ T4 )
= ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% at_within_open_subset
thf(fact_6706_open__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A,X: A,Y: A] :
( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( member @ A @ X @ S3 )
=> ( ( ord_less @ A @ X @ Y )
=> ? [B4: A] :
( ( ord_less @ A @ X @ B4 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ B4 ) @ S3 ) ) ) ) ) ) ).
% open_right
thf(fact_6707_abs__dist__diff__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [A2: A,B2: A,C3: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( real_V557655796197034286t_dist @ A @ A2 @ B2 ) @ ( real_V557655796197034286t_dist @ A @ B2 @ C3 ) ) ) @ ( real_V557655796197034286t_dist @ A @ A2 @ C3 ) ) ) ).
% abs_dist_diff_le
thf(fact_6708_has__field__derivative__transform__within,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,F8: A,A2: A,S3: set @ A,D2: real,G3: A > A] :
( ( has_field_derivative @ A @ F3 @ F8 @ ( topolo174197925503356063within @ A @ A2 @ S3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ( member @ A @ A2 @ S3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ D2 )
=> ( ( F3 @ X3 )
= ( G3 @ X3 ) ) ) )
=> ( has_field_derivative @ A @ G3 @ F8 @ ( topolo174197925503356063within @ A @ A2 @ S3 ) ) ) ) ) ) ) ).
% has_field_derivative_transform_within
thf(fact_6709_has__derivative__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F8: A > B,X: A,S: set @ A,D2: real,G3: A > B] :
( ( has_derivative @ A @ B @ F3 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ( member @ A @ X @ S )
=> ( ! [X18: A] :
( ( member @ A @ X18 @ S )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X18 @ X ) @ D2 )
=> ( ( F3 @ X18 )
= ( G3 @ X18 ) ) ) )
=> ( has_derivative @ A @ B @ G3 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ) ).
% has_derivative_transform_within
thf(fact_6710_metric__CauchyI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M12: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M12 @ M3 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M12 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X6 ) ) ) ).
% metric_CauchyI
thf(fact_6711_metric__CauchyD,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A,E3: real] :
( ( topolo3814608138187158403Cauchy @ A @ X6 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M11: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M11 @ M2 )
=> ! [N9: nat] :
( ( ord_less_eq @ nat @ M11 @ N9 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M2 ) @ ( X6 @ N9 ) ) @ E3 ) ) ) ) ) ) ).
% metric_CauchyD
thf(fact_6712_Cauchy__altdef2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [S8: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N7: nat] :
! [N6: nat] :
( ( ord_less_eq @ nat @ N7 @ N6 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( S8 @ N6 ) @ ( S8 @ N7 ) ) @ E4 ) ) ) ) ) ) ).
% Cauchy_altdef2
thf(fact_6713_Cauchy__def,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M10: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M10 @ M5 )
=> ! [N6: nat] :
( ( ord_less_eq @ nat @ M10 @ N6 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M5 ) @ ( X8 @ N6 ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_def
thf(fact_6714_lim__explicit,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,F0: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ F0 ) @ ( at_top @ nat ) )
= ( ! [S6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S6 )
=> ( ( member @ A @ F0 @ S6 )
=> ? [N7: nat] :
! [N6: nat] :
( ( ord_less_eq @ nat @ N7 @ N6 )
=> ( member @ A @ ( F3 @ N6 ) @ S6 ) ) ) ) ) ) ) ).
% lim_explicit
thf(fact_6715_dist__triangle__half__r,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y: A,X1: A,E3: real,X2: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X1 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X2 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X2 ) @ E3 ) ) ) ) ).
% dist_triangle_half_r
thf(fact_6716_dist__triangle__half__l,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X1: A,Y: A,E3: real,X2: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ Y ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X2 ) @ E3 ) ) ) ) ).
% dist_triangle_half_l
thf(fact_6717_metric__LIM__imp__LIM,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V7819770556892013058_space @ B )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F3: C > A,L: A,A2: C,G3: C > B,M: B] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) )
=> ( ! [X3: C] :
( ( X3 != A2 )
=> ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( G3 @ X3 ) @ M ) @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X3 ) @ L ) ) )
=> ( filterlim @ C @ B @ G3 @ ( topolo7230453075368039082e_nhds @ B @ M ) @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) ) ) ) ) ).
% metric_LIM_imp_LIM
thf(fact_6718_dist__triangle__third,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X1: A,X2: A,E3: real,X33: A,X43: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X2 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ X33 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X33 @ X43 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X43 ) @ E3 ) ) ) ) ) ).
% dist_triangle_third
thf(fact_6719_Lim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,L: B,X: A,S3: set @ A,D2: real,G3: A > B] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [X18: A] :
( ( member @ A @ X18 @ S3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X18 @ X ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X18 @ X ) @ D2 )
=> ( ( F3 @ X18 )
= ( G3 @ X18 ) ) ) ) )
=> ( filterlim @ A @ B @ G3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ) ).
% Lim_transform_within
thf(fact_6720_at__within__nhd,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X: A,S3: set @ A,T4: set @ A,U2: set @ A] :
( ( member @ A @ X @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ T4 @ S3 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ U2 @ S3 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( topolo174197925503356063within @ A @ X @ T4 )
= ( topolo174197925503356063within @ A @ X @ U2 ) ) ) ) ) ) ).
% at_within_nhd
thf(fact_6721_filterlim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [G3: A > B,G6: filter @ B,X: A,S3: set @ A,F5: filter @ B,D2: real,F3: A > B] :
( ( filterlim @ A @ B @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ord_less_eq @ ( filter @ B ) @ G6 @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [X18: A] :
( ( member @ A @ X18 @ S3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X18 @ X ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X18 @ X ) @ D2 )
=> ( ( F3 @ X18 )
= ( G3 @ X18 ) ) ) ) )
=> ( filterlim @ A @ B @ F3 @ F5 @ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ) ) ).
% filterlim_transform_within
thf(fact_6722_CauchyI_H,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M12: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M12 @ M3 )
=> ! [N2: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X6 ) ) ) ).
% CauchyI'
thf(fact_6723_Cauchy__altdef,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [F4: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M10: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M10 @ M5 )
=> ! [N6: nat] :
( ( ord_less @ nat @ M5 @ N6 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F4 @ M5 ) @ ( F4 @ N6 ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_altdef
thf(fact_6724_at__eq__bot__iff,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: A] :
( ( ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( filter @ A ) ) )
= ( topolo1002775350975398744n_open @ A @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% at_eq_bot_iff
thf(fact_6725_metric__LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [G3: A > B,L: B,A2: A,R3: real,F3: A > B] :
( ( filterlim @ A @ B @ G3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ( ! [X3: A] :
( ( X3 != A2 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ R3 )
=> ( ( F3 @ X3 )
= ( G3 @ X3 ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_equal2
thf(fact_6726_metric__LIM__I,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [A2: A,F3: A > B,L5: B] :
( ! [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [S9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S9 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ S9 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X3 ) @ L5 ) @ R ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% metric_LIM_I
thf(fact_6727_metric__LIM__D,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F3: A > B,L5: B,A2: A,R2: real] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [S4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S4 )
& ! [X5: A] :
( ( ( X5 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ A2 ) @ S4 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X5 ) @ L5 ) @ R2 ) ) ) ) ) ) ).
% metric_LIM_D
thf(fact_6728_LIM__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F3: A > B,L5: B,A2: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S8 )
& ! [X4: A] :
( ( ( X4 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ A2 ) @ S8 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X4 ) @ L5 ) @ R5 ) ) ) ) ) ) ) ).
% LIM_def
thf(fact_6729_lim__sequentially,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No2: nat] :
! [N6: nat] :
( ( ord_less_eq @ nat @ No2 @ N6 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ N6 ) @ L5 ) @ R5 ) ) ) ) ) ) ).
% lim_sequentially
thf(fact_6730_metric__LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A,L5: A] :
( ! [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [No3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No3 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ N2 ) @ L5 ) @ R ) ) )
=> ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) ) ) ).
% metric_LIMSEQ_I
thf(fact_6731_metric__LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A,L5: A,R2: real] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ No @ N9 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ N9 ) @ L5 ) @ R2 ) ) ) ) ) ).
% metric_LIMSEQ_D
thf(fact_6732_metric__Cauchy__iff2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [J3: nat] :
? [M10: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M10 @ M5 )
=> ! [N6: nat] :
( ( ord_less_eq @ nat @ M10 @ N6 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M5 ) @ ( X8 @ N6 ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ) ).
% metric_Cauchy_iff2
thf(fact_6733_metric__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,B2: B,A2: A,G3: B > C,C3: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G3 @ ( topolo7230453075368039082e_nhds @ C @ C3 ) @ ( topolo174197925503356063within @ B @ B2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ D4 ) )
=> ( ( F3 @ X3 )
!= B2 ) ) )
=> ( filterlim @ A @ C
@ ^ [X4: A] : ( G3 @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C3 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_compose2
thf(fact_6734_tendsto__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A2: A,S3: set @ A,F3: A > D,L5: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 )
=> ( ( member @ A @ A2 @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( filterlim @ A @ D @ F3 @ ( topolo7230453075368039082e_nhds @ D @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ S3 ) )
= ( filterlim @ A @ D
@ ^ [H4: A] : ( F3 @ ( plus_plus @ A @ A2 @ H4 ) )
@ ( topolo7230453075368039082e_nhds @ D @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% tendsto_offset_zero_iff
thf(fact_6735_LIMSEQ__iff__nz,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ No2 )
& ! [N6: nat] :
( ( ord_less_eq @ nat @ No2 @ N6 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ N6 ) @ L5 ) @ R5 ) ) ) ) ) ) ) ).
% LIMSEQ_iff_nz
thf(fact_6736_totally__bounded__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo6688025880775521714ounded @ A )
= ( ^ [S6: set @ A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [K3: set @ A] :
( ( finite_finite2 @ A @ K3 )
& ( ord_less_eq @ ( set @ A ) @ S6
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image @ A @ ( set @ A )
@ ^ [X4: A] :
( collect @ A
@ ^ [Y5: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ Y5 ) @ E4 ) )
@ K3 ) ) ) ) ) ) ) ) ).
% totally_bounded_metric
thf(fact_6737_lim__zero__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,L: A] :
( ( filterlim @ A @ A
@ ^ [X4: A] : ( F3 @ ( divide_divide @ A @ ( one_one @ A ) @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) ) ) ) ).
% lim_zero_infinity
thf(fact_6738_totally__bounded__empty,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( topolo6688025880775521714ounded @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% totally_bounded_empty
thf(fact_6739_totally__bounded__subset,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [S3: set @ A,T4: set @ A] :
( ( topolo6688025880775521714ounded @ A @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ T4 @ S3 )
=> ( topolo6688025880775521714ounded @ A @ T4 ) ) ) ) ).
% totally_bounded_subset
thf(fact_6740_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_6741_tendsto__mult__filterlim__at__infinity,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: B > A,C3: A,F5: filter @ B,G3: B > A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F5 )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A @ G3 @ ( at_infinity @ A ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( at_infinity @ A )
@ F5 ) ) ) ) ) ).
% tendsto_mult_filterlim_at_infinity
thf(fact_6742_filterlim__power__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [F3: A > B,F5: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ F5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N )
@ ( at_infinity @ B )
@ F5 ) ) ) ) ).
% filterlim_power_at_infinity
thf(fact_6743_tendsto__divide__0,axiom,
! [A: $tType,C: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: C > A,C3: A,F5: filter @ C,G3: C > A] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F5 )
=> ( ( filterlim @ C @ A @ G3 @ ( at_infinity @ A ) @ F5 )
=> ( filterlim @ C @ A
@ ^ [X4: C] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ) ).
% tendsto_divide_0
thf(fact_6744_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_6745_filterlim__inverse__at__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [G3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X4: A] : ( inverse_inverse @ B @ ( G3 @ X4 ) )
@ ( topolo174197925503356063within @ B @ ( zero_zero @ B ) @ ( top_top @ ( set @ B ) ) )
@ F5 )
= ( filterlim @ A @ B @ G3 @ ( at_infinity @ B ) @ F5 ) ) ) ).
% filterlim_inverse_at_iff
thf(fact_6746_filterlim__divide__at__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,C3: A,F5: filter @ A,G3: A > A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F5 )
=> ( ( filterlim @ A @ A @ G3 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) @ F5 )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X4: A] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( at_infinity @ A )
@ F5 ) ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_6747_filterlim__realpow__sequentially__gt1,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X ) @ ( at_infinity @ A ) @ ( at_top @ nat ) ) ) ) ).
% filterlim_realpow_sequentially_gt1
thf(fact_6748_polyfun__extremal,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [C3: nat > A,K: nat,N: nat,B5: real] :
( ( ( C3 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( eventually @ A
@ ^ [Z2: A] :
( ord_less_eq @ real @ B5
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C3 @ I3 ) @ ( power_power @ A @ Z2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
@ ( at_infinity @ A ) ) ) ) ) ) ).
% polyfun_extremal
thf(fact_6749_filterlim__pow__at__bot__odd,axiom,
! [N: nat,F3: real > real,F5: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F3 @ ( at_bot @ real ) @ F5 )
=> ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( power_power @ real @ ( F3 @ X4 ) @ N )
@ ( at_bot @ real )
@ F5 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_6750_eventually__sequentially__Suc,axiom,
! [P2: nat > $o] :
( ( eventually @ nat
@ ^ [I3: nat] : ( P2 @ ( suc @ I3 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P2 @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_Suc
thf(fact_6751_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
! [A: $tType,F3: A > real,F5: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( at_infinity @ real ) @ F5 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ real @ ( F3 @ X4 ) @ ( zero_zero @ real ) )
@ F5 )
=> ( filterlim @ A @ real @ F3 @ ( at_bot @ real ) @ F5 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_bot
thf(fact_6752_at__bot__le__at__infinity,axiom,
ord_less_eq @ ( filter @ real ) @ ( at_bot @ real ) @ ( at_infinity @ real ) ).
% at_bot_le_at_infinity
thf(fact_6753_filterlim__inverse__at__bot,axiom,
! [A: $tType,F3: A > real,F5: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ real @ ( F3 @ X4 ) @ ( zero_zero @ real ) )
@ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( inverse_inverse @ real @ ( F3 @ X4 ) )
@ ( at_bot @ real )
@ F5 ) ) ) ).
% filterlim_inverse_at_bot
thf(fact_6754_eventually__sequentiallyI,axiom,
! [C3: nat,P2: nat > $o] :
( ! [X3: nat] :
( ( ord_less_eq @ nat @ C3 @ X3 )
=> ( P2 @ X3 ) )
=> ( eventually @ nat @ P2 @ ( at_top @ nat ) ) ) ).
% eventually_sequentiallyI
thf(fact_6755_eventually__sequentially,axiom,
! [P2: nat > $o] :
( ( eventually @ nat @ P2 @ ( at_top @ nat ) )
= ( ? [N7: nat] :
! [N6: nat] :
( ( ord_less_eq @ nat @ N7 @ N6 )
=> ( P2 @ N6 ) ) ) ) ).
% eventually_sequentially
thf(fact_6756_eventually__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o] :
( ( eventually @ A @ P2 @ ( at_top @ A ) )
= ( ? [N7: A] :
! [N6: A] :
( ( ord_less_eq @ A @ N7 @ N6 )
=> ( P2 @ N6 ) ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_6757_eventually__at__top__linorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,P2: A > $o] :
( ! [X3: A] :
( ( ord_less_eq @ A @ C3 @ X3 )
=> ( P2 @ X3 ) )
=> ( eventually @ A @ P2 @ ( at_top @ A ) ) ) ) ).
% eventually_at_top_linorderI
thf(fact_6758_eventually__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A] : ( eventually @ A @ ( ord_less_eq @ A @ C3 ) @ ( at_top @ A ) ) ) ).
% eventually_ge_at_top
thf(fact_6759_le__sequentially,axiom,
! [F5: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F5 @ ( at_top @ nat ) )
= ( ! [N7: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N7 ) @ F5 ) ) ) ).
% le_sequentially
thf(fact_6760_eventually__gt__at__top,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C3: A] : ( eventually @ A @ ( ord_less @ A @ C3 ) @ ( at_top @ A ) ) ) ).
% eventually_gt_at_top
thf(fact_6761_eventually__at__top__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P2: A > $o] :
( ( eventually @ A @ P2 @ ( at_top @ A ) )
= ( ? [N7: A] :
! [N6: A] :
( ( ord_less @ A @ N7 @ N6 )
=> ( P2 @ N6 ) ) ) ) ) ).
% eventually_at_top_dense
thf(fact_6762_eventually__at__bot__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P2: A > $o] :
( ( eventually @ A @ P2 @ ( at_bot @ A ) )
= ( ? [N7: A] :
! [N6: A] :
( ( ord_less @ A @ N6 @ N7 )
=> ( P2 @ N6 ) ) ) ) ) ).
% eventually_at_bot_dense
thf(fact_6763_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C3: A] :
( eventually @ A
@ ^ [X4: A] : ( ord_less @ A @ X4 @ C3 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_6764_eventually__le__at__bot,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ A @ X4 @ C3 )
@ ( at_bot @ A ) ) ) ).
% eventually_le_at_bot
thf(fact_6765_le__filter__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( filter @ A ) )
= ( ^ [F11: filter @ A,F15: filter @ A] :
! [P5: A > $o] :
( ( eventually @ A @ P5 @ F15 )
=> ( eventually @ A @ P5 @ F11 ) ) ) ) ).
% le_filter_def
thf(fact_6766_filter__leI,axiom,
! [A: $tType,F7: filter @ A,F5: filter @ A] :
( ! [P: A > $o] :
( ( eventually @ A @ P @ F7 )
=> ( eventually @ A @ P @ F5 ) )
=> ( ord_less_eq @ ( filter @ A ) @ F5 @ F7 ) ) ).
% filter_leI
thf(fact_6767_filter__leD,axiom,
! [A: $tType,F5: filter @ A,F7: filter @ A,P2: A > $o] :
( ( ord_less_eq @ ( filter @ A ) @ F5 @ F7 )
=> ( ( eventually @ A @ P2 @ F7 )
=> ( eventually @ A @ P2 @ F5 ) ) ) ).
% filter_leD
thf(fact_6768_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o] :
( ( eventually @ A @ P2 @ ( at_bot @ A ) )
= ( ? [N7: A] :
! [N6: A] :
( ( ord_less_eq @ A @ N6 @ N7 )
=> ( P2 @ N6 ) ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_6769_filterlim__mono__eventually,axiom,
! [B: $tType,A: $tType,F3: A > B,F5: filter @ B,G6: filter @ A,F7: filter @ B,G8: filter @ A,F8: A > B] :
( ( filterlim @ A @ B @ F3 @ F5 @ G6 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F5 @ F7 )
=> ( ( ord_less_eq @ ( filter @ A ) @ G8 @ G6 )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ( F3 @ X4 )
= ( F8 @ X4 ) )
@ G8 )
=> ( filterlim @ A @ B @ F8 @ F7 @ G8 ) ) ) ) ) ).
% filterlim_mono_eventually
thf(fact_6770_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F5: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F5 )
= ( ! [Z10: B] :
( ( ord_less_eq @ B @ Z10 @ C3 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ Z10 )
@ F5 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_6771_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F5 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ Z10 )
@ F5 ) ) ) ) ).
% filterlim_at_bot
thf(fact_6772_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F5: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F5 )
= ( ! [Z10: B] :
( ( ord_less @ B @ Z10 @ C3 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ Z10 )
@ F5 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_6773_filterlim__at__bot__dense,axiom,
! [A: $tType,B: $tType] :
( ( ( dense_linorder @ B )
& ( no_bot @ B ) )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F5 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ ( F3 @ X4 ) @ Z10 )
@ F5 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_6774_eventually__nhds__top,axiom,
! [A: $tType] :
( ( ( order_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: A,P2: A > $o] :
( ( ord_less @ A @ B2 @ ( top_top @ A ) )
=> ( ( eventually @ A @ P2 @ ( topolo7230453075368039082e_nhds @ A @ ( top_top @ A ) ) )
= ( ? [B8: A] :
( ( ord_less @ A @ B8 @ ( top_top @ A ) )
& ! [Z2: A] :
( ( ord_less @ A @ B8 @ Z2 )
=> ( P2 @ Z2 ) ) ) ) ) ) ) ).
% eventually_nhds_top
thf(fact_6775_filterlim__at__top__at__top,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [Q2: A > $o,F3: A > B,P2: B > $o,G3: B > A] :
( ! [X3: A,Y3: A] :
( ( Q2 @ X3 )
=> ( ( Q2 @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) ) ) )
=> ( ! [X3: B] :
( ( P2 @ X3 )
=> ( ( F3 @ ( G3 @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P2 @ X3 )
=> ( Q2 @ ( G3 @ X3 ) ) )
=> ( ( eventually @ A @ Q2 @ ( at_top @ A ) )
=> ( ( eventually @ B @ P2 @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ ( at_top @ A ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_6776_eventually__at__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [Y: A,X: A,P2: A > $o] :
( ( ord_less @ A @ Y @ X )
=> ( ( eventually @ A @ P2 @ ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) ) )
= ( ? [B8: A] :
( ( ord_less @ A @ B8 @ X )
& ! [Y5: A] :
( ( ord_less @ A @ B8 @ Y5 )
=> ( ( ord_less @ A @ Y5 @ X )
=> ( P2 @ Y5 ) ) ) ) ) ) ) ) ).
% eventually_at_left
thf(fact_6777_eventually__at__left__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P2: A > $o,X: A] :
( ( eventually @ A @ P2 @ ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) ) )
= ( ? [B8: A] :
( ( ord_less @ A @ B8 @ X )
& ! [Y5: A] :
( ( ord_less @ A @ B8 @ Y5 )
=> ( ( ord_less @ A @ Y5 @ X )
=> ( P2 @ Y5 ) ) ) ) ) ) ) ).
% eventually_at_left_field
thf(fact_6778_eventually__at__infinity,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P2: A > $o] :
( ( eventually @ A @ P2 @ ( at_infinity @ A ) )
= ( ? [B8: real] :
! [X4: A] :
( ( ord_less_eq @ real @ B8 @ ( real_V7770717601297561774m_norm @ A @ X4 ) )
=> ( P2 @ X4 ) ) ) ) ) ).
% eventually_at_infinity
thf(fact_6779_tendsto__sandwich,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,G3: B > A,Net: filter @ B,H2: B > A,C3: A] :
( ( eventually @ B
@ ^ [N6: B] : ( ord_less_eq @ A @ ( F3 @ N6 ) @ ( G3 @ N6 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N6: B] : ( ord_less_eq @ A @ ( G3 @ N6 ) @ ( H2 @ N6 ) )
@ Net )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ Net )
=> ( ( filterlim @ B @ A @ H2 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ Net )
=> ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ Net ) ) ) ) ) ) ).
% tendsto_sandwich
thf(fact_6780_order__tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,X: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F5 )
= ( ! [L4: A] :
( ( ord_less @ A @ L4 @ X )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ A @ L4 @ ( F3 @ X4 ) )
@ F5 ) )
& ! [U3: A] :
( ( ord_less @ A @ X @ U3 )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ A @ ( F3 @ X4 ) @ U3 )
@ F5 ) ) ) ) ) ).
% order_tendsto_iff
thf(fact_6781_order__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [Y: A,F3: B > A,F5: filter @ B] :
( ! [A4: A] :
( ( ord_less @ A @ A4 @ Y )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ A @ A4 @ ( F3 @ X4 ) )
@ F5 ) )
=> ( ! [A4: A] :
( ( ord_less @ A @ Y @ A4 )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ A @ ( F3 @ X4 ) @ A4 )
@ F5 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 ) ) ) ) ).
% order_tendstoI
thf(fact_6782_order__tendstoD_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,Y: A,F5: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 )
=> ( ( ord_less @ A @ A2 @ Y )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ A @ A2 @ ( F3 @ X4 ) )
@ F5 ) ) ) ) ).
% order_tendstoD(1)
thf(fact_6783_order__tendstoD_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,Y: A,F5: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 )
=> ( ( ord_less @ A @ Y @ A2 )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ A @ ( F3 @ X4 ) @ A2 )
@ F5 ) ) ) ) ).
% order_tendstoD(2)
thf(fact_6784_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,F5: filter @ B,G3: B > A] :
( ( filterlim @ B @ A @ F3 @ ( at_top @ A ) @ F5 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ F5 )
=> ( filterlim @ B @ A @ G3 @ ( at_top @ A ) @ F5 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_6785_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F5: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F5 )
= ( ! [Z10: B] :
( ( ord_less_eq @ B @ C3 @ Z10 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z10 @ ( F3 @ X4 ) )
@ F5 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_6786_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F5 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z10 @ ( F3 @ X4 ) )
@ F5 ) ) ) ) ).
% filterlim_at_top
thf(fact_6787_filterlim__at__top__dense,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F5 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ Z10 @ ( F3 @ X4 ) )
@ F5 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_6788_real__tendsto__sandwich,axiom,
! [B: $tType,F3: B > real,G3: B > real,Net: filter @ B,H2: B > real,C3: real] :
( ( eventually @ B
@ ^ [N6: B] : ( ord_less_eq @ real @ ( F3 @ N6 ) @ ( G3 @ N6 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N6: B] : ( ord_less_eq @ real @ ( G3 @ N6 ) @ ( H2 @ N6 ) )
@ Net )
=> ( ( filterlim @ B @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ Net )
=> ( ( filterlim @ B @ real @ H2 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ Net )
=> ( filterlim @ B @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ Net ) ) ) ) ) ).
% real_tendsto_sandwich
thf(fact_6789_countable__basis__at__decseq,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X: A] :
~ ! [A7: nat > ( set @ A )] :
( ! [I4: nat] : ( topolo1002775350975398744n_open @ A @ ( A7 @ I4 ) )
=> ( ! [I4: nat] : ( member @ A @ X @ ( A7 @ I4 ) )
=> ~ ! [S10: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S10 )
=> ( ( member @ A @ X @ S10 )
=> ( eventually @ nat
@ ^ [I3: nat] : ( ord_less_eq @ ( set @ A ) @ ( A7 @ I3 ) @ S10 )
@ ( at_top @ nat ) ) ) ) ) ) ) ).
% countable_basis_at_decseq
thf(fact_6790_eventually__Inf__base,axiom,
! [A: $tType,B5: set @ ( filter @ A ),P2: A > $o] :
( ( B5
!= ( bot_bot @ ( set @ ( filter @ A ) ) ) )
=> ( ! [F6: filter @ A] :
( ( member @ ( filter @ A ) @ F6 @ B5 )
=> ! [G5: filter @ A] :
( ( member @ ( filter @ A ) @ G5 @ B5 )
=> ? [X5: filter @ A] :
( ( member @ ( filter @ A ) @ X5 @ B5 )
& ( ord_less_eq @ ( filter @ A ) @ X5 @ ( inf_inf @ ( filter @ A ) @ F6 @ G5 ) ) ) ) )
=> ( ( eventually @ A @ P2 @ ( complete_Inf_Inf @ ( filter @ A ) @ B5 ) )
= ( ? [X4: filter @ A] :
( ( member @ ( filter @ A ) @ X4 @ B5 )
& ( eventually @ A @ P2 @ X4 ) ) ) ) ) ) ).
% eventually_Inf_base
thf(fact_6791_eventually__INF__finite,axiom,
! [B: $tType,A: $tType,A5: set @ A,P2: B > $o,F5: A > ( filter @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( eventually @ B @ P2 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ F5 @ A5 ) ) )
= ( ? [Q6: A > B > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( eventually @ B @ ( Q6 @ X4 ) @ ( F5 @ X4 ) ) )
& ! [Y5: B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( Q6 @ X4 @ Y5 ) )
=> ( P2 @ Y5 ) ) ) ) ) ) ).
% eventually_INF_finite
thf(fact_6792_eventually__at__left__real,axiom,
! [B2: real,A2: real] :
( ( ord_less @ real @ B2 @ A2 )
=> ( eventually @ real
@ ^ [X4: real] : ( member @ real @ X4 @ ( set_or5935395276787703475ssThan @ real @ B2 @ A2 ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ).
% eventually_at_left_real
thf(fact_6793_eventually__at,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P2: A > $o,A2: A,S3: set @ A] :
( ( eventually @ A @ P2 @ ( topolo174197925503356063within @ A @ A2 @ S3 ) )
= ( ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [X4: A] :
( ( member @ A @ X4 @ S3 )
=> ( ( ( X4 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ A2 ) @ D6 ) )
=> ( P2 @ X4 ) ) ) ) ) ) ) ).
% eventually_at
thf(fact_6794_eventually__nhds__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P2: A > $o,A2: A] :
( ( eventually @ A @ P2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) )
= ( ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [X4: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ A2 ) @ D6 )
=> ( P2 @ X4 ) ) ) ) ) ) ).
% eventually_nhds_metric
thf(fact_6795_eventually__at__leftI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A,P2: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
=> ( P2 @ X3 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( eventually @ A @ P2 @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ) ).
% eventually_at_leftI
thf(fact_6796_eventually__at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P2: A > $o,A2: A] :
( ( eventually @ A @ P2 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( eventually @ A
@ ^ [X4: A] : ( P2 @ ( plus_plus @ A @ X4 @ A2 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% eventually_at_to_0
thf(fact_6797_decreasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [L: A,F3: B > A,F5: filter @ B] :
( ( eventually @ B
@ ^ [N6: B] : ( ord_less_eq @ A @ L @ ( F3 @ N6 ) )
@ F5 )
=> ( ! [X3: A] :
( ( ord_less @ A @ L @ X3 )
=> ( eventually @ B
@ ^ [N6: B] : ( ord_less @ A @ ( F3 @ N6 ) @ X3 )
@ F5 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% decreasing_tendsto
thf(fact_6798_increasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,L: A,F5: filter @ B] :
( ( eventually @ B
@ ^ [N6: B] : ( ord_less_eq @ A @ ( F3 @ N6 ) @ L )
@ F5 )
=> ( ! [X3: A] :
( ( ord_less @ A @ X3 @ L )
=> ( eventually @ B
@ ^ [N6: B] : ( ord_less @ A @ X3 @ ( F3 @ N6 ) )
@ F5 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% increasing_tendsto
thf(fact_6799_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F5: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F5 )
= ( ! [Z10: B] :
( ( ord_less @ B @ C3 @ Z10 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z10 @ ( F3 @ X4 ) )
@ F5 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_6800_tendsto__le,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F5: filter @ B,F3: B > A,X: A,G3: B > A,Y: A] :
( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F5 )
=> ( ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( ord_less_eq @ A @ ( G3 @ X4 ) @ ( F3 @ X4 ) )
@ F5 )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ) ).
% tendsto_le
thf(fact_6801_tendsto__lowerbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: B > A,X: A,F5: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F5 )
=> ( ( eventually @ B
@ ^ [I3: B] : ( ord_less_eq @ A @ A2 @ ( F3 @ I3 ) )
@ F5 )
=> ( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ A2 @ X ) ) ) ) ) ).
% tendsto_lowerbound
thf(fact_6802_tendsto__upperbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: B > A,X: A,F5: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F5 )
=> ( ( eventually @ B
@ ^ [I3: B] : ( ord_less_eq @ A @ ( F3 @ I3 ) @ A2 )
@ F5 )
=> ( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ X @ A2 ) ) ) ) ) ).
% tendsto_upperbound
thf(fact_6803_metric__tendsto__imp__tendsto,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( real_V7819770556892013058_space @ B )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F3: C > A,A2: A,F5: filter @ C,G3: C > B,B2: B] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( eventually @ C
@ ^ [X4: C] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( G3 @ X4 ) @ B2 ) @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X4 ) @ A2 ) )
@ F5 )
=> ( filterlim @ C @ B @ G3 @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ F5 ) ) ) ) ).
% metric_tendsto_imp_tendsto
thf(fact_6804_eventually__INF,axiom,
! [A: $tType,B: $tType,P2: A > $o,F5: B > ( filter @ A ),B5: set @ B] :
( ( eventually @ A @ P2 @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ B @ ( filter @ A ) @ F5 @ B5 ) ) )
= ( ? [X8: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ X8 @ B5 )
& ( finite_finite2 @ B @ X8 )
& ( eventually @ A @ P2 @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ B @ ( filter @ A ) @ F5 @ X8 ) ) ) ) ) ) ).
% eventually_INF
thf(fact_6805_eventually__at__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P2: A > $o,A2: A,S3: set @ A] :
( ( eventually @ A @ P2 @ ( topolo174197925503356063within @ A @ A2 @ S3 ) )
= ( ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [X4: A] :
( ( member @ A @ X4 @ S3 )
=> ( ( ( X4 != A2 )
& ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ A2 ) @ D6 ) )
=> ( P2 @ X4 ) ) ) ) ) ) ) ).
% eventually_at_le
thf(fact_6806_eventually__at__infinity__pos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P3: A > $o] :
( ( eventually @ A @ P3 @ ( at_infinity @ A ) )
= ( ? [B8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B8 )
& ! [X4: A] :
( ( ord_less_eq @ real @ B8 @ ( real_V7770717601297561774m_norm @ A @ X4 ) )
=> ( P3 @ X4 ) ) ) ) ) ) ).
% eventually_at_infinity_pos
thf(fact_6807_tendsto__imp__filterlim__at__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F3: A > B,L5: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ F5 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ ( F3 @ X4 ) @ L5 )
@ F5 )
=> ( filterlim @ A @ B @ F3 @ ( topolo174197925503356063within @ B @ L5 @ ( set_ord_lessThan @ B @ L5 ) ) @ F5 ) ) ) ) ).
% tendsto_imp_filterlim_at_left
thf(fact_6808_tendstoD,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F3: B > A,L: A,F5: filter @ B,E3: real] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X4 ) @ L ) @ E3 )
@ F5 ) ) ) ) ).
% tendstoD
thf(fact_6809_tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F3: B > A,L: A,F5: filter @ B] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X4 ) @ L ) @ E2 )
@ F5 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ).
% tendstoI
thf(fact_6810_tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F3: B > A,L: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 )
= ( ! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X4 ) @ L ) @ E4 )
@ F5 ) ) ) ) ) ).
% tendsto_iff
thf(fact_6811_eventually__Inf,axiom,
! [A: $tType,P2: A > $o,B5: set @ ( filter @ A )] :
( ( eventually @ A @ P2 @ ( complete_Inf_Inf @ ( filter @ A ) @ B5 ) )
= ( ? [X8: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X8 @ B5 )
& ( finite_finite2 @ ( filter @ A ) @ X8 )
& ( eventually @ A @ P2 @ ( complete_Inf_Inf @ ( filter @ A ) @ X8 ) ) ) ) ) ).
% eventually_Inf
thf(fact_6812_summable__comparison__test__ev,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,G3: nat > real] :
( ( eventually @ nat
@ ^ [N6: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N6 ) ) @ ( G3 @ N6 ) )
@ ( at_top @ nat ) )
=> ( ( summable @ real @ G3 )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_comparison_test_ev
thf(fact_6813_tendsto__arcosh__strong,axiom,
! [B: $tType,F3: B > real,A2: real,F5: filter @ B] :
( ( filterlim @ B @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ A2 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F3 @ X4 ) )
@ F5 )
=> ( filterlim @ B @ real
@ ^ [X4: B] : ( arcosh @ real @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A2 ) )
@ F5 ) ) ) ) ).
% tendsto_arcosh_strong
thf(fact_6814_filterlim__at__top__at__left,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q2: A > $o,F3: A > B,P2: B > $o,G3: B > A,A2: A] :
( ! [X3: A,Y3: A] :
( ( Q2 @ X3 )
=> ( ( Q2 @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) ) ) )
=> ( ! [X3: B] :
( ( P2 @ X3 )
=> ( ( F3 @ ( G3 @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P2 @ X3 )
=> ( Q2 @ ( G3 @ X3 ) ) )
=> ( ( eventually @ A @ Q2 @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) )
=> ( ! [B4: A] :
( ( Q2 @ B4 )
=> ( ord_less @ A @ B4 @ A2 ) )
=> ( ( eventually @ B @ P2 @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_left
thf(fact_6815_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B5: set @ A,F5: A > ( filter @ B ),P2: B > $o] :
( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ B5 )
=> ! [B4: A] :
( ( member @ A @ B4 @ B5 )
=> ? [X5: A] :
( ( member @ A @ X5 @ B5 )
& ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X5 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ A4 ) @ ( F5 @ B4 ) ) ) ) ) )
=> ( ( eventually @ B @ P2 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ F5 @ B5 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ B5 )
& ( eventually @ B @ P2 @ ( F5 @ X4 ) ) ) ) ) ) ) ).
% eventually_INF_base
thf(fact_6816_filterlim__at__withinI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,C3: A,F5: filter @ B,A5: set @ A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F5 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( member @ A @ ( F3 @ X4 ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ F5 )
=> ( filterlim @ B @ A @ F3 @ ( topolo174197925503356063within @ A @ C3 @ A5 ) @ F5 ) ) ) ) ).
% filterlim_at_withinI
thf(fact_6817_tendsto__0__le,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F5: filter @ A,G3: A > C,K4: real] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G3 @ X4 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) ) @ K4 ) )
@ F5 )
=> ( filterlim @ A @ C @ G3 @ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) ) @ F5 ) ) ) ) ).
% tendsto_0_le
thf(fact_6818_filterlim__tendsto__pos__mult__at__bot,axiom,
! [A: $tType,F3: A > real,C3: real,F5: filter @ A,G3: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ A @ real @ G3 @ ( at_bot @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( times_times @ real @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( at_bot @ real )
@ F5 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_bot
thf(fact_6819_eventually__floor__less,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F3: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ B @ L ) ) @ ( F3 @ X4 ) )
@ F5 ) ) ) ) ).
% eventually_floor_less
thf(fact_6820_eventually__less__ceiling,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F3: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ ( F3 @ X4 ) @ ( ring_1_of_int @ B @ ( archimedean_ceiling @ B @ L ) ) )
@ F5 ) ) ) ) ).
% eventually_less_ceiling
thf(fact_6821_filterlim__at__infinity,axiom,
! [C: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C3: real,F3: C > A,F5: filter @ C] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ C @ A @ F3 @ ( at_infinity @ A ) @ F5 )
= ( ! [R5: real] :
( ( ord_less @ real @ C3 @ R5 )
=> ( eventually @ C
@ ^ [X4: C] : ( ord_less_eq @ real @ R5 @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ X4 ) ) )
@ F5 ) ) ) ) ) ) ).
% filterlim_at_infinity
thf(fact_6822_tendsto__zero__powrI,axiom,
! [A: $tType,F3: A > real,F5: filter @ A,G3: A > real,B2: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 )
=> ( ( filterlim @ A @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X4 ) )
@ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( powr @ real @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 ) ) ) ) ) ).
% tendsto_zero_powrI
thf(fact_6823_tendsto__powr2,axiom,
! [A: $tType,F3: A > real,A2: real,F5: filter @ A,G3: A > real,B2: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 )
=> ( ( filterlim @ A @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X4 ) )
@ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( powr @ real @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F5 ) ) ) ) ) ).
% tendsto_powr2
thf(fact_6824_tendsto__powr_H,axiom,
! [A: $tType,F3: A > real,A2: real,F5: filter @ A,G3: A > real,B2: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 )
=> ( ( filterlim @ A @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 )
=> ( ( ( A2
!= ( zero_zero @ real ) )
| ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
& ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X4 ) )
@ F5 ) ) )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( powr @ real @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F5 ) ) ) ) ).
% tendsto_powr'
thf(fact_6825_DERIV__pos__imp__increasing__at__bot,axiom,
! [B2: real,F3: real > real,Flim: real] :
( ! [X3: real] :
( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F3 @ Y4 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y4 ) ) )
=> ( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_bot @ real ) )
=> ( ord_less @ real @ Flim @ ( F3 @ B2 ) ) ) ) ).
% DERIV_pos_imp_increasing_at_bot
thf(fact_6826_summable__Cauchy_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,G3: nat > real] :
( ( eventually @ nat
@ ^ [M5: nat] :
! [N6: nat] :
( ( ord_less_eq @ nat @ M5 @ N6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M5 @ N6 ) ) ) @ ( G3 @ M5 ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_Cauchy'
thf(fact_6827_Bfun__metric__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F4: A > B,F11: filter @ A] :
? [Y5: B,K5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K5 )
& ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( F4 @ X4 ) @ Y5 ) @ K5 )
@ F11 ) ) ) ) ) ).
% Bfun_metric_def
thf(fact_6828_Bseq__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( bfun @ nat @ A
@ ^ [N6: nat] : ( F3 @ ( suc @ N6 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ).
% Bseq_Suc_iff
thf(fact_6829_at__top__le__at__infinity,axiom,
ord_less_eq @ ( filter @ real ) @ ( at_top @ real ) @ ( at_infinity @ real ) ).
% at_top_le_at_infinity
thf(fact_6830_BseqI_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A,K4: real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X6 @ N2 ) ) @ K4 )
=> ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) ) ) ) ).
% BseqI'
thf(fact_6831_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o] :
( ( eventually @ A @ P2 @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X4: A] :
! [Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
=> ( P2 @ Y5 ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_6832_finite__set__of__finite__funs,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B,D2: B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( finite_finite2 @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F4: A > B] :
! [X4: A] :
( ( ( member @ A @ X4 @ A5 )
=> ( member @ B @ ( F4 @ X4 ) @ B5 ) )
& ( ~ ( member @ A @ X4 @ A5 )
=> ( ( F4 @ X4 )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_6833_Bseq__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F3: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( bfun @ nat @ A
@ ^ [X4: nat] : ( times_times @ A @ C3 @ ( F3 @ X4 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_cmult_iff
thf(fact_6834_Least__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_Least @ A )
= ( ^ [P5: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P5 @ X4 )
& ! [Y5: A] :
( ( P5 @ Y5 )
=> ( ord_less_eq @ A @ X4 @ Y5 ) ) ) ) ) ) ) ).
% Least_def
thf(fact_6835_Bseq__eventually__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: nat > A,G3: nat > B] :
( ( eventually @ nat
@ ^ [N6: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N6 ) ) @ ( real_V7770717601297561774m_norm @ B @ ( G3 @ N6 ) ) )
@ ( at_top @ nat ) )
=> ( ( bfun @ nat @ B @ G3 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_eventually_mono
thf(fact_6836_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P5: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P5 @ X4 )
& ! [Y5: A] :
( ( P5 @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_6837_filterlim__pow__at__top,axiom,
! [A: $tType,N: nat,F3: A > real,F5: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( power_power @ real @ ( F3 @ X4 ) @ N )
@ ( at_top @ real )
@ F5 ) ) ) ).
% filterlim_pow_at_top
thf(fact_6838_filterlim__at__infinity__imp__filterlim__at__top,axiom,
! [A: $tType,F3: A > real,F5: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( at_infinity @ real ) @ F5 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X4 ) )
@ F5 )
=> ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F5 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_top
thf(fact_6839_Bseq__eq__bounded,axiom,
! [F3: nat > real,A2: real,B2: real] :
( ( ord_less_eq @ ( set @ real ) @ ( image @ nat @ real @ F3 @ ( top_top @ ( set @ nat ) ) ) @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
=> ( bfun @ nat @ real @ F3 @ ( at_top @ nat ) ) ) ).
% Bseq_eq_bounded
thf(fact_6840_Bseq__def,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
= ( ? [K5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K5 )
& ! [N6: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X6 @ N6 ) ) @ K5 ) ) ) ) ) ).
% Bseq_def
thf(fact_6841_BseqI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [K4: real,X6: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K4 )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X6 @ N2 ) ) @ K4 )
=> ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) ) ) ) ) ).
% BseqI
thf(fact_6842_BseqE,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
=> ~ ! [K8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K8 )
=> ~ ! [N9: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X6 @ N9 ) ) @ K8 ) ) ) ) ).
% BseqE
thf(fact_6843_BseqD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
=> ? [K8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K8 )
& ! [N9: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X6 @ N9 ) ) @ K8 ) ) ) ) ).
% BseqD
thf(fact_6844_Bseq__iff1a,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
= ( ? [N7: nat] :
! [N6: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( X6 @ N6 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N7 ) ) ) ) ) ) ).
% Bseq_iff1a
thf(fact_6845_Bseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
= ( ? [N7: nat] :
! [N6: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X6 @ N6 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N7 ) ) ) ) ) ) ).
% Bseq_iff
thf(fact_6846_Bseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( bfun @ nat @ real @ ( power_power @ real @ X ) @ ( at_top @ nat ) ) ) ) ).
% Bseq_realpow
thf(fact_6847_filterlim__at__top__mult__tendsto__pos,axiom,
! [A: $tType,F3: A > real,C3: real,F5: filter @ A,G3: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ A @ real @ G3 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( times_times @ real @ ( G3 @ X4 ) @ ( F3 @ X4 ) )
@ ( at_top @ real )
@ F5 ) ) ) ) ).
% filterlim_at_top_mult_tendsto_pos
thf(fact_6848_filterlim__tendsto__pos__mult__at__top,axiom,
! [A: $tType,F3: A > real,C3: real,F5: filter @ A,G3: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ A @ real @ G3 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( times_times @ real @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( at_top @ real )
@ F5 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_top
thf(fact_6849_tendsto__neg__powr,axiom,
! [A: $tType,S: real,F3: A > real,F5: filter @ A] :
( ( ord_less @ real @ S @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( powr @ real @ ( F3 @ X4 ) @ S )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 ) ) ) ).
% tendsto_neg_powr
thf(fact_6850_LIM__at__top__divide,axiom,
! [A: $tType,F3: A > real,A2: real,F5: filter @ A,G3: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( filterlim @ A @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X4 ) )
@ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( divide_divide @ real @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( at_top @ real )
@ F5 ) ) ) ) ) ).
% LIM_at_top_divide
thf(fact_6851_filterlim__inverse__at__top,axiom,
! [A: $tType,F3: A > real,F5: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X4 ) )
@ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( inverse_inverse @ real @ ( F3 @ X4 ) )
@ ( at_top @ real )
@ F5 ) ) ) ).
% filterlim_inverse_at_top
thf(fact_6852_filterlim__inverse__at__top__iff,axiom,
! [A: $tType,F3: A > real,F5: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X4 ) )
@ F5 )
=> ( ( filterlim @ A @ real
@ ^ [X4: A] : ( inverse_inverse @ real @ ( F3 @ X4 ) )
@ ( at_top @ real )
@ F5 )
= ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) ) ) ).
% filterlim_inverse_at_top_iff
thf(fact_6853_filterlim__tendsto__neg__mult__at__bot,axiom,
! [A: $tType,F3: A > real,C3: real,F5: filter @ A,G3: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F5 )
=> ( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ G3 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( times_times @ real @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( at_bot @ real )
@ F5 ) ) ) ) ).
% filterlim_tendsto_neg_mult_at_bot
thf(fact_6854_BfunI,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,K4: real,F5: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) ) @ K4 )
@ F5 )
=> ( bfun @ A @ B @ F3 @ F5 ) ) ) ).
% BfunI
thf(fact_6855_Bseq__iff2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [X4: A] :
! [N6: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X6 @ N6 ) @ ( uminus_uminus @ A @ X4 ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff2
thf(fact_6856_Bseq__iff3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [N7: nat] :
! [N6: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X6 @ N6 ) @ ( uminus_uminus @ A @ ( X6 @ N7 ) ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff3
thf(fact_6857_DERIV__neg__imp__decreasing__at__top,axiom,
! [B2: real,F3: real > real,Flim: real] :
( ! [X3: real] :
( ( ord_less_eq @ real @ B2 @ X3 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F3 @ Y4 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y4 @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_top @ real ) )
=> ( ord_less @ real @ Flim @ ( F3 @ B2 ) ) ) ) ).
% DERIV_neg_imp_decreasing_at_top
thf(fact_6858_Bfun__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: B > A,A2: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( bfun @ B @ A
@ ^ [X4: B] : ( inverse_inverse @ A @ ( F3 @ X4 ) )
@ F5 ) ) ) ) ).
% Bfun_inverse
thf(fact_6859_filterlim__pow__at__bot__even,axiom,
! [N: nat,F3: real > real,F5: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F3 @ ( at_bot @ real ) @ F5 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( power_power @ real @ ( F3 @ X4 ) @ N )
@ ( at_top @ real )
@ F5 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_6860_BfunE,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( bfun @ A @ B @ F3 @ F5 )
=> ~ ! [B9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B9 )
=> ~ ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) ) @ B9 )
@ F5 ) ) ) ) ).
% BfunE
thf(fact_6861_Bfun__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F4: A > B,F11: filter @ A] :
? [K5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K5 )
& ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F4 @ X4 ) ) @ K5 )
@ F11 ) ) ) ) ) ).
% Bfun_def
thf(fact_6862_summable__bounded__partials,axiom,
! [A: $tType] :
( ( ( real_V8037385150606011577_space @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: nat > A,G3: nat > real] :
( ( eventually @ nat
@ ^ [X02: nat] :
! [A8: nat] :
( ( ord_less_eq @ nat @ X02 @ A8 )
=> ! [B8: nat] :
( ( ord_less @ nat @ A8 @ B8 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or3652927894154168847AtMost @ nat @ A8 @ B8 ) ) ) @ ( G3 @ A8 ) ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_bounded_partials
thf(fact_6863_max__ext__eq,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X8: set @ A,Y10: set @ A] :
( ( finite_finite2 @ A @ X8 )
& ( finite_finite2 @ A @ Y10 )
& ( Y10
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ Y10 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R4 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_6864_finite__greaterThanAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% finite_greaterThanAtMost
thf(fact_6865_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_6866_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_6867_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_6868_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_6869_infinite__Ioc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ioc_iff
thf(fact_6870_Sup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanAtMost
thf(fact_6871_cSup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_greaterThanAtMost
thf(fact_6872_Inf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_greaterThanAtMost
thf(fact_6873_cInf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_greaterThanAtMost
thf(fact_6874_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_6875_Ioc__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C3 @ D2 ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% Ioc_subset_iff
thf(fact_6876_Ioc__inj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ( set_or3652927894154168847AtMost @ A @ A2 @ B2 )
= ( set_or3652927894154168847AtMost @ A @ C3 @ D2 ) )
= ( ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ D2 @ C3 ) )
| ( ( A2 = C3 )
& ( B2 = D2 ) ) ) ) ) ).
% Ioc_inj
thf(fact_6877_ivl__disj__int__two_I6_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(6)
thf(fact_6878_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L ) @ U )
= ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_6879_infinite__Ioc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ioc
thf(fact_6880_Ioc__disjoint,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C3 @ D2 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ord_less_eq @ A @ D2 @ C3 )
| ( ord_less_eq @ A @ B2 @ C3 )
| ( ord_less_eq @ A @ D2 @ A2 ) ) ) ) ).
% Ioc_disjoint
thf(fact_6881_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_6882_open__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A,X: A,Y: A] :
( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( member @ A @ X @ S3 )
=> ( ( ord_less @ A @ Y @ X )
=> ? [B4: A] :
( ( ord_less @ A @ B4 @ X )
& ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ B4 @ X ) @ S3 ) ) ) ) ) ) ).
% open_left
thf(fact_6883_ivl__disj__int__two_I8_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(8)
thf(fact_6884_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_6885_ivl__disj__int__one_I3_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(3)
thf(fact_6886_ivl__disj__int__two_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(2)
thf(fact_6887_greaterThanAtMost__upt,axiom,
( ( set_or3652927894154168847AtMost @ nat )
= ( ^ [N6: nat,M5: nat] : ( set2 @ nat @ ( upt @ ( suc @ N6 ) @ ( suc @ M5 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_6888_sum_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G3 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.head
thf(fact_6889_prod_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G3 @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.head
thf(fact_6890_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_6891_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_6892_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_6893_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C3: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C3 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_6894_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A8: A,B8: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A8 @ B8 ) @ ( insert @ A @ A8 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_6895_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_6896_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_6897_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_6898_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_6899_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_6900_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_6901_max__extp__max__ext__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( max_extp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R3 ) )
= ( ^ [X4: set @ A,Y5: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X4 @ Y5 ) @ ( max_ext @ A @ R3 ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_6902_max__ext_Ocases,axiom,
! [A: $tType,A12: set @ A,A23: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A12 @ A23 ) @ ( max_ext @ A @ R3 ) )
=> ~ ( ( finite_finite2 @ A @ A12 )
=> ( ( finite_finite2 @ A @ A23 )
=> ( ( A23
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ A12 )
=> ? [Xa4: A] :
( ( member @ A @ Xa4 @ A23 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Xa4 ) @ R3 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_6903_max__ext_Osimps,axiom,
! [A: $tType,A12: set @ A,A23: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A12 @ A23 ) @ ( max_ext @ A @ R3 ) )
= ( ( finite_finite2 @ A @ A12 )
& ( finite_finite2 @ A @ A23 )
& ( A23
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A12 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A23 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R3 ) ) ) ) ) ).
% max_ext.simps
thf(fact_6904_max__ext_Omax__extI,axiom,
! [A: $tType,X6: set @ A,Y7: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X6 )
=> ( ( finite_finite2 @ A @ Y7 )
=> ( ( Y7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ Y7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Xa ) @ R3 ) ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X6 @ Y7 ) @ ( max_ext @ A @ R3 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_6905_max__ext__def,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ( max_extp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R4 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_6906_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_6907_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa2 )
= Y )
=> ( ( ? [Uu: $o,Uv: $o] :
( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( Y
= ( Xa2
!= ( one_one @ nat ) ) ) )
=> ~ ! [Mima2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
= ( ~ ( ( Deg2 = Xa2 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_6908_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( ? [Uu: $o,Uv: $o] :
( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( Xa2
!= ( one_one @ nat ) ) )
=> ~ ! [Mima2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_6909_finite__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or3652927894154168847AtMost @ code_integer @ L @ U ) ) ).
% finite_greaterThanAtMost_integer
thf(fact_6910_ball__empty,axiom,
! [A: $tType,P2: A > $o,X5: A] :
( ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) )
=> ( P2 @ X5 ) ) ).
% ball_empty
thf(fact_6911_finite__Collect__bounded__ex,axiom,
! [B: $tType,A: $tType,P2: A > $o,Q2: B > A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
? [Y5: A] :
( ( P2 @ Y5 )
& ( Q2 @ X4 @ Y5 ) ) ) )
= ( ! [Y5: A] :
( ( P2 @ Y5 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] : ( Q2 @ X4 @ Y5 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_6912_eq__or__mem__image__simp,axiom,
! [B: $tType,A: $tType,F3: B > A,A2: B,B5: set @ B] :
( ( collect @ A
@ ^ [Uu3: A] :
? [L4: B] :
( ( Uu3
= ( F3 @ L4 ) )
& ( ( L4 = A2 )
| ( member @ B @ L4 @ B5 ) ) ) )
= ( insert @ A @ ( F3 @ A2 )
@ ( collect @ A
@ ^ [Uu3: A] :
? [L4: B] :
( ( Uu3
= ( F3 @ L4 ) )
& ( member @ B @ L4 @ B5 ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_6913_ball__UNIV,axiom,
! [A: $tType,P2: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) )
=> ( P2 @ X4 ) ) )
= ( ! [X8: A] : ( P2 @ X8 ) ) ) ).
% ball_UNIV
thf(fact_6914_bex__UNIV,axiom,
! [A: $tType,P2: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) )
& ( P2 @ X4 ) ) )
= ( ? [X8: A] : ( P2 @ X8 ) ) ) ).
% bex_UNIV
thf(fact_6915_eventually__ball__finite,axiom,
! [A: $tType,B: $tType,A5: set @ A,P2: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( eventually @ B
@ ^ [Y5: B] : ( P2 @ Y5 @ X3 )
@ Net ) )
=> ( eventually @ B
@ ^ [X4: B] :
! [Y5: A] :
( ( member @ A @ Y5 @ A5 )
=> ( P2 @ X4 @ Y5 ) )
@ Net ) ) ) ).
% eventually_ball_finite
thf(fact_6916_eventually__ball__finite__distrib,axiom,
! [B: $tType,A: $tType,A5: set @ A,P2: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( eventually @ B
@ ^ [X4: B] :
! [Y5: A] :
( ( member @ A @ Y5 @ A5 )
=> ( P2 @ X4 @ Y5 ) )
@ Net )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( eventually @ B
@ ^ [Y5: B] : ( P2 @ Y5 @ X4 )
@ Net ) ) ) ) ) ).
% eventually_ball_finite_distrib
thf(fact_6917_listrel1__def,axiom,
! [A: $tType] :
( ( listrel1 @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys3: list @ A] :
? [Us3: list @ A,Z2: A,Z9: A,Vs3: list @ A] :
( ( Xs2
= ( append @ A @ Us3 @ ( cons @ A @ Z2 @ Vs3 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z2 @ Z9 ) @ R5 )
& ( Ys3
= ( append @ A @ Us3 @ ( cons @ A @ Z9 @ Vs3 ) ) ) ) ) ) ) ) ).
% listrel1_def
thf(fact_6918_finite__image__set2,axiom,
! [A: $tType,B: $tType,C: $tType,P2: A > $o,Q2: B > $o,F3: A > B > C] :
( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
=> ( ( finite_finite2 @ B @ ( collect @ B @ Q2 ) )
=> ( finite_finite2 @ C
@ ( collect @ C
@ ^ [Uu3: C] :
? [X4: A,Y5: B] :
( ( Uu3
= ( F3 @ X4 @ Y5 ) )
& ( P2 @ X4 )
& ( Q2 @ Y5 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_6919_finite__image__set,axiom,
! [A: $tType,B: $tType,P2: A > $o,F3: A > B] :
( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Uu3: B] :
? [X4: A] :
( ( Uu3
= ( F3 @ X4 ) )
& ( P2 @ X4 ) ) ) ) ) ).
% finite_image_set
thf(fact_6920_open__superdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X4: A,Y5: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y5 ) )
& ( ord_less @ A @ Y5 @ X4 ) ) ) ) ) ).
% open_superdiagonal
thf(fact_6921_open__subdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X4: A,Y5: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y5 ) )
& ( ord_less @ A @ X4 @ Y5 ) ) ) ) ) ).
% open_subdiagonal
thf(fact_6922_open__diagonal__complement,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X4: A,Y5: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y5 ) )
& ( X4 != Y5 ) ) ) ) ) ).
% open_diagonal_complement
thf(fact_6923_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( collect @ A
@ ^ [Uu3: A] :
? [X4: B] :
( ( Uu3
= ( F3 @ X4 ) )
& ( member @ B @ X4 @ A5 ) ) )
= ( image @ B @ A @ F3 @ A5 ) ) ).
% Setcompr_eq_image
thf(fact_6924_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F3: B > A,P2: B > $o] :
( ( collect @ A
@ ^ [Uu3: A] :
? [X4: B] :
( ( Uu3
= ( F3 @ X4 ) )
& ( P2 @ X4 ) ) )
= ( image @ B @ A @ F3 @ ( collect @ B @ P2 ) ) ) ).
% setcompr_eq_image
thf(fact_6925_full__SetCompr__eq,axiom,
! [A: $tType,B: $tType,F3: B > A] :
( ( collect @ A
@ ^ [U3: A] :
? [X4: B] :
( U3
= ( F3 @ X4 ) ) )
= ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% full_SetCompr_eq
thf(fact_6926_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A6: set @ A,P5: A > $o] :
! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( P5 @ X4 ) ) ) ) ).
% Ball_def
thf(fact_6927_Un__interval,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [B1: A,B22: A,B32: A,F3: A > B] :
( ( ord_less_eq @ A @ B1 @ B22 )
=> ( ( ord_less_eq @ A @ B22 @ B32 )
=> ( ( sup_sup @ ( set @ B )
@ ( collect @ B
@ ^ [Uu3: B] :
? [I3: A] :
( ( Uu3
= ( F3 @ I3 ) )
& ( ord_less_eq @ A @ B1 @ I3 )
& ( ord_less @ A @ I3 @ B22 ) ) )
@ ( collect @ B
@ ^ [Uu3: B] :
? [I3: A] :
( ( Uu3
= ( F3 @ I3 ) )
& ( ord_less_eq @ A @ B22 @ I3 )
& ( ord_less @ A @ I3 @ B32 ) ) ) )
= ( collect @ B
@ ^ [Uu3: B] :
? [I3: A] :
( ( Uu3
= ( F3 @ I3 ) )
& ( ord_less_eq @ A @ B1 @ I3 )
& ( ord_less @ A @ I3 @ B32 ) ) ) ) ) ) ) ).
% Un_interval
thf(fact_6928_Inf__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A )
= ( ^ [A6: set @ A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [B8: A] :
! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ B8 @ X4 ) ) ) ) ) ) ) ).
% Inf_eq_Sup
thf(fact_6929_Sup__eq__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A )
= ( ^ [A6: set @ A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [B8: A] :
! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ X4 @ B8 ) ) ) ) ) ) ) ).
% Sup_eq_Inf
thf(fact_6930_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ? [Xys2: list @ A,X4: A,Y5: A,Xs6: list @ A,Ys6: list @ A] :
( ( Xs2
= ( append @ A @ Xys2 @ ( cons @ A @ X4 @ Xs6 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y5 @ Ys6 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_6931_Ball__fold,axiom,
! [A: $tType,A5: set @ A,P2: A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( P2 @ X4 ) ) )
= ( finite_fold @ A @ $o
@ ^ [K3: A,S8: $o] :
( S8
& ( P2 @ K3 ) )
@ $true
@ A5 ) ) ) ).
% Ball_fold
thf(fact_6932_graph__def,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M5: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [A8: A,B8: B] :
( ( Uu3
= ( product_Pair @ A @ B @ A8 @ B8 ) )
& ( ( M5 @ A8 )
= ( some @ B @ B8 ) ) ) ) ) ) ).
% graph_def
thf(fact_6933_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs2: list @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [I3: nat] :
( ( Uu3
= ( nth @ A @ Xs2 @ I3 ) )
& ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_6934_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or1337092689740270186AtMost @ code_integer @ ( plus_plus @ code_integer @ L @ ( one_one @ code_integer ) ) @ U )
= ( set_or3652927894154168847AtMost @ code_integer @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_integer
thf(fact_6935_set__drop__conv,axiom,
! [A: $tType,N: nat,L: list @ A] :
( ( set2 @ A @ ( drop @ A @ N @ L ) )
= ( collect @ A
@ ^ [Uu3: A] :
? [I3: nat] :
( ( Uu3
= ( nth @ A @ L @ I3 ) )
& ( ord_less_eq @ nat @ N @ I3 )
& ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) ) ) ) ) ).
% set_drop_conv
thf(fact_6936_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
! [Mima3: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima3 @ Deg @ TreeList @ Summary ) @ Deg3 )
= ( ( Deg = Deg3 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima3 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_6937_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( ? [Uu: $o,Uv: $o] :
( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( Xa2
= ( one_one @ nat ) ) )
=> ~ ! [Mima2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X3 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_6938_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) )
=> ( Xa2
= ( one_one @ nat ) ) ) )
=> ~ ! [Mima2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima2 @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X3 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_6939_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) )
=> ( Xa2
!= ( one_one @ nat ) ) ) )
=> ~ ! [Mima2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima2 @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_6940_option__Inf__Sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A5: set @ ( set @ ( option @ A ) )] :
( ord_less_eq @ ( option @ A ) @ ( complete_Inf_Inf @ ( option @ A ) @ ( image @ ( set @ ( option @ A ) ) @ ( option @ A ) @ ( complete_Sup_Sup @ ( option @ A ) ) @ A5 ) )
@ ( complete_Sup_Sup @ ( option @ A )
@ ( image @ ( set @ ( option @ A ) ) @ ( option @ A ) @ ( complete_Inf_Inf @ ( option @ A ) )
@ ( collect @ ( set @ ( option @ A ) )
@ ^ [Uu3: set @ ( option @ A )] :
? [F4: ( set @ ( option @ A ) ) > ( option @ A )] :
( ( Uu3
= ( image @ ( set @ ( option @ A ) ) @ ( option @ A ) @ F4 @ A5 ) )
& ! [X4: set @ ( option @ A )] :
( ( member @ ( set @ ( option @ A ) ) @ X4 @ A5 )
=> ( member @ ( option @ A ) @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% option_Inf_Sup
thf(fact_6941_Inf__Sup__le,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A5: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A5 ) )
@ ( complete_Sup_Sup @ A
@ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu3
= ( image @ ( set @ A ) @ A @ F4 @ A5 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A5 )
=> ( member @ A @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% Inf_Sup_le
thf(fact_6942_Sup__Inf__le,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: 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] :
? [F4: ( set @ A ) > A] :
( ( Uu3
= ( image @ ( set @ A ) @ A @ F4 @ A5 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A5 )
=> ( member @ A @ ( F4 @ X4 ) @ X4 ) ) ) ) ) )
@ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A5 ) ) ) ) ).
% Sup_Inf_le
thf(fact_6943_finite__Inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A5: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A5 ) )
@ ( complete_Sup_Sup @ A
@ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu3
= ( image @ ( set @ A ) @ A @ F4 @ A5 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A5 )
=> ( member @ A @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% finite_Inf_Sup
thf(fact_6944_Inf__filter__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( filter @ A ) )
= ( ^ [S6: set @ ( filter @ A )] :
( complete_Sup_Sup @ ( filter @ A )
@ ( collect @ ( filter @ A )
@ ^ [F11: filter @ A] :
! [X4: filter @ A] :
( ( member @ ( filter @ A ) @ X4 @ S6 )
=> ( ord_less_eq @ ( filter @ A ) @ F11 @ X4 ) ) ) ) ) ) ).
% Inf_filter_def
thf(fact_6945_Pow__Compl,axiom,
! [A: $tType,A5: set @ A] :
( ( pow @ A @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [B7: set @ A] :
( ( Uu3
= ( uminus_uminus @ ( set @ A ) @ B7 ) )
& ( member @ ( set @ A ) @ A5 @ ( pow @ A @ B7 ) ) ) ) ) ).
% Pow_Compl
thf(fact_6946_Sup__real__def,axiom,
( ( complete_Sup_Sup @ real )
= ( ^ [X8: set @ real] :
( ord_Least @ real
@ ^ [Z2: real] :
! [X4: real] :
( ( member @ real @ X4 @ X8 )
=> ( ord_less_eq @ real @ X4 @ Z2 ) ) ) ) ) ).
% Sup_real_def
thf(fact_6947_Sup__int__def,axiom,
( ( complete_Sup_Sup @ int )
= ( ^ [X8: set @ int] :
( the @ int
@ ^ [X4: int] :
( ( member @ int @ X4 @ X8 )
& ! [Y5: int] :
( ( member @ int @ Y5 @ X8 )
=> ( ord_less_eq @ int @ Y5 @ X4 ) ) ) ) ) ) ).
% Sup_int_def
thf(fact_6948_Union__maximal__sets,axiom,
! [A: $tType,F18: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F18 )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [T11: set @ A] :
( ( member @ ( set @ A ) @ T11 @ F18 )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ F18 )
=> ~ ( ord_less @ ( set @ A ) @ T11 @ X4 ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ F18 ) ) ) ).
% Union_maximal_sets
thf(fact_6949_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu: $o,Uv: $o] :
( ( X
= ( vEBT_Leaf @ Uu @ Uv ) )
=> ( ( Y
= ( Xa2
= ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) ) ) )
=> ~ ! [Mima2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( ( Deg2 = Xa2 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima2 @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
thf(fact_6950_set__bits__int__def,axiom,
( ( bit_bi4170147762399595738t_bits @ int )
= ( ^ [F4: nat > $o] :
( if @ int
@ ? [N6: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ N6 @ M5 )
=> ( ( F4 @ M5 )
= ( F4 @ N6 ) ) )
@ ( bit_ri4674362597316999326ke_bit @ int
@ ( ord_Least @ nat
@ ^ [N6: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ N6 @ M5 )
=> ( ( F4 @ M5 )
= ( F4 @ N6 ) ) ) )
@ ( groups4207007520872428315er_sum @ $o @ int @ ( zero_neq_one_of_bool @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) )
@ ( map @ nat @ $o @ F4
@ ( upt @ ( zero_zero @ nat )
@ ( suc
@ ( ord_Least @ nat
@ ^ [N6: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ N6 @ M5 )
=> ( ( F4 @ M5 )
= ( F4 @ N6 ) ) ) ) ) ) ) ) )
@ ( zero_zero @ int ) ) ) ) ).
% set_bits_int_def
thf(fact_6951_set__bits__int__unfold_H,axiom,
( ( bit_bi4170147762399595738t_bits @ int )
= ( ^ [F4: nat > $o] :
( if @ int
@ ? [N6: nat] :
! [N12: nat] :
( ( ord_less_eq @ nat @ N6 @ N12 )
=> ~ ( F4 @ N12 ) )
@ ( groups4207007520872428315er_sum @ $o @ int @ ( zero_neq_one_of_bool @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) )
@ ( map @ nat @ $o @ F4
@ ( upt @ ( zero_zero @ nat )
@ ( ord_Least @ nat
@ ^ [N6: nat] :
! [N12: nat] :
( ( ord_less_eq @ nat @ N6 @ N12 )
=> ~ ( F4 @ N12 ) ) ) ) ) )
@ ( if @ int
@ ? [N6: nat] :
! [N12: nat] :
( ( ord_less_eq @ nat @ N6 @ N12 )
=> ( F4 @ N12 ) )
@ ( bit_ri4674362597316999326ke_bit @ int
@ ( ord_Least @ nat
@ ^ [N6: nat] :
! [N12: nat] :
( ( ord_less_eq @ nat @ N6 @ N12 )
=> ( F4 @ N12 ) ) )
@ ( groups4207007520872428315er_sum @ $o @ int @ ( zero_neq_one_of_bool @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) )
@ ( append @ $o
@ ( map @ nat @ $o @ F4
@ ( upt @ ( zero_zero @ nat )
@ ( ord_Least @ nat
@ ^ [N6: nat] :
! [N12: nat] :
( ( ord_less_eq @ nat @ N6 @ N12 )
=> ( F4 @ N12 ) ) ) ) )
@ ( cons @ $o @ $true @ ( nil @ $o ) ) ) ) )
@ ( zero_zero @ int ) ) ) ) ) ).
% set_bits_int_unfold'
thf(fact_6952_set__bits__False__eq,axiom,
! [A: $tType] :
( ( bit_bi6583157726757044596ension @ A )
=> ( ( bit_bi4170147762399595738t_bits @ A
@ ^ [Uu3: nat] : $false )
= ( zero_zero @ A ) ) ) ).
% set_bits_False_eq
thf(fact_6953_min__ext__def,axiom,
! [A: $tType] :
( ( min_ext @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ^ [Uu3: product_prod @ ( set @ A ) @ ( set @ A )] :
? [X8: set @ A,Y10: set @ A] :
( ( Uu3
= ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X8 @ Y10 ) )
& ( X8
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ Y10 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ X8 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_6954_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),N6: nat] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= N6 )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N6 )
& ? [Xys2: list @ A,X4: A,Y5: A,Xs6: list @ A,Ys6: list @ A] :
( ( Xs2
= ( append @ A @ Xys2 @ ( cons @ A @ X4 @ Xs6 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y5 @ Ys6 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_6955_lexn_Osimps_I1_J,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( lexn @ A @ R2 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) ).
% lexn.simps(1)
thf(fact_6956_brk__rel__def,axiom,
! [B: $tType,A: $tType] :
( ( brk_rel @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
( sup_sup @ ( set @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) ) )
@ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
@ ^ [Uu3: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
? [X4: A,Y5: B] :
( ( Uu3
= ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $false @ X4 ) @ ( product_Pair @ $o @ B @ $false @ Y5 ) ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R4 ) ) )
@ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
@ ^ [Uu3: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
? [X4: A,Y5: B] :
( Uu3
= ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $true @ X4 ) @ ( product_Pair @ $o @ B @ $false @ Y5 ) ) ) ) ) ) ) ).
% brk_rel_def
thf(fact_6957_bin__last__set__bits,axiom,
! [F3: nat > $o] :
( ( bit_wf_set_bits_int @ F3 )
=> ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_bi4170147762399595738t_bits @ int @ F3 ) ) )
= ( F3 @ ( zero_zero @ nat ) ) ) ) ).
% bin_last_set_bits
thf(fact_6958_wf__set__bits__int__Suc,axiom,
! [F3: nat > $o] :
( ( bit_wf_set_bits_int
@ ^ [N6: nat] : ( F3 @ ( suc @ N6 ) ) )
= ( bit_wf_set_bits_int @ F3 ) ) ).
% wf_set_bits_int_Suc
thf(fact_6959_wf__set__bits__int__simps,axiom,
( bit_wf_set_bits_int
= ( ^ [F4: nat > $o] :
? [N6: nat] :
( ! [N12: nat] :
( ( ord_less_eq @ nat @ N6 @ N12 )
=> ~ ( F4 @ N12 ) )
| ! [N12: nat] :
( ( ord_less_eq @ nat @ N6 @ N12 )
=> ( F4 @ N12 ) ) ) ) ) ).
% wf_set_bits_int_simps
thf(fact_6960_zeros,axiom,
! [N: nat,F3: nat > $o] :
( ! [N5: nat] :
( ( ord_less_eq @ nat @ N @ N5 )
=> ~ ( F3 @ N5 ) )
=> ( bit_wf_set_bits_int @ F3 ) ) ).
% zeros
thf(fact_6961_wf__set__bits__int_Osimps,axiom,
( bit_wf_set_bits_int
= ( ^ [F4: nat > $o] :
( ? [N6: nat] :
! [N12: nat] :
( ( ord_less_eq @ nat @ N6 @ N12 )
=> ~ ( F4 @ N12 ) )
| ? [N6: nat] :
! [N12: nat] :
( ( ord_less_eq @ nat @ N6 @ N12 )
=> ( F4 @ N12 ) ) ) ) ) ).
% wf_set_bits_int.simps
thf(fact_6962_wf__set__bits__int_Ocases,axiom,
! [F3: nat > $o] :
( ( bit_wf_set_bits_int @ F3 )
=> ( ! [N2: nat] :
~ ! [N4: nat] :
( ( ord_less_eq @ nat @ N2 @ N4 )
=> ~ ( F3 @ N4 ) )
=> ~ ! [N2: nat] :
~ ! [N4: nat] :
( ( ord_less_eq @ nat @ N2 @ N4 )
=> ( F3 @ N4 ) ) ) ) ).
% wf_set_bits_int.cases
thf(fact_6963_ones,axiom,
! [N: nat,F3: nat > $o] :
( ! [N5: nat] :
( ( ord_less_eq @ nat @ N @ N5 )
=> ( F3 @ N5 ) )
=> ( bit_wf_set_bits_int @ F3 ) ) ).
% ones
thf(fact_6964_pairself__image__eq,axiom,
! [B: $tType,A: $tType,F3: B > A,P2: B > B > $o] :
( ( image @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F3 ) @ ( collect @ ( product_prod @ B @ B ) @ ( product_case_prod @ B @ B @ $o @ P2 ) ) )
= ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [A8: B,B8: B] :
( ( Uu3
= ( product_Pair @ A @ A @ ( F3 @ A8 ) @ ( F3 @ B8 ) ) )
& ( P2 @ A8 @ B8 ) ) ) ) ).
% pairself_image_eq
thf(fact_6965_map__to__set__upd,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),K: A,V3: B] :
( ( map_to_set @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( some @ B @ V3 ) ) )
= ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V3 )
@ ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( map_to_set @ A @ B @ M )
@ ( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [V7: B] :
( Uu3
= ( product_Pair @ A @ B @ K @ V7 ) ) ) ) ) ) ).
% map_to_set_upd
thf(fact_6966_map__to__set__empty,axiom,
! [B: $tType,A: $tType] :
( ( map_to_set @ A @ B
@ ^ [X4: A] : ( none @ B ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% map_to_set_empty
thf(fact_6967_pairself_Oelims,axiom,
! [B: $tType,A: $tType,X: A > B,Xa2: product_prod @ A @ A,Y: product_prod @ B @ B] :
( ( ( pairself @ A @ B @ X @ Xa2 )
= Y )
=> ~ ! [A4: A,B4: A] :
( ( Xa2
= ( product_Pair @ A @ A @ A4 @ B4 ) )
=> ( Y
!= ( product_Pair @ B @ B @ ( X @ A4 ) @ ( X @ B4 ) ) ) ) ) ).
% pairself.elims
thf(fact_6968_pairself_Osimps,axiom,
! [B: $tType,A: $tType,F3: A > B,A2: A,B2: A] :
( ( pairself @ A @ B @ F3 @ ( product_Pair @ A @ A @ A2 @ B2 ) )
= ( product_Pair @ B @ B @ ( F3 @ A2 ) @ ( F3 @ B2 ) ) ) ).
% pairself.simps
thf(fact_6969_map__to__set__inverse,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( set_to_map @ A @ B @ ( map_to_set @ A @ B @ M ) )
= M ) ).
% map_to_set_inverse
thf(fact_6970_map__to__set__empty__iff_I2_J,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
= ( map_to_set @ A @ B @ M ) )
= ( M
= ( ^ [X4: A] : ( none @ B ) ) ) ) ).
% map_to_set_empty_iff(2)
thf(fact_6971_map__to__set__empty__iff_I1_J,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( ( map_to_set @ A @ B @ M )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( M
= ( ^ [X4: A] : ( none @ B ) ) ) ) ).
% map_to_set_empty_iff(1)
thf(fact_6972_map__to__set__def,axiom,
! [B: $tType,A: $tType] :
( ( map_to_set @ A @ B )
= ( ^ [M5: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K3: A,V4: B] :
( ( M5 @ K3 )
= ( some @ B @ V4 ) ) ) ) ) ) ).
% map_to_set_def
thf(fact_6973_pairself_Opelims,axiom,
! [B: $tType,A: $tType,X: A > B,Xa2: product_prod @ A @ A,Y: product_prod @ B @ B] :
( ( ( pairself @ A @ B @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) @ ( pairself_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ X @ Xa2 ) )
=> ~ ! [A4: A,B4: A] :
( ( Xa2
= ( product_Pair @ A @ A @ A4 @ B4 ) )
=> ( ( Y
= ( product_Pair @ B @ B @ ( X @ A4 ) @ ( X @ B4 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) @ ( pairself_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ X @ ( product_Pair @ A @ A @ A4 @ B4 ) ) ) ) ) ) ) ).
% pairself.pelims
thf(fact_6974_int__set__bits__unfold__BIT,axiom,
! [F3: nat > $o] :
( ( bit_wf_set_bits_int @ F3 )
=> ( ( bit_bi4170147762399595738t_bits @ int @ F3 )
= ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( F3 @ ( zero_zero @ nat ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_bi4170147762399595738t_bits @ int @ ( comp @ nat @ $o @ nat @ F3 @ suc ) ) ) ) ) ) ).
% int_set_bits_unfold_BIT
thf(fact_6975_bin__rest__set__bits,axiom,
! [F3: nat > $o] :
( ( bit_wf_set_bits_int @ F3 )
=> ( ( divide_divide @ int @ ( bit_bi4170147762399595738t_bits @ int @ F3 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( bit_bi4170147762399595738t_bits @ int @ ( comp @ nat @ $o @ nat @ F3 @ suc ) ) ) ) ).
% bin_rest_set_bits
thf(fact_6976_surj__fun__eq,axiom,
! [B: $tType,C: $tType,A: $tType,F3: B > A,X6: set @ B,G1: A > C,G22: A > C] :
( ( ( image @ B @ A @ F3 @ X6 )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ X6 )
=> ( ( comp @ A @ C @ B @ G1 @ F3 @ X3 )
= ( comp @ A @ C @ B @ G22 @ F3 @ X3 ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_6977_comp__fun__commute__def,axiom,
! [B: $tType,A: $tType] :
( ( finite6289374366891150609ommute @ A @ B )
= ( ^ [F4: A > B > B] :
! [Y5: A,X4: A] :
( ( comp @ B @ B @ B @ ( F4 @ Y5 ) @ ( F4 @ X4 ) )
= ( comp @ B @ B @ B @ ( F4 @ X4 ) @ ( F4 @ Y5 ) ) ) ) ) ).
% comp_fun_commute_def
thf(fact_6978_comp__fun__commute_Ocomp__fun__commute,axiom,
! [B: $tType,A: $tType,F3: A > B > B,Y: A,X: A] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y ) @ ( F3 @ X ) )
= ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ).
% comp_fun_commute.comp_fun_commute
thf(fact_6979_comp__fun__commute_Ointro,axiom,
! [B: $tType,A: $tType,F3: A > B > B] :
( ! [Y3: A,X3: A] :
( ( comp @ B @ B @ B @ ( F3 @ Y3 ) @ ( F3 @ X3 ) )
= ( comp @ B @ B @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( finite6289374366891150609ommute @ A @ B @ F3 ) ) ).
% comp_fun_commute.intro
thf(fact_6980_comp__fun__idem__on__axioms_Ointro,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( comp @ B @ B @ B @ ( F3 @ X3 ) @ ( F3 @ X3 ) )
= ( F3 @ X3 ) ) )
=> ( finite4980608107308702382axioms @ A @ B @ S3 @ F3 ) ) ).
% comp_fun_idem_on_axioms.intro
thf(fact_6981_comp__fun__idem__on__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( finite4980608107308702382axioms @ A @ B )
= ( ^ [S6: set @ A,F4: A > B > B] :
! [X4: A] :
( ( member @ A @ X4 @ S6 )
=> ( ( comp @ B @ B @ B @ ( F4 @ X4 ) @ ( F4 @ X4 ) )
= ( F4 @ X4 ) ) ) ) ) ).
% comp_fun_idem_on_axioms_def
thf(fact_6982_comp__fun__commute__on_Ointro,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B] :
( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( member @ A @ Y3 @ S3 )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y3 ) @ ( F3 @ X3 ) )
= ( comp @ B @ B @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) ) ) )
=> ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 ) ) ).
% comp_fun_commute_on.intro
thf(fact_6983_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [A: $tType,B: $tType,C: $tType,S3: set @ A,F3: A > B > B,X: A,Y: A,G3: C > B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( member @ A @ X @ S3 )
=> ( ( member @ A @ Y @ S3 )
=> ( ( comp @ B @ B @ C @ ( F3 @ Y ) @ ( comp @ B @ B @ C @ ( F3 @ X ) @ G3 ) )
= ( comp @ B @ B @ C @ ( F3 @ X ) @ ( comp @ B @ B @ C @ ( F3 @ Y ) @ G3 ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_6984_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,X: A,Y: A] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( member @ A @ X @ S3 )
=> ( ( member @ A @ Y @ S3 )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y ) @ ( F3 @ X ) )
= ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
thf(fact_6985_comp__fun__commute__on__def,axiom,
! [B: $tType,A: $tType] :
( ( finite4664212375090638736ute_on @ A @ B )
= ( ^ [S6: set @ A,F4: A > B > B] :
! [X4: A,Y5: A] :
( ( member @ A @ X4 @ S6 )
=> ( ( member @ A @ Y5 @ S6 )
=> ( ( comp @ B @ B @ B @ ( F4 @ Y5 ) @ ( F4 @ X4 ) )
= ( comp @ B @ B @ B @ ( F4 @ X4 ) @ ( F4 @ Y5 ) ) ) ) ) ) ) ).
% comp_fun_commute_on_def
thf(fact_6986_fun__comp__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,G3: A > C,Fg: A > B] :
( ( ( comp @ C @ B @ A @ F3 @ G3 )
= Fg )
= ( ! [X4: A] :
( ( F3 @ ( G3 @ X4 ) )
= ( Fg @ X4 ) ) ) ) ).
% fun_comp_eq_conv
thf(fact_6987_comp__cong__right,axiom,
! [C: $tType,B: $tType,A: $tType,X: A > B,Y: A > B,F3: B > C] :
( ( X = Y )
=> ( ( comp @ B @ C @ A @ F3 @ X )
= ( comp @ B @ C @ A @ F3 @ Y ) ) ) ).
% comp_cong_right
thf(fact_6988_comp__cong__left,axiom,
! [B: $tType,A: $tType,C: $tType,X: A > B,Y: A > B,F3: C > A] :
( ( X = Y )
=> ( ( comp @ A @ B @ C @ X @ F3 )
= ( comp @ A @ B @ C @ Y @ F3 ) ) ) ).
% comp_cong_left
thf(fact_6989_comp__funpow,axiom,
! [B: $tType,A: $tType,N: nat,F3: A > A] :
( ( compow @ ( ( B > A ) > B > A ) @ N @ ( comp @ A @ A @ B @ F3 ) )
= ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N @ F3 ) ) ) ).
% comp_funpow
thf(fact_6990_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: A > B > B,X: A] :
( ( finite673082921795544331dem_on @ A @ B @ S3 @ F3 )
=> ( ( member @ A @ X @ S3 )
=> ( ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ X ) )
= ( F3 @ X ) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
thf(fact_6991_funpow__Suc__right,axiom,
! [A: $tType,N: nat,F3: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N ) @ F3 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ F3 ) ) ).
% funpow_Suc_right
thf(fact_6992_funpow_Osimps_I2_J,axiom,
! [A: $tType,N: nat,F3: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N ) @ F3 )
= ( comp @ A @ A @ A @ F3 @ ( compow @ ( A > A ) @ N @ F3 ) ) ) ).
% funpow.simps(2)
thf(fact_6993_funpow__add,axiom,
! [A: $tType,M: nat,N: nat,F3: A > A] :
( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M @ N ) @ F3 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M @ F3 ) @ ( compow @ ( A > A ) @ N @ F3 ) ) ) ).
% funpow_add
thf(fact_6994_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( comm_monoid_add @ B )
& ( comm_monoid_add @ A ) )
=> ! [H2: B > A,G3: C > B,A5: set @ C] :
( ( ( H2 @ ( zero_zero @ B ) )
= ( zero_zero @ A ) )
=> ( ! [X3: B,Y3: B] :
( ( H2 @ ( plus_plus @ B @ X3 @ Y3 ) )
= ( plus_plus @ A @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H2 @ G3 ) @ A5 )
= ( H2 @ ( groups7311177749621191930dd_sum @ C @ B @ G3 @ A5 ) ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_6995_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_Suc_lessThan_Suc_shift
thf(fact_6996_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_Suc_atMost_Suc_shift
thf(fact_6997_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_Suc_lessThan_Suc_shift
thf(fact_6998_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_Suc_atMost_Suc_shift
thf(fact_6999_sum_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,H2: B > C,G3: C > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X3: B,Y3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ( member @ B @ Y3 @ A5 )
=> ( ( X3 != Y3 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y3 ) )
=> ( ( G3 @ ( H2 @ X3 ) )
= ( zero_zero @ A ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G3 @ ( image @ B @ C @ H2 @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G3 @ H2 ) @ A5 ) ) ) ) ) ).
% sum.reindex_nontrivial
thf(fact_7000_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,H2: B > C,G3: C > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X3: B,Y3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ( member @ B @ Y3 @ A5 )
=> ( ( X3 != Y3 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y3 ) )
=> ( ( G3 @ ( H2 @ X3 ) )
= ( one_one @ A ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G3 @ ( image @ B @ C @ H2 @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G3 @ H2 ) @ A5 ) ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_7001_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [I6: set @ C,G3: A > B,F3: C > A] :
( ( finite_finite2 @ C @ I6 )
=> ( ! [I2: C] :
( ( member @ C @ I2 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G3 @ ( F3 @ I2 ) ) ) )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G3 @ ( image @ C @ A @ F3 @ I6 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G3 @ F3 ) @ I6 ) ) ) ) ) ).
% sum_image_le
thf(fact_7002_sum_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G3 @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% sum.atLeast0_lessThan_Suc_shift
thf(fact_7003_sum_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G3 @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc_shift
thf(fact_7004_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_lessThan_Suc_shift
thf(fact_7005_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc_shift
thf(fact_7006_sum_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% sum.atLeastLessThan_shift_0
thf(fact_7007_prod_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% prod.atLeastLessThan_shift_0
thf(fact_7008_sum_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G3
@ ^ [N6: nat] : ( minus_minus @ nat @ N6 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_lessThan_pred_shift
thf(fact_7009_sum_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G3
@ ^ [N6: nat] : ( minus_minus @ nat @ N6 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_atMost_pred_shift
thf(fact_7010_prod_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G3
@ ^ [N6: nat] : ( minus_minus @ nat @ N6 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_lessThan_pred_shift
thf(fact_7011_prod_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G3
@ ^ [N6: nat] : ( minus_minus @ nat @ N6 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_atMost_pred_shift
thf(fact_7012_sum_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% sum.atLeastAtMost_shift_0
thf(fact_7013_prod_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% prod.atLeastAtMost_shift_0
thf(fact_7014_lim__at__infinity__0,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,L: A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) )
= ( filterlim @ A @ A @ ( comp @ A @ A @ A @ F3 @ ( inverse_inverse @ A ) ) @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% lim_at_infinity_0
thf(fact_7015_tendsto__at__iff__sequentially,axiom,
! [C: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > C,A2: C,X: A,S: set @ A] :
( ( filterlim @ A @ C @ F3 @ ( topolo7230453075368039082e_nhds @ C @ A2 ) @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ! [X8: nat > A] :
( ! [I3: nat] : ( member @ A @ ( X8 @ I3 ) @ ( minus_minus @ ( set @ A ) @ S @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C @ ( comp @ A @ C @ nat @ F3 @ X8 ) @ ( topolo7230453075368039082e_nhds @ C @ A2 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% tendsto_at_iff_sequentially
thf(fact_7016_less__eq__char__simp,axiom,
! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C12: $o,C23: $o,C32: $o,C42: $o,C52: $o,C62: $o,C72: $o] :
( ( ord_less_eq @ char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) @ ( char2 @ C0 @ C12 @ C23 @ C32 @ C42 @ C52 @ C62 @ C72 ) )
= ( ord_less_eq @ nat
@ ( foldr @ $o @ nat
@ ^ [B8: $o,K3: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B8 ) @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( 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 ) ) ) ) ) ) ) ) )
@ ( zero_zero @ nat ) )
@ ( foldr @ $o @ nat
@ ^ [B8: $o,K3: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B8 ) @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
@ ( cons @ $o @ C0 @ ( cons @ $o @ C12 @ ( cons @ $o @ C23 @ ( cons @ $o @ C32 @ ( cons @ $o @ C42 @ ( cons @ $o @ C52 @ ( cons @ $o @ C62 @ ( cons @ $o @ C72 @ ( nil @ $o ) ) ) ) ) ) ) ) )
@ ( zero_zero @ nat ) ) ) ) ).
% less_eq_char_simp
thf(fact_7017_less__char__simp,axiom,
! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C12: $o,C23: $o,C32: $o,C42: $o,C52: $o,C62: $o,C72: $o] :
( ( ord_less @ char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) @ ( char2 @ C0 @ C12 @ C23 @ C32 @ C42 @ C52 @ C62 @ C72 ) )
= ( ord_less @ nat
@ ( foldr @ $o @ nat
@ ^ [B8: $o,K3: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B8 ) @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( 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 ) ) ) ) ) ) ) ) )
@ ( zero_zero @ nat ) )
@ ( foldr @ $o @ nat
@ ^ [B8: $o,K3: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B8 ) @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
@ ( cons @ $o @ C0 @ ( cons @ $o @ C12 @ ( cons @ $o @ C23 @ ( cons @ $o @ C32 @ ( cons @ $o @ C42 @ ( cons @ $o @ C52 @ ( cons @ $o @ C62 @ ( cons @ $o @ C72 @ ( nil @ $o ) ) ) ) ) ) ) ) )
@ ( zero_zero @ nat ) ) ) ) ).
% less_char_simp
thf(fact_7018_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F3: A > C,G3: D > B] :
( ( comp @ C @ ( set @ B ) @ A
@ ^ [Uu3: C] : ( bot_bot @ ( set @ B ) )
@ F3 )
= ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image @ D @ B @ G3 )
@ ^ [Uu3: A] : ( bot_bot @ ( set @ D ) ) ) ) ).
% empty_natural
thf(fact_7019_comp__fun__commute_Ocomp__comp__fun__commute,axiom,
! [B: $tType,A: $tType,C: $tType,F3: A > B > B,G3: C > A] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( finite6289374366891150609ommute @ C @ B @ ( comp @ A @ ( B > B ) @ C @ F3 @ G3 ) ) ) ).
% comp_fun_commute.comp_comp_fun_commute
thf(fact_7020_card_Ocomp__fun__commute__on,axiom,
( ( comp @ nat @ nat @ nat @ suc @ suc )
= ( comp @ nat @ nat @ nat @ suc @ suc ) ) ).
% card.comp_fun_commute_on
thf(fact_7021_set__oo__map__alt,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( comp @ ( list @ B ) @ ( set @ B ) @ ( list @ A ) @ ( set2 @ B ) @ ( map @ A @ B @ F3 ) )
= ( ^ [L4: list @ A] : ( image @ A @ B @ F3 @ ( set2 @ A @ L4 ) ) ) ) ).
% set_oo_map_alt
thf(fact_7022_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B5: set @ ( set @ B ),G3: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ B5 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [A18: set @ B] :
( ( member @ ( set @ B ) @ A18 @ B5 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B5 )
=> ( ( A18 != A25 )
=> ! [X3: B] :
( ( member @ B @ X3 @ A18 )
=> ( ( member @ B @ X3 @ A25 )
=> ( ( G3 @ X3 )
= ( one_one @ A ) ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( complete_Sup_Sup @ ( set @ B ) @ B5 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G3 @ B5 ) ) ) ) ) ).
% prod.Union_comp
thf(fact_7023_sum_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups7311177749621191930dd_sum @ B @ A )
= ( ^ [G4: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( plus_plus @ A ) @ G4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% sum.eq_fold
thf(fact_7024_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
! [B: $tType,A: $tType,C: $tType,S3: set @ A,F3: A > B > B,G3: C > A,R3: set @ C] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ G3 @ ( top_top @ ( set @ C ) ) ) @ S3 )
=> ( finite4664212375090638736ute_on @ C @ B @ R3 @ ( comp @ A @ ( B > B ) @ C @ F3 @ G3 ) ) ) ) ).
% comp_fun_commute_on.comp_comp_fun_commute_on
thf(fact_7025_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
! [B: $tType,A: $tType,C: $tType,S3: set @ A,F3: A > B > B,G3: C > A,R3: set @ C] :
( ( finite673082921795544331dem_on @ A @ B @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ G3 @ ( top_top @ ( set @ C ) ) ) @ S3 )
=> ( finite673082921795544331dem_on @ C @ B @ R3 @ ( comp @ A @ ( B > B ) @ C @ F3 @ G3 ) ) ) ) ).
% comp_fun_idem_on.comp_comp_fun_idem_on
thf(fact_7026_infinite__int__iff__infinite__nat__abs,axiom,
! [S3: set @ int] :
( ( ~ ( finite_finite2 @ int @ S3 ) )
= ( ~ ( finite_finite2 @ nat @ ( image @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ S3 ) ) ) ) ).
% infinite_int_iff_infinite_nat_abs
thf(fact_7027_prod_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C4: set @ ( set @ B ),G3: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C4 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C4 )
=> ! [Xa4: set @ B] :
( ( member @ ( set @ B ) @ Xa4 @ C4 )
=> ( ( X3 != Xa4 )
=> ( ( inf_inf @ ( set @ B ) @ X3 @ Xa4 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( complete_Sup_Sup @ ( set @ B ) @ C4 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G3 @ C4 ) ) ) ) ) ).
% prod.Union_disjoint
thf(fact_7028_sup__SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B5: A,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( sup_sup @ A @ B5 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A5 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F3 ) @ B5 @ A5 ) ) ) ) ).
% sup_SUP_fold_sup
thf(fact_7029_inf__INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B5: A,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( inf_inf @ A @ B5 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A5 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F3 ) @ B5 @ A5 ) ) ) ) ).
% inf_INF_fold_inf
thf(fact_7030_sum_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B5: set @ ( set @ B ),G3: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ B5 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [A18: set @ B] :
( ( member @ ( set @ B ) @ A18 @ B5 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B5 )
=> ( ( A18 != A25 )
=> ! [X3: B] :
( ( member @ B @ X3 @ A18 )
=> ( ( member @ B @ X3 @ A25 )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( complete_Sup_Sup @ ( set @ B ) @ B5 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G3 @ B5 ) ) ) ) ) ).
% sum.Union_comp
thf(fact_7031_SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A5 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F3 ) @ ( bot_bot @ A ) @ A5 ) ) ) ) ).
% SUP_fold_sup
thf(fact_7032_INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A5 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F3 ) @ ( top_top @ A ) @ A5 ) ) ) ) ).
% INF_fold_inf
thf(fact_7033_sum_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C4: set @ ( set @ B ),G3: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C4 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C4 )
=> ! [Xa4: set @ B] :
( ( member @ ( set @ B ) @ Xa4 @ C4 )
=> ( ( X3 != Xa4 )
=> ( ( inf_inf @ ( set @ B ) @ X3 @ Xa4 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( complete_Sup_Sup @ ( set @ B ) @ C4 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G3 @ C4 ) ) ) ) ) ).
% sum.Union_disjoint
thf(fact_7034_filterlim__at__top__iff__inverse__0,axiom,
! [A: $tType,F3: A > real,F5: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X4 ) )
@ F5 )
=> ( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F5 )
= ( filterlim @ A @ real @ ( comp @ real @ real @ A @ ( inverse_inverse @ real ) @ F3 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) ) ) ).
% filterlim_at_top_iff_inverse_0
thf(fact_7035_Code__Numeral_Onegative__def,axiom,
( code_negative
= ( comp @ code_integer @ code_integer @ num @ ( uminus_uminus @ code_integer ) @ ( numeral_numeral @ code_integer ) ) ) ).
% Code_Numeral.negative_def
thf(fact_7036_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G3: B > A,Y7: set @ B,X6: set @ A,F5: filter @ B,F3: A > C] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G3 @ Y7 ) @ X6 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( member @ B @ X4 @ Y7 )
@ F5 )
=> ( ( map_filter_on @ A @ C @ X6 @ F3 @ ( map_filter_on @ B @ A @ Y7 @ G3 @ F5 ) )
= ( map_filter_on @ B @ C @ Y7 @ ( comp @ A @ C @ B @ F3 @ G3 ) @ F5 ) ) ) ) ).
% map_filter_on_comp
thf(fact_7037_cauchy__filter__metric,axiom,
! [A: $tType] :
( ( ( real_V768167426530841204y_dist @ A )
& ( topolo7287701948861334536_space @ A ) )
=> ( ( topolo6773858410816713723filter @ A )
= ( ^ [F11: filter @ A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [P5: A > $o] :
( ( eventually @ A @ P5 @ F11 )
& ! [X4: A,Y5: A] :
( ( ( P5 @ X4 )
& ( P5 @ Y5 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ Y5 ) @ E4 ) ) ) ) ) ) ) ).
% cauchy_filter_metric
thf(fact_7038_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L4: 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 )
@ ( L4
= ( 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 ) @ L4
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( ( sgn_sgn @ code_integer @ K3 )
= ( sgn_sgn @ code_integer @ L4 ) )
@ ( code_divmod_abs @ K3 @ L4 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S8: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S8
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L4 ) @ S8 ) ) )
@ ( code_divmod_abs @ K3 @ L4 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_7039_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > B,X: A,Y: C] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_Pair @ A @ C @ X @ Y ) )
= ( product_Pair @ A @ B @ X @ ( F3 @ Y ) ) ) ).
% apsnd_conv
thf(fact_7040_apsnd__compose,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F3: C > B,G3: D > C,X: product_prod @ A @ D] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_apsnd @ D @ C @ A @ G3 @ X ) )
= ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F3 @ G3 ) @ X ) ) ).
% apsnd_compose
thf(fact_7041_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_7042_nhds__imp__cauchy__filter,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [F5: filter @ A,X: A] :
( ( ord_less_eq @ ( filter @ A ) @ F5 @ ( topolo7230453075368039082e_nhds @ A @ X ) )
=> ( topolo6773858410816713723filter @ A @ F5 ) ) ) ).
% nhds_imp_cauchy_filter
thf(fact_7043_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_7044_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L4: 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 ) @ L4 )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ ( code_divmod_abs @ K3 @ L4 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S8: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S8
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L4 @ S8 ) ) )
@ ( code_divmod_abs @ K3 @ L4 ) ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L4
= ( 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 @ L4 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S8: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S8
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L4 ) @ S8 ) ) )
@ ( code_divmod_abs @ K3 @ L4 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_7045_divmod__abs__def,axiom,
( code_divmod_abs
= ( ^ [K3: code_integer,L4: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L4 ) ) @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L4 ) ) ) ) ) ).
% divmod_abs_def
thf(fact_7046_bit__cut__integer__code,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( if @ ( product_prod @ code_integer @ $o )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ $o @ ( zero_zero @ code_integer ) @ $false )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ $o )
@ ^ [R5: code_integer,S8: code_integer] :
( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ R5 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ S8 ) )
@ ( S8
= ( one_one @ code_integer ) ) )
@ ( code_divmod_abs @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_7047_alignUp__div__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: nat,N: nat,X: word @ A,A2: word @ A] :
( ( ord_less @ nat @ K @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) )
=> ( ( X
= ( times_times @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( semiring_1_of_nat @ ( word @ A ) @ K ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ X )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ( modulo_modulo @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ ( semiring_1_of_nat @ ( word @ A ) @ K ) ) ) ) ) ) ) ) ).
% alignUp_div_helper
thf(fact_7048_CHAR__eq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) ) ) ).
% CHAR_eq_0
thf(fact_7049_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_7050_less__eq__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less_eq @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% less_eq_word_numeral_numeral
thf(fact_7051_less__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% less_word_numeral_numeral
thf(fact_7052_bit__numeral__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W2 ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W2 ) @ N ) ) ) ) ).
% bit_numeral_word_iff
thf(fact_7053_unat__lt2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% unat_lt2p
thf(fact_7054_uint__bounded,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A] : ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% uint_bounded
thf(fact_7055_uint__lt2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% uint_lt2p
thf(fact_7056_exp__eq__zero__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ ( word @ A ) ) )
= ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) ) ).
% exp_eq_zero_iff
thf(fact_7057_signed__take__bit__word__Suc__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ ( word @ A ) @ ( suc @ N ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% signed_take_bit_word_Suc_numeral
thf(fact_7058_scast__sbintr,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W2: num] :
( ( ring_1_signed @ A @ ( word @ B ) @ ( numeral_numeral @ ( word @ A ) @ W2 ) )
= ( ring_1_of_int @ ( word @ B ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ int @ W2 ) ) ) ) ) ).
% scast_sbintr
thf(fact_7059_less__eq__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% less_eq_word_minus_numeral_numeral
thf(fact_7060_less__eq__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less_eq @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% less_eq_word_numeral_minus_numeral
thf(fact_7061_less__eq__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% less_eq_word_minus_numeral_minus_numeral
thf(fact_7062_less__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% less_word_minus_numeral_minus_numeral
thf(fact_7063_less__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% less_word_numeral_minus_numeral
thf(fact_7064_less__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% less_word_minus_numeral_numeral
thf(fact_7065_bit__neg__numeral__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W2 ) ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ N ) ) ) ) ).
% bit_neg_numeral_word_iff
thf(fact_7066_unat__power__lower,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% unat_power_lower
thf(fact_7067_signed__take__bit__word__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ ( word @ A ) @ ( suc @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ) ).
% signed_take_bit_word_Suc_minus_numeral
thf(fact_7068_word__less__sub__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% word_less_sub_le
thf(fact_7069_less__word__numeral__minus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num] :
( ( ord_less @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) ) ) ) ).
% less_word_numeral_minus_1
thf(fact_7070_less__word__minus__numeral__minus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num] :
( ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) ) ) ) ).
% less_word_minus_numeral_minus_1
thf(fact_7071_word__of__nat__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat] :
( ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ M ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) )
= ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) ) ) ).
% word_of_nat_less_iff
thf(fact_7072_two__power__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N )
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
= ( N = M ) ) ) ) ) ).
% two_power_eq
thf(fact_7073_signed__take__bit__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A ) )
=> ! [N: nat,W2: word @ B] :
( ( ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ring_1_signed @ B @ A @ ( bit_se2584673776208193580ke_bit @ ( word @ B ) @ N @ W2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( ring_1_signed @ B @ A @ W2 ) ) ) )
& ( ~ ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ring_1_signed @ B @ A @ ( bit_se2584673776208193580ke_bit @ ( word @ B ) @ N @ W2 ) )
= ( ring_1_signed @ B @ A @ W2 ) ) ) ) ) ).
% signed_take_bit_eq
thf(fact_7074_word__of__int__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: int,L: int] :
( ( ord_less @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ K ) @ ( ring_1_of_int @ ( word @ A ) @ L ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L ) ) ) ) ).
% word_of_int_less_iff
thf(fact_7075_ucast__less__ucast__weak,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) )
= ( ord_less @ ( word @ A ) @ X @ Y ) ) ) ) ).
% ucast_less_ucast_weak
thf(fact_7076_ucast__ucast__id,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A] :
( ( ord_less @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) )
= X ) ) ) ).
% ucast_ucast_id
thf(fact_7077_test__bit__1_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% test_bit_1'
thf(fact_7078_bit__word__eqI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A2 @ N2 )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ B2 @ N2 ) ) )
=> ( A2 = B2 ) ) ) ).
% bit_word_eqI
thf(fact_7079_bit__imp__le__length,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ N )
=> ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% bit_imp_le_length
thf(fact_7080_test__bit__conj__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,M: nat] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ M )
& ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ M ) ) ) ).
% test_bit_conj_lt
thf(fact_7081_word__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ^ [Y6: word @ A,Z4: word @ A] : Y6 = Z4 )
= ( ^ [X4: word @ A,Y5: word @ A] :
! [N6: nat] :
( ( ord_less @ nat @ N6 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X4 @ N6 )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y5 @ N6 ) ) ) ) ) ) ).
% word_eq_iff
thf(fact_7082_test__bit__wi,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ X ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ X @ N ) ) ) ) ).
% test_bit_wi
thf(fact_7083_bit__word__of__int__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ K ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ) ).
% bit_word_of_int_iff
thf(fact_7084_max__test__bit,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ N )
= ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% max_test_bit
thf(fact_7085_bit__uint__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ N ) ) ) ) ).
% bit_uint_iff
thf(fact_7086_bit__ucast__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [A2: word @ B,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ A2 ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ A2 @ N ) ) ) ) ).
% bit_ucast_iff
thf(fact_7087_bin__nth__uint__imp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ N )
=> ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% bin_nth_uint_imp
thf(fact_7088_bit__word__ucast__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W2: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ W2 ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ N ) ) ) ) ).
% bit_word_ucast_iff
thf(fact_7089_test__bit__bin,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [W3: word @ A,N6: nat] :
( ( ord_less @ nat @ N6 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W3 ) @ N6 ) ) ) ) ) ).
% test_bit_bin
thf(fact_7090_nth__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W2: word @ B,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ W2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W2 @ N )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% nth_ucast
thf(fact_7091_nth__slice,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [N: nat,W2: word @ B,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( slice2 @ B @ A @ N @ W2 ) @ M )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W2 @ ( plus_plus @ nat @ M @ N ) )
& ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% nth_slice
thf(fact_7092_word__of__nat__less__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ M ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) )
= ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) ) ) ).
% word_of_nat_less_eq_iff
thf(fact_7093_word__cat__inj,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [A2: word @ A,B2: word @ B,C3: word @ A,D2: word @ B] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) @ ( type_len0_len_of @ C @ ( type2 @ C ) ) )
=> ( ( ( word_cat @ A @ B @ C @ A2 @ B2 )
= ( word_cat @ A @ B @ C @ C3 @ D2 ) )
= ( ( A2 = C3 )
& ( B2 = D2 ) ) ) ) ) ).
% word_cat_inj
thf(fact_7094_ucast__ucast__add,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ A,Y: word @ B] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( plus_plus @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ Y ) )
= ( plus_plus @ ( word @ A ) @ X @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) ) ) ) ) ).
% ucast_ucast_add
thf(fact_7095_wi__bintr,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W2: int] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ W2 ) )
= ( ring_1_of_int @ ( word @ A ) @ W2 ) ) ) ) ).
% wi_bintr
thf(fact_7096_word__of__int__less__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: int,L: int] :
( ( ord_less_eq @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ K ) @ ( ring_1_of_int @ ( word @ A ) @ L ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L ) ) ) ) ).
% word_of_int_less_eq_iff
thf(fact_7097_take__bit__word__beyond__length__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W2: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ N @ W2 )
= W2 ) ) ) ).
% take_bit_word_beyond_length_eq
thf(fact_7098_up__scast__inj__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ( ring_1_signed @ A @ ( word @ B ) @ X )
= ( ring_1_signed @ A @ ( word @ B ) @ Y ) )
= ( X = Y ) ) ) ) ).
% up_scast_inj_eq
thf(fact_7099_up__scast__inj,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ( ring_1_signed @ A @ ( word @ B ) @ X )
= ( ring_1_signed @ A @ ( word @ B ) @ Y ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ X ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( X = Y ) ) ) ) ).
% up_scast_inj
thf(fact_7100_less__ucast__ucast__less,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ A,Y: word @ B] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) )
=> ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ Y ) ) ) ) ).
% less_ucast_ucast_less
thf(fact_7101_ucast__less__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) )
= ( ord_less @ ( word @ A ) @ X @ Y ) ) ) ) ).
% ucast_less_ucast
thf(fact_7102_ucast__up__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) ) ) ) ) ).
% ucast_up_mono
thf(fact_7103_bintr__uint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W2: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( semiring_1_unsigned @ A @ int @ W2 ) )
= ( semiring_1_unsigned @ A @ int @ W2 ) ) ) ) ).
% bintr_uint
thf(fact_7104_unat__ucast__up__simp,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( semiring_1_unsigned @ B @ nat @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) )
= ( semiring_1_unsigned @ A @ nat @ X ) ) ) ) ).
% unat_ucast_up_simp
thf(fact_7105_eq__ucast__ucast__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ A,Y: word @ B] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( X
= ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X )
= Y ) ) ) ) ).
% eq_ucast_ucast_eq
thf(fact_7106_ucast__up__mono__le,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) ) ) ) ) ).
% ucast_up_mono_le
thf(fact_7107_up__ucast__inj__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X )
= ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) )
= ( X = Y ) ) ) ) ).
% up_ucast_inj_eq
thf(fact_7108_ucast__ucast__eq,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [X: word @ A,Y: word @ B] :
( ( ( semiring_1_unsigned @ A @ ( word @ C ) @ X )
= ( semiring_1_unsigned @ A @ ( word @ C ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) ) )
=> ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ C @ ( type2 @ C ) ) )
=> ( X
= ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) ) ) ) ) ) ).
% ucast_ucast_eq
thf(fact_7109_ucast__le__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) )
= ( ord_less_eq @ ( word @ A ) @ X @ Y ) ) ) ) ).
% ucast_le_ucast
thf(fact_7110_up__ucast__inj,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X )
= ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) )
=> ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( X = Y ) ) ) ) ).
% up_ucast_inj
thf(fact_7111_word__of__int__bin__cat__eq__iff,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B )
& ( type_len @ C ) )
=> ! [B2: word @ B,A2: word @ A,D2: word @ B,C3: word @ A] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) @ ( type_len0_len_of @ C @ ( type2 @ C ) ) )
=> ( ( ( ring_1_of_int @ ( word @ C ) @ ( bit_concat_bit @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( semiring_1_unsigned @ B @ int @ B2 ) @ ( semiring_1_unsigned @ A @ int @ A2 ) ) )
= ( ring_1_of_int @ ( word @ C ) @ ( bit_concat_bit @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( semiring_1_unsigned @ B @ int @ D2 ) @ ( semiring_1_unsigned @ A @ int @ C3 ) ) ) )
= ( ( B2 = D2 )
& ( A2 = C3 ) ) ) ) ) ).
% word_of_int_bin_cat_eq_iff
thf(fact_7112_bit__set__bits__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: nat > $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_bi4170147762399595738t_bits @ ( word @ A ) @ P2 ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( P2 @ N ) ) ) ) ).
% bit_set_bits_word_iff
thf(fact_7113_word__test__bit__set__bits,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F3: nat > $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_bi4170147762399595738t_bits @ ( word @ A ) @ F3 ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( F3 @ N ) ) ) ) ).
% word_test_bit_set_bits
thf(fact_7114_bin__nth__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W2: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W2 ) @ N )
= ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% bin_nth_sint
thf(fact_7115_ucast__sub__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ X )
=> ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) ) ) ) ) ) ).
% ucast_sub_ucast
thf(fact_7116_bit__word__cat__iff,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [V3: word @ A,W2: word @ B,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ ( word_cat @ A @ B @ C @ V3 @ W2 ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ C @ ( type2 @ C ) ) )
& ( ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W2 @ N ) )
& ( ~ ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V3 @ ( minus_minus @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ) ) ) ).
% bit_word_cat_iff
thf(fact_7117_signed__scast__eq,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A )
& ( type_len @ C ) )
=> ! [W2: word @ B] :
( ( ring_1_signed @ C @ A @ ( ring_1_signed @ B @ ( word @ C ) @ W2 ) )
= ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ ( type_len0_len_of @ C @ ( type2 @ C ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( ring_1_signed @ B @ A @ W2 ) ) ) ) ).
% signed_scast_eq
thf(fact_7118_signed__take__bit__decr__length__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( bit_ri3973907225187159222ations @ B )
& ( type_len @ A ) )
=> ! [K: B,L: B] :
( ( ( bit_ri4674362597316999326ke_bit @ B @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ K )
= ( bit_ri4674362597316999326ke_bit @ B @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L ) )
= ( ( bit_se2584673776208193580ke_bit @ B @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K )
= ( bit_se2584673776208193580ke_bit @ B @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L ) ) ) ) ).
% signed_take_bit_decr_length_iff
thf(fact_7119_size__0__same,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,V3: word @ A] :
( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) )
=> ( W2 = V3 ) ) ) ).
% size_0_same
thf(fact_7120_unsigned__less,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( unique1627219031080169319umeral @ A ) )
=> ! [W2: word @ B] : ( ord_less @ A @ ( semiring_1_unsigned @ B @ A @ W2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ).
% unsigned_less
thf(fact_7121_not__degenerate__imp__2__neq__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% not_degenerate_imp_2_neq_0
thf(fact_7122_bit__word__scast__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W2: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( ring_1_signed @ A @ ( word @ B ) @ W2 ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ N )
| ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% bit_word_scast_iff
thf(fact_7123_word__of__nat__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ( semiring_1_of_nat @ ( word @ A ) @ X )
= ( semiring_1_of_nat @ ( word @ A ) @ Y ) )
=> ( X = Y ) ) ) ) ) ).
% word_of_nat_inj
thf(fact_7124_of__nat__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ( semiring_1_of_nat @ ( word @ A ) @ X )
= ( semiring_1_of_nat @ ( word @ A ) @ Y ) )
= ( X = Y ) ) ) ) ) ).
% of_nat_inj
thf(fact_7125_word__nchotomy,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W4: word @ A] :
? [N2: nat] :
( ( W4
= ( semiring_1_of_nat @ ( word @ A ) @ N2 ) )
& ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% word_nchotomy
thf(fact_7126_word__nat__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ! [N2: nat] :
( ( X
= ( semiring_1_of_nat @ ( word @ A ) @ N2 ) )
=> ~ ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% word_nat_cases
thf(fact_7127_More__Word_Opower__not__zero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% More_Word.power_not_zero
thf(fact_7128_word__power__increasing,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ X ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Y ) )
=> ( ( ord_less @ nat @ X @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ nat @ Y @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ nat @ X @ Y ) ) ) ) ) ).
% word_power_increasing
thf(fact_7129_power__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% power_overflow
thf(fact_7130_nth__w2p__same,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ N )
= ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% nth_w2p_same
thf(fact_7131_nth__w2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ M )
= ( ( M = N )
& ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% nth_w2p
thf(fact_7132_take__bit__word__eq__self__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W2: word @ A] :
( ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ N @ W2 )
= W2 )
= ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
| ( ord_less @ ( word @ A ) @ W2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% take_bit_word_eq_self_iff
thf(fact_7133_sint__uint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_signed @ A @ int )
= ( ^ [W3: word @ A] : ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( semiring_1_unsigned @ A @ int @ W3 ) ) ) ) ) ).
% sint_uint
thf(fact_7134_ucast__ucast__len,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) )
= X ) ) ) ).
% ucast_ucast_len
thf(fact_7135_ucast__mono,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ B,Y: word @ B] :
( ( ord_less @ ( word @ B ) @ X @ Y )
=> ( ( ord_less @ ( word @ B ) @ Y @ ( power_power @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ X ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) ) ) ) ) ).
% ucast_mono
thf(fact_7136_nth__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W2 ) @ N )
= ( ( ( ord_less @ nat @ N @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ N ) )
& ( ~ ( ord_less @ nat @ N @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% nth_sint
thf(fact_7137_bit__sint__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W2 ) @ N )
= ( ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ N ) ) ) ) ).
% bit_sint_iff
thf(fact_7138_less__Suc__unat__less__bound,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less @ nat @ N @ ( suc @ ( semiring_1_unsigned @ A @ nat @ X ) ) )
=> ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% less_Suc_unat_less_bound
thf(fact_7139_uint__2__id,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) )
= ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% uint_2_id
thf(fact_7140_lt2p__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W2: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% lt2p_lem
thf(fact_7141_power__le__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat] :
( ( ord_less_eq @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less_eq @ nat @ N @ M ) ) ) ) ) ).
% power_le_mono
thf(fact_7142_two__power__increasing,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% two_power_increasing
thf(fact_7143_of__nat__mono__maybe_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y @ X )
= ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ Y ) @ ( semiring_1_of_nat @ ( word @ A ) @ X ) ) ) ) ) ) ).
% of_nat_mono_maybe'
thf(fact_7144_of__nat__mono__maybe,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y @ X )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ Y ) @ ( semiring_1_of_nat @ ( word @ A ) @ X ) ) ) ) ) ).
% of_nat_mono_maybe
thf(fact_7145_unat__of__nat__len,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ X ) )
= X ) ) ) ).
% unat_of_nat_len
thf(fact_7146_unat__eq__of__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ nat @ X )
= N )
= ( X
= ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% unat_eq_of_nat
thf(fact_7147_unat__split,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: nat > $o,X: word @ A] :
( ( P2 @ ( semiring_1_unsigned @ A @ nat @ X ) )
= ( ! [N6: nat] :
( ( ( ( semiring_1_of_nat @ ( word @ A ) @ N6 )
= X )
& ( ord_less @ nat @ N6 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( P2 @ N6 ) ) ) ) ) ).
% unat_split
thf(fact_7148_of__nat__inverse,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [R2: nat,A2: word @ A] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ R2 )
= A2 )
=> ( ( ord_less @ nat @ R2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ A2 )
= R2 ) ) ) ) ).
% of_nat_inverse
thf(fact_7149_unat__split__asm,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: nat > $o,X: word @ A] :
( ( P2 @ ( semiring_1_unsigned @ A @ nat @ X ) )
= ( ~ ? [N6: nat] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N6 )
= X )
& ( ord_less @ nat @ N6 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
& ~ ( P2 @ N6 ) ) ) ) ) ).
% unat_split_asm
thf(fact_7150_x__less__2__0__1_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
!= ( one_one @ nat ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) )
=> ( ( X
= ( zero_zero @ ( word @ A ) ) )
| ( X
= ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% x_less_2_0_1'
thf(fact_7151_test__bit__2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ M )
= ( ( M = N )
& ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% test_bit_2p
thf(fact_7152_word__1__le__power,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% word_1_le_power
thf(fact_7153_ucast__of__nat__small,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( semiring_1_of_nat @ ( word @ A ) @ X ) )
= ( semiring_1_of_nat @ ( word @ B ) @ X ) ) ) ) ).
% ucast_of_nat_small
thf(fact_7154_uint__sub__lt2p,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ B] : ( ord_less @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ B @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% uint_sub_lt2p
thf(fact_7155_p2__gt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% p2_gt_0
thf(fact_7156_bit__last__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ord_less @ int @ ( ring_1_signed @ A @ int @ W2 ) @ ( zero_zero @ int ) ) ) ) ).
% bit_last_iff
thf(fact_7157_scast__1_H,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ( ( ring_1_signed @ A @ ( word @ B ) @ ( one_one @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ B ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( one_one @ int ) ) ) ) ) ).
% scast_1'
thf(fact_7158_ucast__less,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ B] :
( ( ord_less @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ X ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ) ).
% ucast_less
thf(fact_7159_signed__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [N: int] :
( ( ring_1_signed @ B @ A @ ( ring_1_of_int @ ( word @ B ) @ N ) )
= ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ N ) ) ) ) ).
% signed_of_int
thf(fact_7160_of__nat__n__less__equal__power__2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% of_nat_n_less_equal_power_2
thf(fact_7161_signed__ucast__eq,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A )
& ( type_len @ C ) )
=> ! [W2: word @ B] :
( ( ring_1_signed @ C @ A @ ( semiring_1_unsigned @ B @ ( word @ C ) @ W2 ) )
= ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ ( type_len0_len_of @ C @ ( type2 @ C ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( semiring_1_unsigned @ B @ A @ W2 ) ) ) ) ).
% signed_ucast_eq
thf(fact_7162_upper__trivial,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( X
!= ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% upper_trivial
thf(fact_7163_word__power__less__diff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,Q3: word @ A,M: nat] :
( ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ Q3 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less @ ( word @ A ) @ Q3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) )
=> ( ord_less @ ( word @ A ) @ Q3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ) ).
% word_power_less_diff
thf(fact_7164_ucast__mono__le,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ( ord_less @ ( word @ A ) @ Y @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
=> ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) ) ) ) ) ).
% ucast_mono_le
thf(fact_7165_unat__add__lem_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ) ).
% unat_add_lem'
thf(fact_7166_unat__add__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ) ).
% unat_add_lem
thf(fact_7167_More__Word_Oof__nat__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N )
= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% More_Word.of_nat_0
thf(fact_7168_Word_Oof__nat__neq__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less @ nat @ K @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_of_nat @ ( word @ A ) @ K )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% Word.of_nat_neq_0
thf(fact_7169_of__nat__mono__maybe__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less_eq @ nat @ Y @ X )
= ( ord_less_eq @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ Y ) @ ( semiring_1_of_nat @ ( word @ A ) @ X ) ) ) ) ) ) ).
% of_nat_mono_maybe_le
thf(fact_7170_unat__word__ariths_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ nat @ ( zero_zero @ ( word @ A ) ) )
= ( modulo_modulo @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_word_ariths(4)
thf(fact_7171_uint__range_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_unsigned @ A @ int @ X ) )
& ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_range'
thf(fact_7172_sint__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( ord_less @ int @ ( ring_1_signed @ A @ int @ X ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ).
% sint_lt
thf(fact_7173_ucast__mono__le_H,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
=> ( ( ord_less @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) ) ) ) ) ) ).
% ucast_mono_le'
thf(fact_7174_word__int__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ! [N2: int] :
( ( X
= ( ring_1_of_int @ ( word @ A ) @ N2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ~ ( ord_less @ int @ N2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% word_int_cases
thf(fact_7175_word__of__int__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int,Y: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
& ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( ( ( ring_1_of_int @ ( word @ A ) @ X )
= ( ring_1_of_int @ ( word @ A ) @ Y ) )
= ( X = Y ) ) ) ) ) ).
% word_of_int_inj
thf(fact_7176_unat__mult__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ nat @ ( times_times @ ( word @ A ) @ X @ Y ) )
= ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ) ).
% unat_mult_lem
thf(fact_7177_unat__ucast__no__overflow__le,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [B2: word @ B,F3: word @ A] :
( ( ord_less @ nat @ ( semiring_1_unsigned @ B @ nat @ B2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ F3 ) @ B2 )
= ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ F3 ) @ ( semiring_1_unsigned @ B @ nat @ B2 ) ) ) ) ) ) ).
% unat_ucast_no_overflow_le
thf(fact_7178_uint__m2p__not__non__neg,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_m2p_not_non_neg
thf(fact_7179_uint__m2p__neg,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( ord_less @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( zero_zero @ int ) ) ) ).
% uint_m2p_neg
thf(fact_7180_unat__ucast__less__no__overflow__simp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,F3: word @ A] :
( ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ F3 ) @ N )
= ( ord_less @ ( word @ A ) @ F3 @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% unat_ucast_less_no_overflow_simp
thf(fact_7181_unat__ucast__less__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,F3: word @ A] :
( ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ F3 ) @ N )
=> ( ord_less @ ( word @ A ) @ F3 @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% unat_ucast_less_no_overflow
thf(fact_7182_uint__power__lower,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% uint_power_lower
thf(fact_7183_less__2p__is__upper__bits__unset,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P3: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ P3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ! [N12: nat] :
( ( ord_less_eq @ nat @ N @ N12 )
=> ( ( ord_less @ nat @ N12 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P3 @ N12 ) ) ) ) ) ) ).
% less_2p_is_upper_bits_unset
thf(fact_7184_nth__bounded,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
=> ( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less_eq @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ nat @ N @ M ) ) ) ) ) ).
% nth_bounded
thf(fact_7185_upper__bits__unset__is__l2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,P3: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ! [N12: nat] :
( ( ord_less_eq @ nat @ N @ N12 )
=> ( ( ord_less @ nat @ N12 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P3 @ N12 ) ) ) )
= ( ord_less @ ( word @ A ) @ P3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% upper_bits_unset_is_l2p
thf(fact_7186_uint__add__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ).
% uint_add_lem
thf(fact_7187_wi__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: int,M: int] :
( ( ord_less_eq @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ N ) @ ( ring_1_of_int @ ( word @ A ) @ M ) )
= ( ord_less_eq @ int @ ( modulo_modulo @ int @ N @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( modulo_modulo @ int @ M @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% wi_le
thf(fact_7188_word__2p__mult__inc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat] :
( ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% word_2p_mult_inc
thf(fact_7189_wi__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: int,M: int] :
( ( ord_less @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ N ) @ ( ring_1_of_int @ ( word @ A ) @ M ) )
= ( ord_less @ int @ ( modulo_modulo @ int @ N @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( modulo_modulo @ int @ M @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% wi_less
thf(fact_7190_word__power__less__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Sz: nat] :
( ( ord_less @ nat @ Sz @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Sz ) @ ( one_one @ ( word @ A ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Sz ) ) ) ) ).
% word_power_less_1
thf(fact_7191_power__2__ge__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% power_2_ge_iff
thf(fact_7192_sint__1__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
=> ( ( A2
= ( zero_zero @ ( word @ A ) ) )
=> ( ( ring_1_signed @ A @ int @ A2 )
!= ( zero_zero @ int ) ) ) )
=> ( ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
=> ( ( A2
= ( one_one @ ( word @ A ) ) )
=> ( ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) )
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) )
=> ~ ( ( ord_less @ nat @ ( one_one @ nat ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) )
!= ( one_one @ int ) ) ) ) ) ) ).
% sint_1_cases
thf(fact_7193_word__set__bits__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_bi4170147762399595738t_bits @ ( word @ A ) )
= ( ^ [P5: nat > $o] : ( groups4207007520872428315er_sum @ $o @ ( word @ A ) @ ( zero_neq_one_of_bool @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( map @ nat @ $o @ P5 @ ( upt @ ( zero_zero @ nat ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% word_set_bits_def
thf(fact_7194_uint__mult__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ int @ ( times_times @ ( word @ A ) @ X @ Y ) )
= ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ).
% uint_mult_lem
thf(fact_7195_signed__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [N: nat] :
( ( ring_1_signed @ B @ A @ ( semiring_1_of_nat @ ( word @ B ) @ N ) )
= ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( semiring_1_of_nat @ int @ N ) ) ) ) ) ).
% signed_of_nat
thf(fact_7196_word__power__mod__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,X: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( divide_divide @ ( word @ A ) @ ( modulo_modulo @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ).
% word_power_mod_div
thf(fact_7197_ucast__range__less,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( ord_less @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( image @ ( word @ A ) @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) ) @ ( top_top @ ( set @ ( word @ A ) ) ) )
= ( collect @ ( word @ B )
@ ^ [X4: word @ B] : ( ord_less @ ( word @ B ) @ X4 @ ( power_power @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% ucast_range_less
thf(fact_7198_unat__plus__if_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) ) )
& ( ~ ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% unat_plus_if'
thf(fact_7199_unat__word__ariths_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ nat @ ( one_one @ ( word @ A ) ) )
= ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_word_ariths(5)
thf(fact_7200_unat__sub__if_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) )
& ( ~ ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ) ) ).
% unat_sub_if'
thf(fact_7201_no__olen__add__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% no_olen_add_nat
thf(fact_7202_word__add__le__mono2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,J: word @ A,K: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ J )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ K @ I ) @ ( plus_plus @ ( word @ A ) @ K @ J ) ) ) ) ) ).
% word_add_le_mono2
thf(fact_7203_word__add__le__mono1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,J: word @ A,K: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ J )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ K ) @ ( plus_plus @ ( word @ A ) @ J @ K ) ) ) ) ) ).
% word_add_le_mono1
thf(fact_7204_word__add__le__dest,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,J: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ K ) @ ( plus_plus @ ( word @ A ) @ J @ K ) )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ I @ J ) ) ) ) ) ).
% word_add_le_dest
thf(fact_7205_word__add__le__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,J: word @ A] :
( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ K ) @ ( plus_plus @ ( word @ A ) @ J @ K ) )
= ( ord_less_eq @ ( word @ A ) @ I @ J ) ) ) ) ) ).
% word_add_le_iff
thf(fact_7206_sint__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A] : ( ord_less @ int @ ( ring_1_signed @ A @ int @ W2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sint_less
thf(fact_7207_word__add__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,J: word @ A] :
( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ K ) @ ( plus_plus @ ( word @ A ) @ J @ K ) )
= ( ord_less @ ( word @ A ) @ I @ J ) ) ) ) ) ).
% word_add_less_iff
thf(fact_7208_word__add__less__dest,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,J: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ K ) @ ( plus_plus @ ( word @ A ) @ J @ K ) )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ I @ J ) ) ) ) ) ).
% word_add_less_dest
thf(fact_7209_word__add__less__mono1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,J: word @ A,K: word @ A] :
( ( ord_less @ ( word @ A ) @ I @ J )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ K ) @ ( plus_plus @ ( word @ A ) @ J @ K ) ) ) ) ) ).
% word_add_less_mono1
thf(fact_7210_sint__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) @ ( ring_1_signed @ A @ int @ X ) ) ) ).
% sint_ge
thf(fact_7211_unat__less__power,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Sz: nat,K: word @ A] :
( ( ord_less @ nat @ Sz @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ K @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Sz ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ K ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Sz ) ) ) ) ) ).
% unat_less_power
thf(fact_7212_uint__split,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: int > $o,X: word @ A] :
( ( P2 @ ( semiring_1_unsigned @ A @ int @ X ) )
= ( ! [I3: int] :
( ( ( ( ring_1_of_int @ ( word @ A ) @ I3 )
= X )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ I3 )
& ( ord_less @ int @ I3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( P2 @ I3 ) ) ) ) ) ).
% uint_split
thf(fact_7213_uint__split__asm,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: int > $o,X: word @ A] :
( ( P2 @ ( semiring_1_unsigned @ A @ int @ X ) )
= ( ~ ? [I3: int] :
( ( ( ring_1_of_int @ ( word @ A ) @ I3 )
= X )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ I3 )
& ( ord_less @ int @ I3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
& ~ ( P2 @ I3 ) ) ) ) ) ).
% uint_split_asm
thf(fact_7214_word__of__int__inverse,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [R2: int,A2: word @ A] :
( ( ( ring_1_of_int @ ( word @ A ) @ R2 )
= A2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ A2 )
= R2 ) ) ) ) ) ).
% word_of_int_inverse
thf(fact_7215_word__mult__less__dest,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,J: word @ A] :
( ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ K ) @ ( times_times @ ( word @ A ) @ J @ K ) )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ I @ J ) ) ) ) ) ).
% word_mult_less_dest
thf(fact_7216_div__lt_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X ) )
=> ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ X ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt'
thf(fact_7217_div__lt_H_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X ) )
=> ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ X ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt''
thf(fact_7218_double__eq__zero__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ A2 )
= ( zero_zero @ ( word @ A ) ) )
= ( ( A2
= ( zero_zero @ ( word @ A ) ) )
| ( A2
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% double_eq_zero_iff
thf(fact_7219_no__olen__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% no_olen_add
thf(fact_7220_no__olen__add_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ Y @ X ) )
= ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ Y ) @ ( semiring_1_unsigned @ A @ int @ X ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% no_olen_add'
thf(fact_7221_word__le__exists_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ? [Z3: word @ A] :
( ( Y
= ( plus_plus @ ( word @ A ) @ X @ Z3 ) )
& ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Z3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% word_le_exists'
thf(fact_7222_More__Word_Oof__nat__power,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P3: nat,X: nat] :
( ( ord_less @ nat @ P3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X ) )
=> ( ( ord_less @ nat @ X @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ P3 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ X ) ) ) ) ) ).
% More_Word.of_nat_power
thf(fact_7223_uint__plus__if_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) ) )
& ( ~ ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
= ( minus_minus @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% uint_plus_if'
thf(fact_7224_word__less__power__trans,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,M: nat,K: nat] :
( ( ord_less @ ( word @ A ) @ N @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ K ) ) )
=> ( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ K ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ).
% word_less_power_trans
thf(fact_7225_word__less__power__trans2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,M: nat,K: nat] :
( ( ord_less @ ( word @ A ) @ N @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ K ) ) )
=> ( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ N @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ).
% word_less_power_trans2
thf(fact_7226_uint__sub__if_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B2: word @ A,A2: word @ A] :
( ( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ B2 ) @ ( semiring_1_unsigned @ A @ int @ A2 ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ B2 ) @ ( semiring_1_unsigned @ A @ int @ A2 ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) )
= ( plus_plus @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% uint_sub_if'
thf(fact_7227_word__less__two__pow__divI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,M: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ X @ ( divide_divide @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ) ).
% word_less_two_pow_divI
thf(fact_7228_word__power__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( times_times @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% word_power_nonzero
thf(fact_7229_div__lt__uint_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X ) )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ I ) @ ( semiring_1_unsigned @ A @ int @ X ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt_uint'
thf(fact_7230_div__lt__uint_H_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X ) )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ I ) @ ( semiring_1_unsigned @ A @ int @ X ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt_uint''
thf(fact_7231_even__mult__exp__div__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,M: nat,N: nat] :
( ( dvd_dvd @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ~ ( ( ord_less_eq @ nat @ M @ N )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ~ ( dvd_dvd @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( divide_divide @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_word_iff
thf(fact_7232_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_7233_CHAR__eqI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C3: nat] :
( ( ( semiring_1_of_nat @ A @ C3 )
= ( zero_zero @ A ) )
=> ( ! [X3: nat] :
( ( ( semiring_1_of_nat @ A @ X3 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ nat @ C3 @ X3 ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C3 ) ) ) ) ).
% CHAR_eqI
thf(fact_7234_UNIV__word__eq__word__of__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( top_top @ ( set @ ( word @ A ) ) )
= ( image @ nat @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% UNIV_word_eq_word_of_nat
thf(fact_7235_sint__greater__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( ring_1_signed @ A @ int @ W2 ) ) ) ).
% sint_greater_eq
thf(fact_7236_int__eq__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ X ) )
= ( semiring_1_of_nat @ int @ X ) ) ) ) ).
% int_eq_sint
thf(fact_7237_word__mult__less__mono1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,J: word @ A,K: word @ A] :
( ( ord_less @ ( word @ A ) @ I @ J )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ K ) @ ( times_times @ ( word @ A ) @ J @ K ) ) ) ) ) ) ).
% word_mult_less_mono1
thf(fact_7238_word__mult__less__cancel,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: word @ A,I: word @ A,J: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ K ) @ ( times_times @ ( word @ A ) @ J @ K ) )
= ( ord_less @ ( word @ A ) @ I @ J ) ) ) ) ) ) ).
% word_mult_less_cancel
thf(fact_7239_le2p__bits__unset,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P3: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ P3 @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less @ nat @ N4 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P3 @ N4 ) ) ) ) ) ).
% le2p_bits_unset
thf(fact_7240_le__2p__upper__bits,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P3: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ P3 @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less @ nat @ N4 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P3 @ N4 ) ) ) ) ) ) ).
% le_2p_upper_bits
thf(fact_7241_word__add__offset__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,N: nat,X: word @ A,M: nat,Sz: nat] :
( ( ord_less @ ( word @ A ) @ Y @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less @ nat @ Sz @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) )
=> ( ( Sz
= ( plus_plus @ nat @ M @ N ) )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( times_times @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ Y ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Sz ) ) ) ) ) ) ) ) ).
% word_add_offset_less
thf(fact_7242_bit__horner__sum__uint__exp__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Ws: list @ ( word @ A ),N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( groups4207007520872428315er_sum @ ( word @ A ) @ int @ ( semiring_1_unsigned @ A @ int ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ Ws ) @ N )
= ( ( ord_less @ nat @ ( divide_divide @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( size_size @ ( list @ ( word @ A ) ) @ Ws ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( nth @ ( word @ A ) @ Ws @ ( divide_divide @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( modulo_modulo @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% bit_horner_sum_uint_exp_iff
thf(fact_7243_div__power__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less_eq @ nat @ X @ Y )
=> ( ( ord_less @ nat @ Y @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( divide_divide @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Y ) @ ( one_one @ ( word @ A ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ X ) )
= ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Y @ X ) ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% div_power_helper
thf(fact_7244_sint__of__nat__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B2: nat,A2: nat] :
( ( ord_less @ nat @ B2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ A2 ) ) @ ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ B2 ) ) ) ) ) ) ).
% sint_of_nat_le
thf(fact_7245_sint__of__nat__ge__zero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ X ) ) ) ) ) ).
% sint_of_nat_ge_zero
thf(fact_7246_sint__int__max__plus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_signed @ A @ int @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) )
= ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% sint_int_max_plus_1
thf(fact_7247_sint__of__int__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( ( ring_1_signed @ A @ int @ ( ring_1_of_int @ ( word @ A ) @ X ) )
= X ) ) ) ) ).
% sint_of_int_eq
thf(fact_7248_word__mult__le__mono1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,J: word @ A,K: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ J )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ K ) @ ( times_times @ ( word @ A ) @ J @ K ) ) ) ) ) ) ).
% word_mult_le_mono1
thf(fact_7249_word__mult__le__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: word @ A,I: word @ A,J: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ K ) @ ( times_times @ ( word @ A ) @ J @ K ) )
= ( ord_less_eq @ ( word @ A ) @ I @ J ) ) ) ) ) ) ).
% word_mult_le_iff
thf(fact_7250_CHAR__pos__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( ? [N6: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N6 )
& ( ( semiring_1_of_nat @ A @ N6 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_pos_iff
thf(fact_7251_CHAR__eq__posI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C3 )
=> ( ( ( semiring_1_of_nat @ A @ C3 )
= ( zero_zero @ A ) )
=> ( ! [X3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 )
=> ( ( ord_less @ nat @ X3 @ C3 )
=> ( ( semiring_1_of_nat @ A @ X3 )
!= ( zero_zero @ A ) ) ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C3 ) ) ) ) ) ).
% CHAR_eq_posI
thf(fact_7252_CHAR__eq0__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) )
= ( ! [N6: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N6 )
=> ( ( semiring_1_of_nat @ A @ N6 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_eq0_iff
thf(fact_7253_Word_Oword__div__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ X @ Y ) @ Y )
= X ) ) ) ) ).
% Word.word_div_mult
thf(fact_7254_sint__int__min,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_signed @ A @ int @ ( uminus_uminus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
= ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% sint_int_min
thf(fact_7255_word__less__power__trans__ofnat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,K: nat] :
( ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ K ) ) )
=> ( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ).
% word_less_power_trans_ofnat
thf(fact_7256_word__bit__induct,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: ( word @ A ) > $o,A2: word @ A] :
( ( P2 @ ( zero_zero @ ( word @ A ) ) )
=> ( ! [A4: word @ A] :
( ( P2 @ A4 )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ A4 )
=> ( ( ord_less @ ( word @ A ) @ A4 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ( P2 @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) )
=> ( ! [A4: word @ A] :
( ( P2 @ A4 )
=> ( ( ord_less @ ( word @ A ) @ A4 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ( P2 @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ) ).
% word_bit_induct
thf(fact_7257_unat__mult__power__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: nat,Sz: nat] :
( ( ord_less @ nat @ K @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ Sz ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( times_times @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Sz ) @ ( semiring_1_of_nat @ ( word @ A ) @ K ) ) )
= ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Sz ) @ K ) ) ) ) ).
% unat_mult_power_lem
thf(fact_7258_bit__word__half__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: $o] :
( ( ord_less @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ( ( divide_divide @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( zero_neq_one_of_bool @ ( word @ A ) @ B2 ) @ ( times_times @ ( word @ A ) @ A2 @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) )
= A2 ) ) ) ).
% bit_word_half_eq
thf(fact_7259_of__nat__less__two__pow__div__set,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( collect @ ( word @ A )
@ ^ [X4: word @ A] : ( ord_less @ ( word @ A ) @ X4 @ ( divide_divide @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) )
= ( image @ nat @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) )
@ ( collect @ nat
@ ^ [K3: nat] : ( ord_less @ nat @ K3 @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ) ) ).
% of_nat_less_two_pow_div_set
thf(fact_7260_Suc__div__unat__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Sz: nat,Us2: nat] :
( ( ord_less @ nat @ Sz @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less_eq @ nat @ Us2 @ Sz )
=> ( ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Sz @ Us2 ) )
= ( suc @ ( semiring_1_unsigned @ A @ nat @ ( divide_divide @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Sz ) @ ( one_one @ ( word @ A ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Us2 ) ) ) ) ) ) ) ) ).
% Suc_div_unat_helper
thf(fact_7261_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_7262_less__eq__decr__length__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% less_eq_decr_length_iff
thf(fact_7263_decr__length__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ N )
= ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) ) ).
% decr_length_less_iff
thf(fact_7264_len__gt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ).
% len_gt_0
thf(fact_7265_length__not__greater__eq__2__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ~ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) ) ) ) ).
% length_not_greater_eq_2_iff
thf(fact_7266_len__num0,axiom,
( ( type_len0_len_of @ numeral_num0 )
= ( ^ [Uu4: itself @ numeral_num0] : ( zero_zero @ nat ) ) ) ).
% len_num0
thf(fact_7267_len__not__eq__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( type_len0_len_of @ A @ ( type2 @ A ) )
!= ( zero_zero @ nat ) ) ) ).
% len_not_eq_0
thf(fact_7268_two__less__eq__exp__length,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( linordered_idom @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ).
% two_less_eq_exp_length
thf(fact_7269_word__int__split__asm,axiom,
! [A: $tType,B: $tType] :
( ( type_len @ B )
=> ! [P2: A > $o,F3: int > A,X: word @ B] :
( ( P2 @ ( word_int_case @ A @ B @ F3 @ X ) )
= ( ~ ? [N6: int] :
( ( X
= ( ring_1_of_int @ ( word @ B ) @ N6 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ N6 )
& ( ord_less @ int @ N6 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
& ~ ( P2 @ ( F3 @ N6 ) ) ) ) ) ) ).
% word_int_split_asm
thf(fact_7270_word__int__split,axiom,
! [A: $tType,B: $tType] :
( ( type_len @ B )
=> ! [P2: A > $o,F3: int > A,X: word @ B] :
( ( P2 @ ( word_int_case @ A @ B @ F3 @ X ) )
= ( ! [I3: int] :
( ( ( X
= ( ring_1_of_int @ ( word @ B ) @ I3 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ I3 )
& ( ord_less @ int @ I3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) )
=> ( P2 @ ( F3 @ I3 ) ) ) ) ) ) ).
% word_int_split
thf(fact_7271_word__reverse__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_reverse @ A )
= ( ^ [W3: word @ A] : ( groups4207007520872428315er_sum @ $o @ ( word @ A ) @ ( zero_neq_one_of_bool @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( rev @ $o @ ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W3 ) @ ( upt @ ( zero_zero @ nat ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ) ).
% word_reverse_def
thf(fact_7272_smod__word__min,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) ) ) ).
% smod_word_min
thf(fact_7273_bit__word__reverse__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_reverse @ A @ W2 ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ N ) ) ) ) ) ) ).
% bit_word_reverse_iff
thf(fact_7274_smod__word__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] : ( ord_less @ int @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% smod_word_max
thf(fact_7275_smod__int__compares_I3_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less @ int @ ( uminus_uminus @ int @ B2 ) @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) ) ) ) ).
% smod_int_compares(3)
thf(fact_7276_smod__int__compares_I5_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less @ int @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) @ ( uminus_uminus @ int @ B2 ) ) ) ) ).
% smod_int_compares(5)
thf(fact_7277_smod__int__compares_I1_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less @ int @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) @ B2 ) ) ) ).
% smod_int_compares(1)
thf(fact_7278_smod__int__compares_I2_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) ) ) ) ).
% smod_int_compares(2)
thf(fact_7279_smod__int__compares_I4_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% smod_int_compares(4)
thf(fact_7280_smod__int__compares_I6_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) ) ) ) ).
% smod_int_compares(6)
thf(fact_7281_smod__int__compares_I7_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% smod_int_compares(7)
thf(fact_7282_smod__int__compares_I8_J,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ B2 @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) ) ) ) ).
% smod_int_compares(8)
thf(fact_7283_smod__mod__positive,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( signed6721504322012087516modulo @ int @ A2 @ B2 )
= ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ).
% smod_mod_positive
thf(fact_7284_bit__horner__sum__bit__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Bs2: list @ $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( groups4207007520872428315er_sum @ $o @ ( word @ A ) @ ( zero_neq_one_of_bool @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Bs2 ) @ N )
= ( ( ord_less @ nat @ N @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( size_size @ ( list @ $o ) @ Bs2 ) ) )
& ( nth @ $o @ Bs2 @ N ) ) ) ) ).
% bit_horner_sum_bit_word_iff
thf(fact_7285_bit__word__roti__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: int,W2: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_roti @ A @ K @ W2 ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( nat2 @ ( modulo_modulo @ int @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ) ).
% bit_word_roti_iff
thf(fact_7286_min__eq__arg_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ( ord_min @ A @ M @ N )
= M )
= ( ord_less_eq @ A @ M @ N ) ) ) ).
% min_eq_arg(1)
thf(fact_7287_min__eq__arg_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ( ord_min @ A @ M @ N )
= N )
= ( ord_less_eq @ A @ N @ M ) ) ) ).
% min_eq_arg(2)
thf(fact_7288_min__arg__le_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N: A,M: A] :
( ( ord_less_eq @ A @ N @ ( ord_min @ A @ M @ N ) )
= ( ( ord_min @ A @ M @ N )
= N ) ) ) ).
% min_arg_le(1)
thf(fact_7289_min__arg__le_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [M: A,N: A] :
( ( ord_less_eq @ A @ M @ ( ord_min @ A @ M @ N ) )
= ( ( ord_min @ A @ M @ N )
= M ) ) ) ).
% min_arg_le(2)
thf(fact_7290_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C3 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% min.bounded_iff
thf(fact_7291_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_7292_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_7293_min__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ ( ord_min @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z @ X )
& ( ord_less @ A @ Z @ Y ) ) ) ) ).
% min_less_iff_conj
thf(fact_7294_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_7295_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_7296_min__simps_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= B2 ) ) ) ).
% min_simps(2)
thf(fact_7297_min__simps_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= A2 ) ) ) ).
% min_simps(1)
thf(fact_7298_min__less__self__conv_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ B2 )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% min_less_self_conv(2)
thf(fact_7299_min__less__self__conv_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ A2 )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% min_less_self_conv(1)
thf(fact_7300_min__arg__not__ge_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ~ ( ord_less @ A @ ( ord_min @ A @ M @ N ) @ N ) )
= ( ( ord_min @ A @ M @ N )
= N ) ) ) ).
% min_arg_not_ge(2)
thf(fact_7301_min__arg__not__ge_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ~ ( ord_less @ A @ ( ord_min @ A @ M @ N ) @ M ) )
= ( ( ord_min @ A @ M @ N )
= M ) ) ) ).
% min_arg_not_ge(1)
thf(fact_7302_min__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% min_bot2
thf(fact_7303_min__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% min_bot
thf(fact_7304_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_7305_min__0R,axiom,
! [N: nat] :
( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_7306_min__0L,axiom,
! [N: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_7307_min__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(4)
thf(fact_7308_min__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(3)
thf(fact_7309_min__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V3: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V3 ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V3 ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V3 ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V3 ) )
= ( numeral_numeral @ A @ V3 ) ) ) ) ) ).
% min_number_of(1)
thf(fact_7310_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_7311_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_7312_min__Suc__gt_I2_J,axiom,
! [A2: nat,B2: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( ( ord_min @ nat @ B2 @ ( suc @ A2 ) )
= ( suc @ A2 ) ) ) ).
% min_Suc_gt(2)
thf(fact_7313_min__Suc__gt_I1_J,axiom,
! [A2: nat,B2: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( ( ord_min @ nat @ ( suc @ A2 ) @ B2 )
= ( suc @ A2 ) ) ) ).
% min_Suc_gt(1)
thf(fact_7314_min__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V3: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) ) ) ) ) ).
% min_number_of(4)
thf(fact_7315_min__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V3: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V3 ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V3 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V3 ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V3 ) )
= ( numeral_numeral @ A @ V3 ) ) ) ) ) ).
% min_number_of(3)
thf(fact_7316_min__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V3: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) ) ) ) ) ).
% min_number_of(2)
thf(fact_7317_zip__replicate,axiom,
! [A: $tType,B: $tType,I: nat,X: A,J: nat,Y: B] :
( ( zip @ A @ B @ ( replicate @ A @ I @ X ) @ ( replicate @ B @ J @ Y ) )
= ( replicate @ ( product_prod @ A @ B ) @ ( ord_min @ nat @ I @ J ) @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_replicate
thf(fact_7318_Min__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A5 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min_insert
thf(fact_7319_take__bit__word__Suc__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( suc @ N ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ N ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ).
% take_bit_word_Suc_numeral
thf(fact_7320_take__bit__word__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( suc @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ N ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% take_bit_word_Suc_minus_numeral
thf(fact_7321_of__nat__min,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X @ Y ) )
= ( ord_min @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_min
thf(fact_7322_minus__max__eq__min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_max_eq_min
thf(fact_7323_minus__min__eq__max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_min_eq_max
thf(fact_7324_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_7325_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_7326_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Z ) @ ( plus_plus @ A @ Y @ Z ) ) ) ) ).
% min_add_distrib_left
thf(fact_7327_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ X @ ( ord_min @ A @ Y @ Z ) )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z ) ) ) ) ).
% min_add_distrib_right
thf(fact_7328_Min_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ).
% Min.in_idem
thf(fact_7329_inf__nat__def,axiom,
( ( inf_inf @ nat )
= ( ord_min @ nat ) ) ).
% inf_nat_def
thf(fact_7330_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_7331_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A,Z: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ord_min @ A @ ( minus_minus @ A @ X @ Z ) @ ( minus_minus @ A @ Y @ Z ) ) ) ) ).
% min_diff_distrib_left
thf(fact_7332_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A8: A,B8: A] : ( if @ A @ ( ord_less_eq @ A @ A8 @ B8 ) @ A8 @ B8 ) ) ) ) ).
% min_def_raw
thf(fact_7333_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C3: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ ( ord_min @ A @ C3 @ D2 ) ) ) ) ) ).
% min.mono
thf(fact_7334_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_7335_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_7336_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C3 ) )
=> ~ ( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% min.boundedE
thf(fact_7337_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C3 ) ) ) ) ) ).
% min.boundedI
thf(fact_7338_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B8: A] :
( A8
= ( ord_min @ A @ A8 @ B8 ) ) ) ) ) ).
% min.order_iff
thf(fact_7339_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_7340_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_7341_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B8: A] :
( ( ord_min @ A @ A8 @ B8 )
= A8 ) ) ) ) ).
% min.absorb_iff1
thf(fact_7342_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B8: A,A8: A] :
( ( ord_min @ A @ A8 @ B8 )
= B8 ) ) ) ) ).
% min.absorb_iff2
thf(fact_7343_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% min.coboundedI1
thf(fact_7344_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% min.coboundedI2
thf(fact_7345_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ( ord_less_eq @ A @ X @ Z )
| ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% min_le_iff_disj
thf(fact_7346_min__absorb2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_min @ A @ X @ Y )
= Y ) ) ) ).
% min_absorb2
thf(fact_7347_min__absorb1,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_min @ A @ X @ Y )
= X ) ) ) ).
% min_absorb1
thf(fact_7348_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A8: A,B8: A] : ( if @ A @ ( ord_less_eq @ A @ A8 @ B8 ) @ A8 @ B8 ) ) ) ) ).
% min_def
thf(fact_7349_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% min.strict_coboundedI2
thf(fact_7350_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ A2 @ C3 )
=> ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% min.strict_coboundedI1
thf(fact_7351_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A8: A,B8: A] :
( ( A8
= ( ord_min @ A @ A8 @ B8 ) )
& ( A8 != B8 ) ) ) ) ) ).
% min.strict_order_iff
thf(fact_7352_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ ( ord_min @ A @ B2 @ C3 ) )
=> ~ ( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% min.strict_boundedE
thf(fact_7353_min__less__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ( ord_less @ A @ X @ Z )
| ( ord_less @ A @ Y @ Z ) ) ) ) ).
% min_less_iff_disj
thf(fact_7354_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P3: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ P3 @ ( ord_max @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P3 @ X ) @ ( times_times @ A @ P3 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ P3 @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P3 @ X ) @ ( times_times @ A @ P3 @ Y ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_7355_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P3: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ P3 @ ( ord_min @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P3 @ X ) @ ( times_times @ A @ P3 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ P3 @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P3 @ X ) @ ( times_times @ A @ P3 @ Y ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_7356_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P3: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P3 )
= ( ord_max @ A @ ( times_times @ A @ X @ P3 ) @ ( times_times @ A @ Y @ P3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P3 )
= ( ord_min @ A @ ( times_times @ A @ X @ P3 ) @ ( times_times @ A @ Y @ P3 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_7357_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P3: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P3 )
= ( ord_min @ A @ ( times_times @ A @ X @ P3 ) @ ( times_times @ A @ Y @ P3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P3 )
= ( ord_max @ A @ ( times_times @ A @ X @ P3 ) @ ( times_times @ A @ Y @ P3 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_7358_max__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P3: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y ) @ P3 )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P3 ) @ ( divide_divide @ A @ Y @ P3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y ) @ P3 )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P3 ) @ ( divide_divide @ A @ Y @ P3 ) ) ) ) ) ) ).
% max_divide_distrib_right
thf(fact_7359_min__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P3: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y ) @ P3 )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P3 ) @ ( divide_divide @ A @ Y @ P3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y ) @ P3 )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P3 ) @ ( divide_divide @ A @ Y @ P3 ) ) ) ) ) ) ).
% min_divide_distrib_right
thf(fact_7360_Inf__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S3: set @ A,X: A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( ( S3
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ X @ S3 ) )
= X ) )
& ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ X @ S3 ) )
= ( ord_min @ A @ X @ ( complete_Inf_Inf @ A @ S3 ) ) ) ) ) ) ) ).
% Inf_insert_finite
thf(fact_7361_hom__Min__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H2: A > A,N8: set @ A] :
( ! [X3: A,Y3: A] :
( ( H2 @ ( ord_min @ A @ X3 @ Y3 ) )
= ( ord_min @ A @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N8 )
=> ( ( N8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic643756798350308766er_Min @ A @ N8 ) )
= ( lattic643756798350308766er_Min @ A @ ( image @ A @ A @ H2 @ N8 ) ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_7362_Min_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ B5 ) @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min.subset
thf(fact_7363_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A5 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_7364_Min_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] : ( member @ A @ ( ord_min @ A @ X3 @ Y3 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Min.closed
thf(fact_7365_Min_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ ( lattic643756798350308766er_Min @ A @ B5 ) ) ) ) ) ) ) ) ).
% Min.union
thf(fact_7366_Min_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A5 ) )
= ( finite_fold @ A @ A @ ( ord_min @ A ) @ X @ A5 ) ) ) ) ).
% Min.eq_fold
thf(fact_7367_Min_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_7368_Min_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A5 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_7369_Min_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798350308766er_Min @ A )
= ( ^ [A6: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y5: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( ord_min @ A @ X4 ) @ Y5 ) )
@ ( none @ A )
@ A6 ) ) ) ) ) ).
% Min.eq_fold'
thf(fact_7370_bit__slice__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [M: nat,W2: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( slice2 @ A @ B @ M @ W2 ) @ N )
= ( ( ord_less @ nat @ N @ ( ord_min @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) ) ) ) ) ) ).
% bit_slice_iff
thf(fact_7371_set__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [I3: nat] :
( ( Uu3
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I3 ) @ ( nth @ B @ Ys @ I3 ) ) )
& ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% set_zip
thf(fact_7372_bit__slice1__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [M: nat,W2: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( slice1 @ A @ B @ M @ W2 ) @ N )
= ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ N )
& ( ord_less @ nat @ N @ ( ord_min @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ M ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) ) @ ( minus_minus @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% bit_slice1_iff
thf(fact_7373_signed__drop__bit__beyond,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W2: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ( signed_drop_bit @ A @ N @ W2 )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) )
& ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ( signed_drop_bit @ A @ N @ W2 )
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% signed_drop_bit_beyond
thf(fact_7374_int__of__integer__min,axiom,
! [K: code_integer,L: code_integer] :
( ( code_int_of_integer @ ( ord_min @ code_integer @ K @ L ) )
= ( ord_min @ int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).
% int_of_integer_min
thf(fact_7375_signed__drop__bit__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A] :
( ( signed_drop_bit @ A @ ( zero_zero @ nat ) @ W2 )
= W2 ) ) ).
% signed_drop_bit_0
thf(fact_7376_signed__drop__bit__of__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( signed_drop_bit @ A @ N @ ( one_one @ ( word @ A ) ) )
= ( zero_neq_one_of_bool @ ( word @ A )
@ ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% signed_drop_bit_of_1
thf(fact_7377_bit__signed__drop__bit__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,W2: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( signed_drop_bit @ A @ M @ W2 ) @ N )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2
@ ( if @ nat
@ ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) @ N )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
@ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ M @ N ) ) ) ) ) ).
% bit_signed_drop_bit_iff
thf(fact_7378_signed__drop__bit__word__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,K: num] :
( ( signed_drop_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ) ).
% signed_drop_bit_word_Suc_minus_numeral
thf(fact_7379_sless__eq__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( word_sle @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% sless_eq_word_minus_numeral_minus_numeral
thf(fact_7380_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_7381_signed_Olength__insort,axiom,
! [A: $tType,B: $tType] :
( ( type_len @ A )
=> ! [F3: B > ( word @ A ),X: B,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( insort_key @ ( word @ A ) @ B @ ( word_sle @ A ) @ F3 @ X @ Xs ) )
= ( suc @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ).
% signed.length_insort
thf(fact_7382_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_7383_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_7384_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_7385_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_7386_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_7387_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_7388_word__sle__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) ) ) ) ).
% word_sle_no
thf(fact_7389_signed_OMax__const,axiom,
! [B: $tType,A: $tType] :
( ( type_len @ A )
=> ! [A5: set @ B,C3: word @ A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattices_Max @ ( word @ A ) @ ( word_sle @ A )
@ ( image @ B @ ( word @ A )
@ ^ [Uu3: B] : C3
@ A5 ) )
= C3 ) ) ) ) ).
% signed.Max_const
thf(fact_7390_signed_OMin__const,axiom,
! [B: $tType,A: $tType] :
( ( type_len @ A )
=> ! [A5: set @ B,C3: word @ A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattices_Min @ ( word @ A ) @ ( word_sle @ A )
@ ( image @ B @ ( word @ A )
@ ^ [Uu3: B] : C3
@ A5 ) )
= C3 ) ) ) ) ).
% signed.Min_const
thf(fact_7391_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_7392_extra__sle__sless__unfolds_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num] :
( ( word_sle @ A @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ N ) )
= ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N ) ) ) ) ) ).
% extra_sle_sless_unfolds(4)
thf(fact_7393_extra__sle__sless__unfolds_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num] :
( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ N ) @ ( one_one @ ( word @ A ) ) )
= ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% extra_sle_sless_unfolds(6)
thf(fact_7394_extra__sle__sless__unfolds_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num] :
( ( word_sle @ A @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ N ) )
= ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N ) ) ) ) ) ).
% extra_sle_sless_unfolds(2)
thf(fact_7395_extra__sle__sless__unfolds_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num] :
( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ N ) @ ( zero_zero @ ( word @ A ) ) )
= ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% extra_sle_sless_unfolds(5)
thf(fact_7396_extra__sle__sless__unfolds_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sle @ A @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) )
= ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% extra_sle_sless_unfolds(3)
thf(fact_7397_extra__sle__sless__unfolds_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sle @ A @ ( zero_zero @ ( word @ A ) ) @ ( one_one @ ( word @ A ) ) )
= ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% extra_sle_sless_unfolds(1)
thf(fact_7398_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_7399_sless__eq__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% sless_eq_word_numeral_numeral
thf(fact_7400_sless__eq__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( word_sle @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% sless_eq_word_minus_numeral_numeral
thf(fact_7401_sless__eq__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% sless_eq_word_numeral_minus_numeral
thf(fact_7402_signed__drop__bit__word__Suc__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,K: num] :
( ( signed_drop_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% signed_drop_bit_word_Suc_numeral
thf(fact_7403_signed_Olift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F3: nat > ( word @ A ),N: nat,N3: nat] :
( ! [N2: nat] : ( word_sle @ A @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
=> ( ( ord_less_eq @ nat @ N @ N3 )
=> ( word_sle @ A @ ( F3 @ N3 ) @ ( F3 @ N ) ) ) ) ) ).
% signed.lift_Suc_antimono_le
thf(fact_7404_signed_Olift__Suc__mono__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F3: nat > ( word @ A ),N: nat,N3: nat] :
( ! [N2: nat] : ( word_sle @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N3 )
=> ( word_sle @ A @ ( F3 @ N ) @ ( F3 @ N3 ) ) ) ) ) ).
% signed.lift_Suc_mono_le
thf(fact_7405_signed_OMax__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M7: set @ ( word @ A ),N8: set @ ( word @ A )] :
( ( ord_less_eq @ ( set @ ( word @ A ) ) @ M7 @ N8 )
=> ( ( M7
!= ( bot_bot @ ( set @ ( word @ A ) ) ) )
=> ( ( finite_finite2 @ ( word @ A ) @ N8 )
=> ( word_sle @ A @ ( lattices_Max @ ( word @ A ) @ ( word_sle @ A ) @ M7 ) @ ( lattices_Max @ ( word @ A ) @ ( word_sle @ A ) @ N8 ) ) ) ) ) ) ).
% signed.Max_mono
thf(fact_7406_signed_OMin__antimono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M7: set @ ( word @ A ),N8: set @ ( word @ A )] :
( ( ord_less_eq @ ( set @ ( word @ A ) ) @ M7 @ N8 )
=> ( ( M7
!= ( bot_bot @ ( set @ ( word @ A ) ) ) )
=> ( ( finite_finite2 @ ( word @ A ) @ N8 )
=> ( word_sle @ A @ ( lattices_Min @ ( word @ A ) @ ( word_sle @ A ) @ N8 ) @ ( lattices_Min @ ( word @ A ) @ ( word_sle @ A ) @ M7 ) ) ) ) ) ) ).
% signed.Min_antimono
thf(fact_7407_signed_OMax_Osubset__imp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A5: set @ ( word @ A ),B5: set @ ( word @ A )] :
( ( ord_less_eq @ ( set @ ( word @ A ) ) @ A5 @ B5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( word @ A ) ) ) )
=> ( ( finite_finite2 @ ( word @ A ) @ B5 )
=> ( word_sle @ A @ ( lattices_Max @ ( word @ A ) @ ( word_sle @ A ) @ A5 ) @ ( lattices_Max @ ( word @ A ) @ ( word_sle @ A ) @ B5 ) ) ) ) ) ) ).
% signed.Max.subset_imp
thf(fact_7408_signed_OMin_Osubset__imp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A5: set @ ( word @ A ),B5: set @ ( word @ A )] :
( ( ord_less_eq @ ( set @ ( word @ A ) ) @ A5 @ B5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( word @ A ) ) ) )
=> ( ( finite_finite2 @ ( word @ A ) @ B5 )
=> ( word_sle @ A @ ( lattices_Min @ ( word @ A ) @ ( word_sle @ A ) @ B5 ) @ ( lattices_Min @ ( word @ A ) @ ( word_sle @ A ) @ A5 ) ) ) ) ) ) ).
% signed.Min.subset_imp
thf(fact_7409_signed_OMin_Osubset,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A5: set @ ( word @ A ),B5: set @ ( word @ A )] :
( ( finite_finite2 @ ( word @ A ) @ A5 )
=> ( ( B5
!= ( bot_bot @ ( set @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( word @ A ) ) @ B5 @ A5 )
=> ( ( min @ ( word @ A ) @ ( word_sle @ A ) @ ( lattices_Min @ ( word @ A ) @ ( word_sle @ A ) @ B5 ) @ ( lattices_Min @ ( word @ A ) @ ( word_sle @ A ) @ A5 ) )
= ( lattices_Min @ ( word @ A ) @ ( word_sle @ A ) @ A5 ) ) ) ) ) ) ).
% signed.Min.subset
thf(fact_7410_word__sle__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sle @ A )
= ( ^ [A8: word @ A,B8: word @ A] : ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ A8 ) @ ( ring_1_signed @ A @ int @ B8 ) ) ) ) ) ).
% word_sle_eq
thf(fact_7411_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_7412_signed_Ofinite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( type_len @ A )
=> ! [S3: set @ B,P2: ( set @ B ) > $o,F3: B > ( word @ A )] :
( ( finite_finite2 @ B @ S3 )
=> ( ( P2 @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B,S5: set @ B] :
( ( finite_finite2 @ B @ S5 )
=> ( ! [Y4: B] :
( ( member @ B @ Y4 @ S5 )
=> ( word_sle @ A @ ( F3 @ Y4 ) @ ( F3 @ X3 ) ) )
=> ( ( P2 @ S5 )
=> ( P2 @ ( insert @ B @ X3 @ S5 ) ) ) ) )
=> ( P2 @ S3 ) ) ) ) ) ).
% signed.finite_ranking_induct
thf(fact_7413_signed_Osorted01,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xs: list @ ( word @ A )] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ ( word @ A ) ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ ( word @ A ) @ ( word_sle @ A ) @ Xs ) ) ) ).
% signed.sorted01
thf(fact_7414_signed_Osorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xs: list @ ( word @ A )] :
( ( sorted_wrt @ ( word @ A ) @ ( word_sle @ A ) @ Xs )
= ( ! [I3: nat,J3: nat] :
( ( ord_less @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ ( word @ A ) ) @ Xs ) )
=> ( word_sle @ A @ ( nth @ ( word @ A ) @ Xs @ I3 ) @ ( nth @ ( word @ A ) @ Xs @ J3 ) ) ) ) ) ) ) ).
% signed.sorted_iff_nth_mono_less
thf(fact_7415_signed_Osorted__iff__nth__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xs: list @ ( word @ A )] :
( ( sorted_wrt @ ( word @ A ) @ ( word_sle @ A ) @ Xs )
= ( ! [I3: nat] :
( ( ord_less @ nat @ ( suc @ I3 ) @ ( size_size @ ( list @ ( word @ A ) ) @ Xs ) )
=> ( word_sle @ A @ ( nth @ ( word @ A ) @ Xs @ I3 ) @ ( nth @ ( word @ A ) @ Xs @ ( suc @ I3 ) ) ) ) ) ) ) ).
% signed.sorted_iff_nth_Suc
thf(fact_7416_signed_Osorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xs: list @ ( word @ A )] :
( ( sorted_wrt @ ( word @ A ) @ ( word_sle @ A ) @ ( rev @ ( word @ A ) @ Xs ) )
= ( ! [I3: nat] :
( ( ord_less @ nat @ ( suc @ I3 ) @ ( size_size @ ( list @ ( word @ A ) ) @ Xs ) )
=> ( word_sle @ A @ ( nth @ ( word @ A ) @ Xs @ ( suc @ I3 ) ) @ ( nth @ ( word @ A ) @ Xs @ I3 ) ) ) ) ) ) ).
% signed.sorted_rev_iff_nth_Suc
thf(fact_7417_signed_Osorted__rev__nth__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xs: list @ ( word @ A ),I: nat,J: nat] :
( ( sorted_wrt @ ( word @ A ) @ ( word_sle @ A ) @ ( rev @ ( word @ A ) @ Xs ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ ( word @ A ) ) @ Xs ) )
=> ( word_sle @ A @ ( nth @ ( word @ A ) @ Xs @ J ) @ ( nth @ ( word @ A ) @ Xs @ I ) ) ) ) ) ) ).
% signed.sorted_rev_nth_mono
thf(fact_7418_signed_Osorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xs: list @ ( word @ A )] :
( ( sorted_wrt @ ( word @ A ) @ ( word_sle @ A ) @ ( rev @ ( word @ A ) @ Xs ) )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ ( word @ A ) ) @ Xs ) )
=> ( word_sle @ A @ ( nth @ ( word @ A ) @ Xs @ J3 ) @ ( nth @ ( word @ A ) @ Xs @ I3 ) ) ) ) ) ) ) ).
% signed.sorted_rev_iff_nth_mono
thf(fact_7419_signed_Osorted__iff__nth__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xs: list @ ( word @ A )] :
( ( sorted_wrt @ ( word @ A ) @ ( word_sle @ A ) @ Xs )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ ( word @ A ) ) @ Xs ) )
=> ( word_sle @ A @ ( nth @ ( word @ A ) @ Xs @ I3 ) @ ( nth @ ( word @ A ) @ Xs @ J3 ) ) ) ) ) ) ) ).
% signed.sorted_iff_nth_mono
thf(fact_7420_signed_Osorted__nth__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xs: list @ ( word @ A ),I: nat,J: nat] :
( ( sorted_wrt @ ( word @ A ) @ ( word_sle @ A ) @ Xs )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ ( word @ A ) ) @ Xs ) )
=> ( word_sle @ A @ ( nth @ ( word @ A ) @ Xs @ I ) @ ( nth @ ( word @ A ) @ Xs @ J ) ) ) ) ) ) ).
% signed.sorted_nth_mono
thf(fact_7421_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_7422_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N6: nat,A8: A] :
( if @ A
@ ( N6
= ( zero_zero @ nat ) )
@ A8
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N6 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_7423_word__0__sle__from__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( word_sle @ A @ ( zero_zero @ ( word @ A ) ) @ X ) ) ) ).
% word_0_sle_from_less
thf(fact_7424_unat__word__ariths_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( word_succ @ A @ A2 ) )
= ( modulo_modulo @ nat @ ( suc @ ( semiring_1_unsigned @ A @ nat @ A2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_word_ariths(3)
thf(fact_7425_sless__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( word_sless @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% sless_word_minus_numeral_numeral
thf(fact_7426_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_7427_drop__bit__word__beyond,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W2: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ N @ W2 )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% drop_bit_word_beyond
thf(fact_7428_extra__sle__sless__unfolds_I12_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num] :
( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ N ) @ ( one_one @ ( word @ A ) ) )
= ( ord_less @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% extra_sle_sless_unfolds(12)
thf(fact_7429_extra__sle__sless__unfolds_I10_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num] :
( ( word_sless @ A @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ N ) )
= ( ord_less @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N ) ) ) ) ) ).
% extra_sle_sless_unfolds(10)
thf(fact_7430_extra__sle__sless__unfolds_I11_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num] :
( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ N ) @ ( zero_zero @ ( word @ A ) ) )
= ( ord_less @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% extra_sle_sless_unfolds(11)
thf(fact_7431_extra__sle__sless__unfolds_I8_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: num] :
( ( word_sless @ A @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ N ) )
= ( ord_less @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N ) ) ) ) ) ).
% extra_sle_sless_unfolds(8)
thf(fact_7432_extra__sle__sless__unfolds_I9_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sless @ A @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) )
= ( ord_less @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% extra_sle_sless_unfolds(9)
thf(fact_7433_extra__sle__sless__unfolds_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sless @ A @ ( zero_zero @ ( word @ A ) ) @ ( one_one @ ( word @ A ) ) )
= ( ord_less @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% extra_sle_sless_unfolds(7)
thf(fact_7434_drop__bit__word__Suc__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( suc @ N ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% drop_bit_word_Suc_numeral
thf(fact_7435_sless__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
= ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% sless_word_numeral_numeral
thf(fact_7436_drop__bit__numeral__bit0__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: num] :
( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( suc @ ( zero_zero @ nat ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).
% drop_bit_numeral_bit0_1
thf(fact_7437_drop__bit__word__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( suc @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ) ).
% drop_bit_word_Suc_minus_numeral
thf(fact_7438_sless__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( word_sless @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% sless_word_minus_numeral_minus_numeral
thf(fact_7439_sless__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: num,B2: num] :
( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
= ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% sless_word_numeral_minus_numeral
thf(fact_7440_drop__bit__integer_Oabs__eq,axiom,
! [Xa2: nat,X: int] :
( ( bit_se4197421643247451524op_bit @ code_integer @ Xa2 @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se4197421643247451524op_bit @ int @ Xa2 @ X ) ) ) ).
% drop_bit_integer.abs_eq
thf(fact_7441_drop__bit__integer_Orep__eq,axiom,
! [X: nat,Xa2: code_integer] :
( ( code_int_of_integer @ ( bit_se4197421643247451524op_bit @ code_integer @ X @ Xa2 ) )
= ( bit_se4197421643247451524op_bit @ int @ X @ ( code_int_of_integer @ Xa2 ) ) ) ).
% drop_bit_integer.rep_eq
thf(fact_7442_word__sless__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sless @ A )
= ( ^ [A8: word @ A,B8: word @ A] : ( ord_less @ int @ ( ring_1_signed @ A @ int @ A8 ) @ ( ring_1_signed @ A @ int @ B8 ) ) ) ) ) ).
% word_sless_alt
thf(fact_7443_signed_Olift__Suc__mono__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F3: nat > ( word @ A ),N: nat,N3: nat] :
( ! [N2: nat] : ( word_sless @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N3 )
=> ( word_sless @ A @ ( F3 @ N ) @ ( F3 @ N3 ) ) ) ) ) ).
% signed.lift_Suc_mono_less
thf(fact_7444_signed_Olift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F3: nat > ( word @ A ),N: nat,M: nat] :
( ! [N2: nat] : ( word_sless @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( word_sless @ A @ ( F3 @ N ) @ ( F3 @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% signed.lift_Suc_mono_less_iff
thf(fact_7445_Abs__fnat__hom__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: nat] :
( ( word_succ @ A @ ( semiring_1_of_nat @ ( word @ A ) @ A2 ) )
= ( semiring_1_of_nat @ ( word @ A ) @ ( suc @ A2 ) ) ) ) ).
% Abs_fnat_hom_Suc
thf(fact_7446_ucast__drop__bit__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [N: nat,W2: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ N @ W2 ) )
= ( bit_se4197421643247451524op_bit @ ( word @ B ) @ N @ ( semiring_1_unsigned @ A @ ( word @ B ) @ W2 ) ) ) ) ) ).
% ucast_drop_bit_eq
thf(fact_7447_word__sless__sint__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( word_sless @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ X ) @ ( minus_minus @ int @ ( ring_1_signed @ A @ int @ Y ) @ ( one_one @ int ) ) ) ) ) ).
% word_sless_sint_le
thf(fact_7448_word__arith__nat__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_succ @ A )
= ( ^ [A8: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( suc @ ( semiring_1_unsigned @ A @ nat @ A8 ) ) ) ) ) ) ).
% word_arith_nat_Suc
thf(fact_7449_drop__bit__eq__zero__iff__not__bit__last,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A] :
( ( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ W2 )
= ( zero_zero @ ( word @ A ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% drop_bit_eq_zero_iff_not_bit_last
thf(fact_7450_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_7451_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_7452_possible__bit__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat] :
( ( bit_se6407376104438227557le_bit @ ( word @ A ) @ ( type2 @ ( word @ A ) ) @ M )
= ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% possible_bit_word
thf(fact_7453_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_7454_possible__bit__less__imp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [Tyrep: itself @ A,I: nat,J: nat] :
( ( bit_se6407376104438227557le_bit @ A @ Tyrep @ I )
=> ( ( ord_less_eq @ nat @ J @ I )
=> ( bit_se6407376104438227557le_bit @ A @ Tyrep @ J ) ) ) ) ).
% possible_bit_less_imp
thf(fact_7455_possible__bit__def,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se6407376104438227557le_bit @ A )
= ( ^ [Tyrep2: itself @ A,N6: nat] :
( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N6 )
!= ( zero_zero @ A ) ) ) ) ) ).
% possible_bit_def
thf(fact_7456_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_7457_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_7458_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_7459_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_7460_bit__signed__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A ) )
=> ! [W2: word @ B,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( ring_1_signed @ B @ A @ W2 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W2 @ ( ord_min @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ N ) ) ) ) ) ).
% bit_signed_iff
thf(fact_7461_div__half__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( Y
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ Y ) @ ( modulo_modulo @ ( word @ A ) @ X @ Y ) )
= ( if @ ( product_prod @ ( word @ A ) @ ( word @ A ) ) @ ( ord_less_eq @ ( word @ A ) @ Y @ ( minus_minus @ ( word @ A ) @ X @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ ( one_one @ ( word @ A ) ) ) @ ( minus_minus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) @ Y ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ ( minus_minus @ ( word @ A ) @ X @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) ) ) ) ) ) ).
% div_half_word
thf(fact_7462_Suc__2p__unat__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ K ) @ ( semiring_1_unsigned @ A @ nat @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) )
= ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_2p_unat_mask
thf(fact_7463_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_7464_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_7465_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_7466_mask__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% mask_0
thf(fact_7467_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_7468_Word_Omask__Suc__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ A ) ) ) ).
% Word.mask_Suc_0
thf(fact_7469_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_7470_push__bit__word__beyond,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W2: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ N @ W2 )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% push_bit_word_beyond
thf(fact_7471_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_7472_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_7473_less__eq__mask__iff__take__bit__eq__self,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ W2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ N @ W2 )
= W2 ) ) ) ).
% less_eq_mask_iff_take_bit_eq_self
thf(fact_7474_More__Word_Omask__Suc__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ ( word @ A ) ) ) ) ).
% More_Word.mask_Suc_0
thf(fact_7475_nth__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,I: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) @ I )
= ( ( ord_less @ nat @ I @ N )
& ( ord_less @ nat @ I @ ( size_size @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ) ) ).
% nth_mask
thf(fact_7476_signed__push__bit__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A ) )
=> ! [N: nat,W2: word @ B] :
( ( ring_1_signed @ B @ A @ ( bit_se4730199178511100633sh_bit @ ( word @ B ) @ N @ W2 ) )
= ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( ring_1_signed @ B @ A @ W2 ) ) ) ) ) ).
% signed_push_bit_eq
thf(fact_7477_le__mask__high__bits,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ W2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( ! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or7035219750837199246ssThan @ nat @ N @ ( size_size @ ( word @ A ) @ W2 ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ X4 ) ) ) ) ) ).
% le_mask_high_bits
thf(fact_7478_mask__over__length,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% mask_over_length
thf(fact_7479_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_7480_Word_Obit__mask__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) @ N )
= ( ord_less @ nat @ N @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) ) ) ) ).
% Word.bit_mask_iff
thf(fact_7481_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_7482_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_7483_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N6: nat,A8: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( bit_se4730199178511100633sh_bit @ A @ K3 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A8 @ K3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N6 ) ) ) ) ) ).
% take_bit_sum
thf(fact_7484_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_7485_mask__Suc__rec,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( suc @ N ) )
= ( plus_plus @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% mask_Suc_rec
thf(fact_7486_slice1__def,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( slice1 @ A @ B )
= ( ^ [N6: nat,W3: word @ A] : ( if @ ( word @ B ) @ ( ord_less @ nat @ N6 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N6 ) @ W3 ) ) @ ( bit_se4730199178511100633sh_bit @ ( word @ B ) @ ( minus_minus @ nat @ N6 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ W3 ) ) ) ) ) ) ).
% slice1_def
thf(fact_7487_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N6: nat,A8: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N6 ) @ A8 ) @ N6 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N6 ) @ A8 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N6 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N6 ) @ A8 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_7488_mask__lt__2pn,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% mask_lt_2pn
thf(fact_7489_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_7490_le__mask__iff__lt__2n,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W2: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
= ( ( ord_less_eq @ ( word @ A ) @ W2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( ord_less @ ( word @ A ) @ W2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% le_mask_iff_lt_2n
thf(fact_7491_push__bit__word__eq__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,M: nat,N: nat] :
( ( ord_less @ ( word @ A ) @ W2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ M @ N ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( W2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ N @ W2 )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% push_bit_word_eq_nonzero
thf(fact_7492_mult__pow2__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat,X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ N ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) )
=> ( ( ( times_times @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ ( word @ A ) @ Y @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) )
=> ( X = Y ) ) ) ) ) ) ).
% mult_pow2_inj
thf(fact_7493_divmod__via__sdivmod,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( Y
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ( ord_less_eq @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( one_one @ ( word @ A ) ) ) @ Y )
=> ( ( ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ Y ) @ ( modulo_modulo @ ( word @ A ) @ X @ Y ) )
= ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X ) ) )
& ( ~ ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ Y ) @ ( modulo_modulo @ ( word @ A ) @ X @ Y ) )
= ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) ) ) ) ) )
& ( ~ ( ord_less_eq @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( one_one @ ( word @ A ) ) ) @ Y )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ Y ) @ ( modulo_modulo @ ( word @ A ) @ X @ Y ) )
= ( if @ ( product_prod @ ( word @ A ) @ ( word @ A ) ) @ ( ord_less_eq @ ( word @ A ) @ Y @ ( minus_minus @ ( word @ A ) @ X @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ ( one_one @ ( word @ A ) ) ) @ ( minus_minus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) @ Y ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ ( minus_minus @ ( word @ A ) @ X @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) ) ) ) ) ) ) ) ).
% divmod_via_sdivmod
thf(fact_7494_set__bits__aux__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F3: nat > $o,N: nat,W2: word @ A] :
( ( code_T2661198915054445665ts_aux @ A @ F3 @ ( suc @ N ) @ W2 )
= ( code_T2661198915054445665ts_aux @ A @ F3 @ N @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ W2 ) @ ( if @ ( word @ A ) @ ( F3 @ N ) @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% set_bits_aux_Suc
thf(fact_7495_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_7496_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_7497_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_7498_set__bits__aux__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F3: nat > $o,W2: word @ A] :
( ( code_T2661198915054445665ts_aux @ A @ F3 @ ( zero_zero @ nat ) @ W2 )
= W2 ) ) ).
% set_bits_aux_0
thf(fact_7499_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_7500_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_7501_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( zero_zero @ int ) ) ).
% not_mask_negative_int
thf(fact_7502_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2239418461657761734s_mask @ int @ N ) ) ).
% mask_nonnegative_int
thf(fact_7503_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% less_eq_mask
thf(fact_7504_mask__integer_Orep__eq,axiom,
! [X: nat] :
( ( code_int_of_integer @ ( bit_se2239418461657761734s_mask @ code_integer @ X ) )
= ( bit_se2239418461657761734s_mask @ int @ X ) ) ).
% mask_integer.rep_eq
thf(fact_7505_push__bit__integer_Orep__eq,axiom,
! [X: nat,Xa2: code_integer] :
( ( code_int_of_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ X @ Xa2 ) )
= ( bit_se4730199178511100633sh_bit @ int @ X @ ( code_int_of_integer @ Xa2 ) ) ) ).
% push_bit_integer.rep_eq
thf(fact_7506_mask__integer_Oabs__eq,axiom,
( ( bit_se2239418461657761734s_mask @ code_integer )
= ( ^ [X4: nat] : ( code_integer_of_int @ ( bit_se2239418461657761734s_mask @ int @ X4 ) ) ) ) ).
% mask_integer.abs_eq
thf(fact_7507_push__bit__integer_Oabs__eq,axiom,
! [Xa2: nat,X: int] :
( ( bit_se4730199178511100633sh_bit @ code_integer @ Xa2 @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se4730199178511100633sh_bit @ int @ Xa2 @ X ) ) ) ).
% push_bit_integer.abs_eq
thf(fact_7508_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_7509_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_7510_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_7511_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_7512_bit__set__bit__aux,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F3: nat > $o,N: nat,W2: word @ A,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( code_T2661198915054445665ts_aux @ A @ F3 @ N @ W2 ) @ M )
= ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( ( ord_less @ nat @ M @ N )
=> ( F3 @ M ) )
& ( ~ ( ord_less @ nat @ M @ N )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ) ).
% bit_set_bit_aux
thf(fact_7513_set__bits__aux__rec,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( code_T2661198915054445665ts_aux @ A )
= ( ^ [F4: nat > $o,N6: nat,W3: word @ A] :
( if @ ( word @ A )
@ ( N6
= ( zero_zero @ nat ) )
@ W3
@ ( code_T2661198915054445665ts_aux @ A @ F4 @ ( minus_minus @ nat @ N6 @ ( one_one @ nat ) ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ W3 ) @ ( if @ ( word @ A ) @ ( F4 @ ( minus_minus @ nat @ N6 @ ( one_one @ nat ) ) ) @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ) ) ).
% set_bits_aux_rec
thf(fact_7514_and__mask__arith_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W2: word @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ W2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W2 ) @ N ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W2 ) @ N ) ) ) ) ) ) ).
% and_mask_arith'
thf(fact_7515_word__of__int__via__signed,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Mask: int,Shift: int,Index: nat,Overflow: int,Least: int,I: int,Arbitrary1: int > ( word @ A ),Arbitrary2: int > ( word @ A )] :
( ( Mask
= ( bit_se2239418461657761734s_mask @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( Shift
= ( bit_se4730199178511100633sh_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ int ) ) )
=> ( ( Index
= ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) )
=> ( ( Overflow
= ( bit_se4730199178511100633sh_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( one_one @ int ) ) )
=> ( ( Least
= ( uminus_uminus @ int @ Overflow ) )
=> ( ( ring_1_of_int @ ( word @ A ) @ I )
= ( if @ ( word @ A ) @ ( bit_se5641148757651400278ts_bit @ int @ ( bit_se5824344872417868541ns_and @ int @ I @ Mask ) @ Index )
@ ( if @ ( word @ A )
@ ( ( ord_less @ int @ ( minus_minus @ int @ ( bit_se5824344872417868541ns_and @ int @ I @ Mask ) @ Shift ) @ Least )
| ( ord_less_eq @ int @ Overflow @ ( minus_minus @ int @ ( bit_se5824344872417868541ns_and @ int @ I @ Mask ) @ Shift ) ) )
@ ( Arbitrary1 @ ( bit_se5824344872417868541ns_and @ int @ I @ Mask ) )
@ ( ring_1_of_int @ ( word @ A ) @ ( minus_minus @ int @ ( bit_se5824344872417868541ns_and @ int @ I @ Mask ) @ Shift ) ) )
@ ( if @ ( word @ A )
@ ( ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ I @ Mask ) @ Least )
| ( ord_less_eq @ int @ Overflow @ ( bit_se5824344872417868541ns_and @ int @ I @ Mask ) ) )
@ ( Arbitrary2 @ ( bit_se5824344872417868541ns_and @ int @ I @ Mask ) )
@ ( ring_1_of_int @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ int @ I @ Mask ) ) ) ) ) ) ) ) ) ) ) ).
% word_of_int_via_signed
thf(fact_7516_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_7517_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_7518_bit_Oconj__zero__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ X )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_left
thf(fact_7519_bit_Oconj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_right
thf(fact_7520_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_7521_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_7522_and__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(5)
thf(fact_7523_and__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(1)
thf(fact_7524_and__mask__eq__iff__le__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= W2 )
= ( ord_less_eq @ ( word @ A ) @ W2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% and_mask_eq_iff_le_mask
thf(fact_7525_le__mask__imp__and__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= X ) ) ) ).
% le_mask_imp_and_mask
thf(fact_7526_mask__twice2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,X: word @ A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ) ).
% mask_twice2
thf(fact_7527_AND__upper2_H_H,axiom,
! [Y: int,Z: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less @ int @ Y @ Z )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Z ) ) ) ).
% AND_upper2''
thf(fact_7528_AND__upper1_H_H,axiom,
! [Y: int,Z: int,Ya2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less @ int @ Y @ Z )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y @ Ya2 ) @ Z ) ) ) ).
% AND_upper1''
thf(fact_7529_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_7530_word__unat__and__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,Y: word @ A] :
( ( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ N )
| ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ N ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) ) @ N ) ) ) ).
% word_unat_and_lt
thf(fact_7531_word__and__le1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,A2: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y @ A2 ) @ A2 ) ) ).
% word_and_le1
thf(fact_7532_word__and__le2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,Y: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A2 @ Y ) @ A2 ) ) ).
% word_and_le2
thf(fact_7533_AND__upper2_H,axiom,
! [Y: int,Z: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ Y @ Z )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Z ) ) ) ).
% AND_upper2'
thf(fact_7534_AND__upper1_H,axiom,
! [Y: int,Z: int,Ya2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ Y @ Z )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ Y @ Ya2 ) @ Z ) ) ) ).
% AND_upper1'
thf(fact_7535_AND__upper2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Y ) ) ).
% AND_upper2
thf(fact_7536_AND__upper1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ X ) ) ).
% AND_upper1
thf(fact_7537_AND__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) ) ) ).
% AND_lower
thf(fact_7538_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ X )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( X = Y ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_7539_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A8: A,N6: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A8 @ ( bit_se4730199178511100633sh_bit @ A @ N6 @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_7540_word__and__1,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [N: word @ B] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ N @ ( zero_zero @ nat ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ B ) @ N @ ( one_one @ ( word @ B ) ) )
= ( one_one @ ( word @ B ) ) ) )
& ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ N @ ( zero_zero @ nat ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ B ) @ N @ ( one_one @ ( word @ B ) ) )
= ( zero_zero @ ( word @ B ) ) ) ) ) ) ).
% word_and_1
thf(fact_7541_bit__twiddle__min,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ Y @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ ( if @ ( word @ A ) @ ( ord_less @ ( word @ A ) @ X @ Y ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( zero_zero @ ( word @ A ) ) ) ) )
= ( ord_min @ ( word @ A ) @ X @ Y ) ) ) ).
% bit_twiddle_min
thf(fact_7542_bit__twiddle__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ ( if @ ( word @ A ) @ ( ord_less @ ( word @ A ) @ X @ Y ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( zero_zero @ ( word @ A ) ) ) ) )
= ( ord_max @ ( word @ A ) @ X @ Y ) ) ) ).
% bit_twiddle_max
thf(fact_7543_ucast__mask__drop,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [N: nat,X: word @ B] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ B ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ B ) @ N ) ) )
= ( semiring_1_unsigned @ B @ ( word @ A ) @ X ) ) ) ) ).
% ucast_mask_drop
thf(fact_7544_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_7545_less__mask__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= X ) ) ) ).
% less_mask_eq
thf(fact_7546_and__mask__lt__2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat] : ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% and_mask_lt_2p
thf(fact_7547_mask__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= W2 )
= ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% mask_eq_iff
thf(fact_7548_is__aligned__AND__less__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U: word @ A,N: nat,V3: word @ A] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ U @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ V3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ U @ V3 )
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% is_aligned_AND_less_0
thf(fact_7549_mask__eq__iff__w2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W2: word @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W2 ) )
=> ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= W2 )
= ( ord_less @ ( word @ A ) @ W2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% mask_eq_iff_w2p
thf(fact_7550_and__mask__less__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ X ) )
=> ( ord_less @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% and_mask_less_size
thf(fact_7551_word__and__mask__le__2pm1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat] : ( ord_less_eq @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_and_mask_le_2pm1
thf(fact_7552_word__mod__2p__is__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( modulo_modulo @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ) ).
% word_mod_2p_is_mask
thf(fact_7553_bool__mask_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) )
= ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
= ( one_one @ ( word @ A ) ) ) ) ) ) ).
% bool_mask'
thf(fact_7554_and__mask__less_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W2: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% and_mask_less'
thf(fact_7555_eq__mask__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat] :
( ( W2
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ W2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% eq_mask_less
thf(fact_7556_word__unat__mask__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,W2: word @ A] :
( ( ord_less_eq @ nat @ M @ ( size_size @ ( word @ A ) @ W2 ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% word_unat_mask_lt
thf(fact_7557_drop__bit__int__code_I2_J,axiom,
! [N: nat] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% drop_bit_int_code(2)
thf(fact_7558_neg__mask__is__div_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W2: word @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W2 ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ W2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% neg_mask_is_div'
thf(fact_7559_bit_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_right
thf(fact_7560_bit_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_left
thf(fact_7561_and__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(1)
thf(fact_7562_and__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(3)
thf(fact_7563_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_7564_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_7565_NOT__mask__AND__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [W2: A,N: nat] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se5824344872417868541ns_and @ A @ W2 @ ( bit_se2239418461657761734s_mask @ A @ N ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) )
= ( zero_zero @ A ) ) ) ).
% NOT_mask_AND_mask
thf(fact_7566_and__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(4)
thf(fact_7567_and__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(2)
thf(fact_7568_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_7569_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_7570_mask__out__first__mask__some,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,Y: word @ A,M: nat] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= Y )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ) ).
% mask_out_first_mask_some
thf(fact_7571_mask__lower__twice,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,X: word @ A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ).
% mask_lower_twice
thf(fact_7572_and__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( bit_se5824344872417868541ns_and @ code_integer @ X @ Xa2 ) )
= ( bit_se5824344872417868541ns_and @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% and_integer.rep_eq
thf(fact_7573_and__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( bit_se5824344872417868541ns_and @ code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se5824344872417868541ns_and @ int @ Xa2 @ X ) ) ) ).
% and_integer.abs_eq
thf(fact_7574_neg__test__bit,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% neg_test_bit
thf(fact_7575_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_7576_multiple__mask__trivia,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat,X: word @ A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ).
% multiple_mask_trivia
thf(fact_7577_and__mask__0__iff__le__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= ( zero_zero @ ( word @ A ) ) )
= ( ord_less_eq @ ( word @ A ) @ W2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% and_mask_0_iff_le_mask
thf(fact_7578_mask__subsume,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ).
% mask_subsume
thf(fact_7579_bit_Ocompl__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ X @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ X @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( bit_ri4277139882892585799ns_not @ A @ X )
= Y ) ) ) ) ).
% bit.compl_unique
thf(fact_7580_word__ops__nth__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A,Y: word @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ X ) )
=> ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N ) ) ) ) ) ) ).
% word_ops_nth_size
thf(fact_7581_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_7582_neg__mask__test__bit,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ M )
= ( ( ord_less_eq @ nat @ N @ M )
& ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% neg_mask_test_bit
thf(fact_7583_mask__exceed,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% mask_exceed
thf(fact_7584_msb0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,I: nat] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) ) ) ) ).
% msb0
thf(fact_7585_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M5: nat,N6: nat] :
( if @ nat
@ ( ( M5
= ( zero_zero @ nat ) )
| ( N6
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_7586_drop__bit__int__code_I1_J,axiom,
! [I: int] :
( ( bit_se4197421643247451524op_bit @ int @ ( zero_zero @ nat ) @ I )
= I ) ).
% drop_bit_int_code(1)
thf(fact_7587_complement__nth__w2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N3: nat,N: nat] :
( ( ord_less @ nat @ N3 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ N3 )
= ( N3 != N ) ) ) ) ).
% complement_nth_w2p
thf(fact_7588_push__bit__int__code_I1_J,axiom,
! [I: int] :
( ( bit_se4730199178511100633sh_bit @ int @ ( zero_zero @ nat ) @ I )
= I ) ).
% push_bit_int_code(1)
thf(fact_7589_msb1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% msb1
thf(fact_7590_word__ops__lsb,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ ( zero_zero @ nat ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( zero_zero @ nat ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ ( zero_zero @ nat ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( zero_zero @ nat ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ ( zero_zero @ nat ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( zero_zero @ nat ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ ( zero_zero @ nat ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) ) ) ) ) ) ).
% word_ops_lsb
thf(fact_7591_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_7592_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_7593_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_7594_not__integer_Oabs__eq,axiom,
! [X: int] :
( ( bit_ri4277139882892585799ns_not @ code_integer @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_ri4277139882892585799ns_not @ int @ X ) ) ) ).
% not_integer.abs_eq
thf(fact_7595_not__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( bit_ri4277139882892585799ns_not @ code_integer @ X ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( code_int_of_integer @ X ) ) ) ).
% not_integer.rep_eq
thf(fact_7596_bit__word__rotl__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,W2: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_rotl @ A @ M @ W2 ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% bit_word_rotl_iff
thf(fact_7597_bit__sshiftr__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_Sh8784991116023147202shiftr @ A @ W2 @ M ) @ N )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2
@ ( if @ nat
@ ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) @ N )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
@ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ M @ N ) ) ) ) ) ).
% bit_sshiftr_iff
thf(fact_7598_sshiftr__of__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A] :
( ( bit_Sh8784991116023147202shiftr @ A @ W2 @ ( zero_zero @ nat ) )
= W2 ) ) ).
% sshiftr_of_0
thf(fact_7599_sshiftr__numeral__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: num,N: nat] :
( ( bit_Sh8784991116023147202shiftr @ A @ ( numeral_numeral @ ( word @ A ) @ M ) @ ( suc @ N ) )
= ( signed_drop_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ).
% sshiftr_numeral_Suc
thf(fact_7600_sshiftr__minus__numeral__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: num,N: nat] :
( ( bit_Sh8784991116023147202shiftr @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) @ ( suc @ N ) )
= ( signed_drop_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ).
% sshiftr_minus_numeral_Suc
thf(fact_7601_sshiftr__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( bit_Sh8784991116023147202shiftr @ A @ ( one_one @ ( word @ A ) ) @ N )
= ( zero_neq_one_of_bool @ ( word @ A )
@ ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% sshiftr_1
thf(fact_7602_shiftl__Suc__0,axiom,
! [N: nat] :
( ( bit_Sh4282982442137083160shiftl @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% shiftl_Suc_0
thf(fact_7603_word__roti__eq__word__rotr__word__rotl,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_roti @ A )
= ( ^ [I3: int,W3: word @ A] : ( if @ ( word @ A ) @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I3 ) @ ( word_rotr @ A @ ( nat2 @ I3 ) @ W3 ) @ ( word_rotl @ A @ ( nat2 @ ( uminus_uminus @ int @ I3 ) ) @ W3 ) ) ) ) ) ).
% word_roti_eq_word_rotr_word_rotl
thf(fact_7604_shiftl__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ).
% shiftl_0
thf(fact_7605_shiftl__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_Sh4282982442137083160shiftl @ A @ A2 @ ( zero_zero @ nat ) )
= A2 ) ) ).
% shiftl_of_0
thf(fact_7606_shiftl__numeral__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N: nat] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( numeral_numeral @ A @ M ) @ ( suc @ N ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ M ) ) ) ) ).
% shiftl_numeral_Suc
thf(fact_7607_shiftl__minus__numeral__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: num,N: nat] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( suc @ N ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ) ).
% shiftl_minus_numeral_Suc
thf(fact_7608_shiftl__of__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( bit_Sh4282982442137083160shiftl @ A @ A2 @ ( suc @ N ) )
= ( bit_Sh4282982442137083160shiftl @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) ) ).
% shiftl_of_Suc
thf(fact_7609_bit__word__rotr__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,W2: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_rotr @ A @ M @ W2 ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ N @ M ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% bit_word_rotr_iff
thf(fact_7610_bit__shiftl__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_Sh4282982442137083160shiftl @ A @ A2 @ M ) @ 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_shiftl_iff
thf(fact_7611_shiftr__minus__numeral__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: num,N: nat] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( suc @ N ) )
= ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ) ).
% shiftr_minus_numeral_Suc
thf(fact_7612_bit__revcast__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W2: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( revcast @ A @ B @ W2 ) @ N )
= ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ N )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% bit_revcast_iff
thf(fact_7613_shiftr__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ).
% shiftr_0
thf(fact_7614_shiftr__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_Sh4282982442137083166shiftr @ A @ A2 @ ( zero_zero @ nat ) )
= A2 ) ) ).
% shiftr_of_0
thf(fact_7615_shiftr__Suc__0,axiom,
! [N: nat] :
( ( bit_Sh4282982442137083166shiftr @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( zero_neq_one_of_bool @ nat
@ ( N
= ( zero_zero @ nat ) ) ) ) ).
% shiftr_Suc_0
thf(fact_7616_shiftr__numeral__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N: nat] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( numeral_numeral @ A @ M ) @ ( suc @ N ) )
= ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ M ) ) ) ) ).
% shiftr_numeral_Suc
thf(fact_7617_shiftr__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( one_one @ A ) @ N )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% shiftr_1
thf(fact_7618_signed_Orsp,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ( bNF_rel_fun @ int @ int @ A @ A
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ L4 ) )
@ ^ [Y6: A,Z4: A] : Y6 = Z4
@ ( comp @ int @ A @ int @ ( ring_1_of_int @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( comp @ int @ A @ int @ ( ring_1_of_int @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% signed.rsp
thf(fact_7619_inj__on__word__of__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( inj_on @ nat @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% inj_on_word_of_nat
thf(fact_7620_inj__on__empty,axiom,
! [B: $tType,A: $tType,F3: A > B] : ( inj_on @ A @ B @ F3 @ ( bot_bot @ ( set @ A ) ) ) ).
% inj_on_empty
thf(fact_7621_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_7622_inj__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( inj_on @ A @ A
@ ^ [B8: A] : ( divide_divide @ A @ B8 @ A2 )
@ ( top_top @ ( set @ A ) ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% inj_divide_right
thf(fact_7623_inj__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F3: B > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ B @ C @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% inj_apsnd
thf(fact_7624_inj__on__insert,axiom,
! [B: $tType,A: $tType,F3: A > B,A2: A,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( insert @ A @ A2 @ A5 ) )
= ( ( inj_on @ A @ B @ F3 @ A5 )
& ~ ( member @ B @ ( F3 @ A2 ) @ ( image @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_7625_not__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bit_ri4277139882892585799ns_not @ int )
@ ( bit_ri4277139882892585799ns_not @ int ) ) ).
% not_integer.rsp
thf(fact_7626_xor__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ ( bit_se5824344971392196577ns_xor @ int )
@ ( bit_se5824344971392196577ns_xor @ int ) ) ).
% xor_integer.rsp
thf(fact_7627_inj__fn,axiom,
! [A: $tType,F3: A > A,N: nat] :
( ( inj_on @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_fn
thf(fact_7628_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [F3: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( F3 @ X3 )
!= ( F3 @ Y3 ) ) )
=> ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% linorder_injI
thf(fact_7629_less__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ( ord_less @ int )
@ ( ord_less @ int ) ) ).
% less_integer.rsp
thf(fact_7630_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,F3: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( F3 @ X3 )
!= ( F3 @ Y3 ) ) ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X3 ) ) ) )
=> ( inj_on @ A @ B @ F3 @ A5 ) ) ) ) ).
% linorder_inj_onI
thf(fact_7631_less__eq__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ( ord_less_eq @ int )
@ ( ord_less_eq @ int ) ) ).
% less_eq_integer.rsp
thf(fact_7632_subset__inj__on,axiom,
! [B: $tType,A: $tType,F3: A > B,B5: set @ A,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( inj_on @ A @ B @ F3 @ A5 ) ) ) ).
% subset_inj_on
thf(fact_7633_inj__on__subset,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( inj_on @ A @ B @ F3 @ B5 ) ) ) ).
% inj_on_subset
thf(fact_7634_minus__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ ( minus_minus @ int )
@ ( minus_minus @ int ) ) ).
% minus_integer.rsp
thf(fact_7635_dup_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 )
@ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 ) ) ).
% dup.rsp
thf(fact_7636_bit__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: nat > $o,Z4: nat > $o] : Y6 = Z4
@ ( bit_se5641148757651400278ts_bit @ int )
@ ( bit_se5641148757651400278ts_bit @ int ) ) ).
% bit_integer.rsp
thf(fact_7637_abs__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( abs_abs @ int )
@ ( abs_abs @ int ) ) ).
% abs_integer.rsp
thf(fact_7638_sgn__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( sgn_sgn @ int )
@ ( sgn_sgn @ int ) ) ).
% sgn_integer.rsp
thf(fact_7639_natural__of__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ nat @ nat
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ nat2
@ nat2 ) ).
% natural_of_integer.rsp
thf(fact_7640_divide__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ ( divide_divide @ int )
@ ( divide_divide @ int ) ) ).
% divide_integer.rsp
thf(fact_7641_modulo__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ ( modulo_modulo @ int )
@ ( modulo_modulo @ int ) ) ).
% modulo_integer.rsp
thf(fact_7642_uminus__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( uminus_uminus @ int )
@ ( uminus_uminus @ int ) ) ).
% uminus_integer.rsp
thf(fact_7643_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,Z4: int] : Y6 = Z4
@ R3
@ ( ring_1_of_int @ A )
@ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% transfer_rule_of_int
thf(fact_7644_plus__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ ( plus_plus @ int )
@ ( plus_plus @ int ) ) ).
% plus_integer.rsp
thf(fact_7645_times__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ ( times_times @ int )
@ ( times_times @ int ) ) ).
% times_integer.rsp
thf(fact_7646_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,A5: set @ A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A2 ) @ A5 ) ) ) ).
% inj_on_mult
thf(fact_7647_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,Z4: num] : Y6 = Z4
@ R3
@ ( numeral_numeral @ A )
@ ( numeral_numeral @ B ) ) ) ) ) ) ).
% transfer_rule_numeral
thf(fact_7648_or__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ ( bit_se1065995026697491101ons_or @ int )
@ ( bit_se1065995026697491101ons_or @ int ) ) ).
% or_integer.rsp
thf(fact_7649_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_7650_the__inv__into__into,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,X: B,B5: set @ A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( member @ B @ X @ ( image @ A @ B @ F3 @ A5 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( member @ A @ ( the_inv_into @ A @ B @ A5 @ F3 @ X ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_7651_inj__on__image__Int,axiom,
! [B: $tType,A: $tType,F3: A > B,C4: set @ A,A5: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F3 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ( image @ A @ B @ F3 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) )
= ( inf_inf @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ ( image @ A @ B @ F3 @ B5 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_7652_fold__image,axiom,
! [C: $tType,B: $tType,A: $tType,G3: A > B,A5: set @ A,F3: B > C > C,Z: C] :
( ( inj_on @ A @ B @ G3 @ A5 )
=> ( ( finite_fold @ B @ C @ F3 @ Z @ ( image @ A @ B @ G3 @ A5 ) )
= ( finite_fold @ A @ C @ ( comp @ B @ ( C > C ) @ A @ F3 @ G3 ) @ Z @ A5 ) ) ) ).
% fold_image
thf(fact_7653_finite__inverse__image__gen,axiom,
! [A: $tType,B: $tType,A5: set @ A,F3: B > A,D5: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inj_on @ B @ A @ F3 @ D5 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ D5 )
& ( member @ A @ ( F3 @ J3 ) @ A5 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_7654_finite__Collect,axiom,
! [A: $tType,B: $tType,S3: set @ A,F3: B > A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( inj_on @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [A8: B] : ( member @ A @ ( F3 @ A8 ) @ S3 ) ) ) ) ) ).
% finite_Collect
thf(fact_7655_finite__inverse__image,axiom,
! [A: $tType,B: $tType,A5: set @ A,F3: B > A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inj_on @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] : ( member @ A @ ( F3 @ J3 ) @ A5 ) ) ) ) ) ).
% finite_inverse_image
thf(fact_7656_range__inj__infinite,axiom,
! [A: $tType,F3: nat > A] :
( ( inj_on @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) )
=> ~ ( finite_finite2 @ A @ ( image @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% range_inj_infinite
thf(fact_7657_finite__UNIV__inj__surj,axiom,
! [A: $tType,F3: A > A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( inj_on @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( image @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_7658_finite__UNIV__surj__inj,axiom,
! [A: $tType,F3: A > A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_7659_inj__on__image__Fpow,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( inj_on @ ( set @ A ) @ ( set @ B ) @ ( image @ A @ B @ F3 ) @ ( finite_Fpow @ A @ A5 ) ) ) ).
% inj_on_image_Fpow
thf(fact_7660_finite__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F3 @ A5 ) )
=> ( ( inj_on @ B @ A @ F3 @ A5 )
=> ( finite_finite2 @ B @ A5 ) ) ) ).
% finite_imageD
thf(fact_7661_finite__image__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( finite_finite2 @ B @ ( image @ A @ B @ F3 @ A5 ) )
= ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_image_iff
thf(fact_7662_card__image,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( finite_card @ B @ ( image @ A @ B @ F3 @ A5 ) )
= ( finite_card @ A @ A5 ) ) ) ).
% card_image
thf(fact_7663_inj__on__iff__eq__card,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inj_on @ A @ B @ F3 @ A5 )
= ( ( finite_card @ B @ ( image @ A @ B @ F3 @ A5 ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_7664_eq__card__imp__inj__on,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( finite_card @ B @ ( image @ A @ B @ F3 @ A5 ) )
= ( finite_card @ A @ A5 ) )
=> ( inj_on @ A @ B @ F3 @ A5 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_7665_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A5: set @ A,A10: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A5 )
& ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A5 ) @ A10 ) ) )
= ( ? [G4: B > A] :
( ( image @ B @ A @ G4 @ A10 )
= A5 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_7666_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,C4: set @ A,A5: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F3 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ( ( image @ A @ B @ F3 @ A5 )
= ( image @ A @ B @ F3 @ B5 ) )
= ( A5 = B5 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_7667_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,B5: set @ A,A2: A,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ B5 )
=> ( ( member @ A @ A2 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( member @ B @ ( F3 @ A2 ) @ ( image @ A @ B @ F3 @ A5 ) )
= ( member @ A @ A2 @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_7668_subset__image__inj,axiom,
! [A: $tType,B: $tType,S3: set @ A,F3: B > A,T4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ ( image @ B @ A @ F3 @ T4 ) )
= ( ? [U7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ U7 @ T4 )
& ( inj_on @ B @ A @ F3 @ U7 )
& ( S3
= ( image @ B @ A @ F3 @ U7 ) ) ) ) ) ).
% subset_image_inj
thf(fact_7669_inj__on__strict__subset,axiom,
! [B: $tType,A: $tType,F3: A > B,B5: set @ A,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ B5 )
=> ( ( ord_less @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ ( image @ A @ B @ F3 @ B5 ) ) ) ) ).
% inj_on_strict_subset
thf(fact_7670_endo__inj__surj,axiom,
! [A: $tType,A5: set @ A,F3: A > A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ F3 @ A5 ) @ A5 )
=> ( ( inj_on @ A @ A @ F3 @ A5 )
=> ( ( image @ A @ A @ F3 @ A5 )
= A5 ) ) ) ) ).
% endo_inj_surj
thf(fact_7671_inj__on__finite,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B5: set @ B] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ B5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( finite_finite2 @ A @ A5 ) ) ) ) ).
% inj_on_finite
thf(fact_7672_finite__surj__inj,axiom,
! [A: $tType,A5: set @ A,F3: A > A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( image @ A @ A @ F3 @ A5 ) )
=> ( inj_on @ A @ A @ F3 @ A5 ) ) ) ).
% finite_surj_inj
thf(fact_7673_inj__image__subset__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ ( image @ A @ B @ F3 @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% inj_image_subset_iff
thf(fact_7674_pigeonhole,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( ord_less @ nat @ ( finite_card @ A @ ( image @ B @ A @ F3 @ A5 ) ) @ ( finite_card @ B @ A5 ) )
=> ~ ( inj_on @ B @ A @ F3 @ A5 ) ) ).
% pigeonhole
thf(fact_7675_inj__on__image__set__diff,axiom,
! [B: $tType,A: $tType,F3: A > B,C4: set @ A,A5: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F3 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ( image @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
= ( minus_minus @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ ( image @ A @ B @ F3 @ B5 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_7676_inj__enumerate,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A] :
( ~ ( finite_finite2 @ A @ S3 )
=> ( inj_on @ nat @ A @ ( infini527867602293511546merate @ A @ S3 ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% inj_enumerate
thf(fact_7677_less__word_Orsp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) )
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) )
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ^ [A8: int,B8: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ A8 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ B8 ) )
@ ^ [A8: int,B8: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ A8 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ B8 ) ) ) ) ).
% less_word.rsp
thf(fact_7678_less__eq__word_Orsp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) )
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) )
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ^ [A8: int,B8: int] : ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ A8 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ B8 ) )
@ ^ [A8: int,B8: int] : ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ A8 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ B8 ) ) ) ) ).
% less_eq_word.rsp
thf(fact_7679_and__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ ( bit_se5824344872417868541ns_and @ int )
@ ( bit_se5824344872417868541ns_and @ int ) ) ).
% and_integer.rsp
thf(fact_7680_inj__on__UNION__chain,axiom,
! [C: $tType,B: $tType,A: $tType,I6: set @ A,A5: A > ( set @ B ),F3: B > C] :
( ! [I2: A,J2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ( member @ A @ J2 @ I6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A5 @ I2 ) @ ( A5 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A5 @ J2 ) @ ( A5 @ I2 ) ) ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( inj_on @ B @ C @ F3 @ ( A5 @ I2 ) ) )
=> ( inj_on @ B @ C @ F3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A5 @ I6 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_7681_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [N2: nat,F2: nat > A] :
( ( A5
= ( image @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) ) )
& ( inj_on @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_7682_inj__on__INTER,axiom,
! [C: $tType,B: $tType,A: $tType,I6: set @ A,F3: B > C,A5: A > ( set @ B )] :
( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( inj_on @ B @ C @ F3 @ ( A5 @ I2 ) ) )
=> ( inj_on @ B @ C @ F3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A5 @ I6 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_7683_word__sle_Orsp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) )
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) )
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ^ [K3: int,L4: int] : ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ K3 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L4 ) )
@ ^ [K3: int,L4: int] : ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ K3 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L4 ) ) ) ) ).
% word_sle.rsp
thf(fact_7684_word__sless_Orsp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) )
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) )
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ^ [K3: int,L4: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ K3 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L4 ) )
@ ^ [K3: int,L4: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ K3 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L4 ) ) ) ) ).
% word_sless.rsp
thf(fact_7685_signed__modulo__word_Orsp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) )
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) )
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) ) )
@ ^ [K3: int,L4: int] : ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ K3 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L4 ) )
@ ^ [K3: int,L4: int] : ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ K3 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L4 ) ) ) ) ).
% signed_modulo_word.rsp
thf(fact_7686_signed__divide__word_Orsp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) )
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) )
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) ) )
@ ^ [K3: int,L4: int] : ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ K3 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L4 ) )
@ ^ [K3: int,L4: int] : ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ K3 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L4 ) ) ) ) ).
% signed_divide_word.rsp
thf(fact_7687_surjective__iff__injective__gen,axiom,
! [B: $tType,A: $tType,S3: set @ A,T4: set @ B,F3: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ( ( finite_card @ A @ S3 )
= ( finite_card @ B @ T4 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ S3 ) @ T4 )
=> ( ( ! [X4: B] :
( ( member @ B @ X4 @ T4 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ S3 )
& ( ( F3 @ Y5 )
= X4 ) ) ) )
= ( inj_on @ A @ B @ F3 @ S3 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_7688_card__bij__eq,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B5: set @ B,G3: B > A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ B5 )
=> ( ( inj_on @ B @ A @ G3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G3 @ B5 ) @ A5 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( finite_card @ A @ A5 )
= ( finite_card @ B @ B5 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_7689_inj__on__nth,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ( distinct @ A @ Xs )
=> ( ! [X3: nat] :
( ( member @ nat @ X3 @ I6 )
=> ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( inj_on @ nat @ A @ ( nth @ A @ Xs ) @ I6 ) ) ) ).
% inj_on_nth
thf(fact_7690_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_7691_infinite__iff__countable__subset,axiom,
! [A: $tType,S3: set @ A] :
( ( ~ ( finite_finite2 @ A @ S3 ) )
= ( ? [F4: nat > A] :
( ( inj_on @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) ) @ S3 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_7692_infinite__countable__subset,axiom,
! [A: $tType,S3: set @ A] :
( ~ ( finite_finite2 @ A @ S3 )
=> ? [F2: nat > A] :
( ( inj_on @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) @ S3 ) ) ) ).
% infinite_countable_subset
thf(fact_7693_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,G3: A > B,B5: set @ A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( inj_on @ A @ B @ G3 @ B5 )
=> ( ( ( inf_inf @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ ( image @ A @ B @ G3 @ B5 ) )
= ( bot_bot @ ( set @ B ) ) )
=> ( inj_on @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ A5 ) @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_7694_image__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F3: A > B,C4: set @ A,A5: set @ C,B5: C > ( set @ A ),J: C] :
( ( inj_on @ A @ B @ F3 @ C4 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( B5 @ X3 ) @ C4 ) )
=> ( ( member @ C @ J @ A5 )
=> ( ( image @ A @ B @ F3 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ C @ ( set @ A ) @ B5 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image @ C @ ( set @ B )
@ ^ [X4: C] : ( image @ A @ B @ F3 @ ( B5 @ X4 ) )
@ A5 ) ) ) ) ) ) ).
% image_INT
thf(fact_7695_inj__on__funpow__least,axiom,
! [A: $tType,N: nat,F3: A > A,S: A] :
( ( ( compow @ ( A > A ) @ N @ F3 @ S )
= S )
=> ( ! [M3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M3 )
=> ( ( ord_less @ nat @ M3 @ N )
=> ( ( compow @ ( A > A ) @ M3 @ F3 @ S )
!= S ) ) )
=> ( inj_on @ nat @ A
@ ^ [K3: nat] : ( compow @ ( A > A ) @ K3 @ F3 @ S )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% inj_on_funpow_least
thf(fact_7696_inj__on__iff__card__le,axiom,
! [A: $tType,B: $tType,A5: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A5 )
& ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A5 ) @ B5 ) ) )
= ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B5 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_7697_card__inj__on__le,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B5: set @ B] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ B5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B5 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_7698_card__le__inj,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B5 ) )
=> ? [F2: A > B] :
( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A5 ) @ B5 )
& ( inj_on @ A @ B @ F2 @ A5 ) ) ) ) ) ).
% card_le_inj
thf(fact_7699_inj__on__Un,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( ( inj_on @ A @ B @ F3 @ A5 )
& ( inj_on @ A @ B @ F3 @ B5 )
& ( ( inf_inf @ ( set @ B ) @ ( image @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) @ ( image @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ B5 @ A5 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% inj_on_Un
thf(fact_7700_bit__word_Orsp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ int @ int @ ( nat > $o ) @ ( nat > $o )
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) )
@ ^ [Y6: nat > $o,Z4: nat > $o] : Y6 = Z4
@ ^ [K3: int,N6: nat] :
( ( ord_less @ nat @ N6 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ K3 @ N6 ) )
@ ^ [K3: int,N6: nat] :
( ( ord_less @ nat @ N6 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ K3 @ N6 ) ) ) ) ).
% bit_word.rsp
thf(fact_7701_map__sorted__distinct__set__unique,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,Ys: list @ B] :
( ( inj_on @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys ) ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Ys ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F3 @ Ys ) )
=> ( ( ( set2 @ B @ Xs )
= ( set2 @ B @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_7702_funpow__inj__finite,axiom,
! [A: $tType,P3: A > A,X: A] :
( ( inj_on @ A @ A @ P3 @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y5: A] :
? [N6: nat] :
( Y5
= ( compow @ ( A > A ) @ N6 @ P3 @ X ) ) ) )
=> ~ ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( compow @ ( A > A ) @ N2 @ P3 @ X )
!= X ) ) ) ) ).
% funpow_inj_finite
thf(fact_7703_the__signed__int_Orsp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) )
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% the_signed_int.rsp
thf(fact_7704_signed__cast_Orsp,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) )
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ L4 ) )
@ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% signed_cast.rsp
thf(fact_7705_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,Z4: nat] : Y6 = Z4
@ R3
@ ( semiring_1_of_nat @ A )
@ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% transfer_rule_of_nat
thf(fact_7706_inj__on__map__inv__f,axiom,
! [B: $tType,A: $tType,L: list @ A,A5: set @ A,F3: A > B] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ L ) @ A5 )
=> ( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( map @ B @ A @ ( inv_on @ A @ B @ F3 @ A5 ) @ ( map @ A @ B @ F3 @ L ) )
= L ) ) ) ).
% inj_on_map_inv_f
thf(fact_7707_inv__on__f__f,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,X: A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( inv_on @ A @ B @ F3 @ A5 @ ( F3 @ X ) )
= X ) ) ) ).
% inv_on_f_f
thf(fact_7708_xor__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ ( bit_se5824344971392196577ns_xor @ nat )
@ ( bit_se5824344971392196577ns_xor @ nat ) ) ).
% xor_natural.rsp
thf(fact_7709_drop__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ ( bit_se4197421643247451524op_bit @ int )
@ ( bit_se4197421643247451524op_bit @ int ) ) ).
% drop_bit_integer.rsp
thf(fact_7710_drop__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ ( bit_se4197421643247451524op_bit @ nat )
@ ( bit_se4197421643247451524op_bit @ nat ) ) ).
% drop_bit_natural.rsp
thf(fact_7711_take__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ ( bit_se2584673776208193580ke_bit @ int )
@ ( bit_se2584673776208193580ke_bit @ int ) ) ).
% take_bit_integer.rsp
thf(fact_7712_take__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ ( bit_se2584673776208193580ke_bit @ nat )
@ ( bit_se2584673776208193580ke_bit @ nat ) ) ).
% take_bit_natural.rsp
thf(fact_7713_inj__graph,axiom,
! [B: $tType,A: $tType] :
( inj_on @ ( A > B ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [F4: A > B] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y5: B] :
( Y5
= ( F4 @ X4 ) ) ) )
@ ( top_top @ ( set @ ( A > B ) ) ) ) ).
% inj_graph
thf(fact_7714_less__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ( ord_less @ nat )
@ ( ord_less @ nat ) ) ).
% less_natural.rsp
thf(fact_7715_less__eq__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ( ord_less_eq @ nat )
@ ( ord_less_eq @ nat ) ) ).
% less_eq_natural.rsp
thf(fact_7716_inj__on__diff__nat,axiom,
! [N8: set @ nat,K: nat] :
( ! [N2: nat] :
( ( member @ nat @ N2 @ N8 )
=> ( ord_less_eq @ nat @ K @ N2 ) )
=> ( inj_on @ nat @ nat
@ ^ [N6: nat] : ( minus_minus @ nat @ N6 @ K )
@ N8 ) ) ).
% inj_on_diff_nat
thf(fact_7717_minus__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ ( minus_minus @ nat )
@ ( minus_minus @ nat ) ) ).
% minus_natural.rsp
thf(fact_7718_sub_Orsp,axiom,
( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > int )
@ ^ [Y6: num,Z4: num] : Y6 = Z4
@ ( bNF_rel_fun @ num @ num @ int @ int
@ ^ [Y6: num,Z4: num] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ ^ [M5: num,N6: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N6 ) )
@ ^ [M5: num,N6: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N6 ) ) ) ).
% sub.rsp
thf(fact_7719_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,Z4: $o] : Y6 = Z4
@ R3
@ ( zero_neq_one_of_bool @ A )
@ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% transfer_rule_of_bool
thf(fact_7720_bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat > $o,Z4: nat > $o] : Y6 = Z4
@ ( bit_se5641148757651400278ts_bit @ nat )
@ ( bit_se5641148757651400278ts_bit @ nat ) ) ).
% bit_natural.rsp
thf(fact_7721_unset__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ ( bit_se2638667681897837118et_bit @ int )
@ ( bit_se2638667681897837118et_bit @ int ) ) ).
% unset_bit_integer.rsp
thf(fact_7722_flip__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ ( bit_se8732182000553998342ip_bit @ int )
@ ( bit_se8732182000553998342ip_bit @ int ) ) ).
% flip_bit_integer.rsp
thf(fact_7723_inj__Suc,axiom,
! [N8: set @ nat] : ( inj_on @ nat @ nat @ suc @ N8 ) ).
% inj_Suc
thf(fact_7724_Suc_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ suc
@ suc ) ).
% Suc.rsp
thf(fact_7725_VEBTi_Ocase__transfer,axiom,
! [A: $tType,B: $tType,S3: A > B > $o] :
( bNF_rel_fun @ ( ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > A ) @ ( ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > B ) @ ( ( $o > $o > A ) > vEBT_VEBTi > A ) @ ( ( $o > $o > B ) > vEBT_VEBTi > B )
@ ( bNF_rel_fun @ ( option @ ( product_prod @ nat @ nat ) ) @ ( option @ ( product_prod @ nat @ nat ) ) @ ( nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > A ) @ ( nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > B )
@ ^ [Y6: option @ ( product_prod @ nat @ nat ),Z4: option @ ( product_prod @ nat @ nat )] : Y6 = Z4
@ ( bNF_rel_fun @ nat @ nat @ ( ( array @ vEBT_VEBTi ) > vEBT_VEBTi > A ) @ ( ( array @ vEBT_VEBTi ) > vEBT_VEBTi > B )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ ( array @ vEBT_VEBTi ) @ ( array @ vEBT_VEBTi ) @ ( vEBT_VEBTi > A ) @ ( vEBT_VEBTi > B )
@ ^ [Y6: array @ vEBT_VEBTi,Z4: array @ vEBT_VEBTi] : Y6 = Z4
@ ( bNF_rel_fun @ vEBT_VEBTi @ vEBT_VEBTi @ A @ B
@ ^ [Y6: vEBT_VEBTi,Z4: vEBT_VEBTi] : Y6 = Z4
@ S3 ) ) ) )
@ ( bNF_rel_fun @ ( $o > $o > A ) @ ( $o > $o > B ) @ ( vEBT_VEBTi > A ) @ ( vEBT_VEBTi > B )
@ ( bNF_rel_fun @ $o @ $o @ ( $o > A ) @ ( $o > B )
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4
@ ( bNF_rel_fun @ $o @ $o @ A @ B
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4
@ S3 ) )
@ ( bNF_rel_fun @ vEBT_VEBTi @ vEBT_VEBTi @ A @ B
@ ^ [Y6: vEBT_VEBTi,Z4: vEBT_VEBTi] : Y6 = Z4
@ S3 ) )
@ ( vEBT_case_VEBTi @ A )
@ ( vEBT_case_VEBTi @ B ) ) ).
% VEBTi.case_transfer
thf(fact_7726_divide__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ ( divide_divide @ nat )
@ ( divide_divide @ nat ) ) ).
% divide_natural.rsp
thf(fact_7727_modulo__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ ( modulo_modulo @ nat )
@ ( modulo_modulo @ nat ) ) ).
% modulo_natural.rsp
thf(fact_7728_Code__Numeral_Oset__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ ( bit_se5668285175392031749et_bit @ int )
@ ( bit_se5668285175392031749et_bit @ int ) ) ).
% Code_Numeral.set_bit_integer.rsp
thf(fact_7729_plus__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ ( plus_plus @ nat )
@ ( plus_plus @ nat ) ) ).
% plus_natural.rsp
thf(fact_7730_times__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ ( times_times @ nat )
@ ( times_times @ nat ) ) ).
% times_natural.rsp
thf(fact_7731_or__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ ( bit_se1065995026697491101ons_or @ nat )
@ ( bit_se1065995026697491101ons_or @ nat ) ) ).
% or_natural.rsp
thf(fact_7732_integer__of__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ int @ int
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( semiring_1_of_nat @ int )
@ ( semiring_1_of_nat @ int ) ) ).
% integer_of_natural.rsp
thf(fact_7733_swap__inj__on,axiom,
! [B: $tType,A: $tType,A5: set @ ( product_prod @ A @ B )] :
( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [I3: A,J3: B] : ( product_Pair @ B @ A @ J3 @ I3 ) )
@ A5 ) ).
% swap_inj_on
thf(fact_7734_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F3: A > B,X6: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ ( F3 @ X4 ) )
@ X6 ) ).
% inj_on_convol_ident
thf(fact_7735_inj__Pair_I2_J,axiom,
! [B: $tType,A: $tType,C3: A > B,S3: set @ A] :
( inj_on @ A @ ( product_prod @ B @ A )
@ ^ [X4: A] : ( product_Pair @ B @ A @ ( C3 @ X4 ) @ X4 )
@ S3 ) ).
% inj_Pair(2)
thf(fact_7736_inj__Pair_I1_J,axiom,
! [B: $tType,A: $tType,C3: A > B,S3: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ ( C3 @ X4 ) )
@ S3 ) ).
% inj_Pair(1)
thf(fact_7737_inj__singleton,axiom,
! [A: $tType,A5: set @ A] :
( inj_on @ A @ ( set @ A )
@ ^ [X4: A] : ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) )
@ A5 ) ).
% inj_singleton
thf(fact_7738_inj__on__set__encode,axiom,
inj_on @ ( set @ nat ) @ nat @ nat_set_encode @ ( collect @ ( set @ nat ) @ ( finite_finite2 @ nat ) ) ).
% inj_on_set_encode
thf(fact_7739_mask__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ int @ int
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bit_se2239418461657761734s_mask @ int )
@ ( bit_se2239418461657761734s_mask @ int ) ) ).
% mask_integer.rsp
thf(fact_7740_mask__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bit_se2239418461657761734s_mask @ nat )
@ ( bit_se2239418461657761734s_mask @ nat ) ) ).
% mask_natural.rsp
thf(fact_7741_push__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ^ [Y6: int,Z4: int] : Y6 = Z4 )
@ ( bit_se4730199178511100633sh_bit @ int )
@ ( bit_se4730199178511100633sh_bit @ int ) ) ).
% push_bit_integer.rsp
thf(fact_7742_push__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ ( bit_se4730199178511100633sh_bit @ nat )
@ ( bit_se4730199178511100633sh_bit @ nat ) ) ).
% push_bit_natural.rsp
thf(fact_7743_and__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ ( bit_se5824344872417868541ns_and @ nat )
@ ( bit_se5824344872417868541ns_and @ nat ) ) ).
% and_natural.rsp
thf(fact_7744_f__inv__on__f,axiom,
! [B: $tType,A: $tType,Y: A,F3: B > A,A5: set @ B] :
( ( member @ A @ Y @ ( image @ B @ A @ F3 @ A5 ) )
=> ( ( F3 @ ( inv_on @ B @ A @ F3 @ A5 @ Y ) )
= Y ) ) ).
% f_inv_on_f
thf(fact_7745_inv__on__f__range,axiom,
! [A: $tType,B: $tType,Y: A,F3: B > A,A5: set @ B] :
( ( member @ A @ Y @ ( image @ B @ A @ F3 @ A5 ) )
=> ( member @ B @ ( inv_on @ B @ A @ F3 @ A5 @ Y ) @ A5 ) ) ).
% inv_on_f_range
thf(fact_7746_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [F2: A > nat,N2: nat] :
( ( ( image @ A @ nat @ F2 @ A5 )
= ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) )
& ( inj_on @ A @ nat @ F2 @ A5 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_7747_finite__imp__inj__to__nat__seg_H,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ~ ! [F2: A > nat] :
( ? [N2: nat] :
( ( image @ A @ nat @ F2 @ A5 )
= ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) )
=> ~ ( inj_on @ A @ nat @ F2 @ A5 ) ) ) ).
% finite_imp_inj_to_nat_seg'
thf(fact_7748_summable__reindex,axiom,
! [F3: nat > real,G3: nat > nat] :
( ( summable @ real @ F3 )
=> ( ( inj_on @ nat @ nat @ G3 @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X3 ) )
=> ( summable @ real @ ( comp @ nat @ real @ nat @ F3 @ G3 ) ) ) ) ) ).
% summable_reindex
thf(fact_7749_is__up_Orsp,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ( bNF_rel_fun @ ( int > int ) @ ( int > int ) @ $o @ $o
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) )
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ L4 ) ) )
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4
@ ^ [Uu3: int > int] : ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
@ ^ [Uu3: int > int] : ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ).
% is_up.rsp
thf(fact_7750_is__down_Orsp,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( bNF_rel_fun @ ( int > int ) @ ( int > int ) @ $o @ $o
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) )
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ L4 ) ) )
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4
@ ^ [Uu3: int > int] : ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
@ ^ [Uu3: int > int] : ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% is_down.rsp
thf(fact_7751_suminf__reindex__mono,axiom,
! [F3: nat > real,G3: nat > nat] :
( ( summable @ real @ F3 )
=> ( ( inj_on @ nat @ nat @ G3 @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X3 ) )
=> ( ord_less_eq @ real @ ( suminf @ real @ ( comp @ nat @ real @ nat @ F3 @ G3 ) ) @ ( suminf @ real @ F3 ) ) ) ) ) ).
% suminf_reindex_mono
thf(fact_7752_suminf__reindex,axiom,
! [F3: nat > real,G3: nat > nat] :
( ( summable @ real @ F3 )
=> ( ( inj_on @ nat @ nat @ G3 @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X3 ) )
=> ( ! [X3: nat] :
( ~ ( member @ nat @ X3 @ ( image @ nat @ nat @ G3 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F3 @ X3 )
= ( zero_zero @ real ) ) )
=> ( ( suminf @ real @ ( comp @ nat @ real @ nat @ F3 @ G3 ) )
= ( suminf @ real @ F3 ) ) ) ) ) ) ).
% suminf_reindex
thf(fact_7753_inj__sgn__power,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( inj_on @ real @ real
@ ^ [Y5: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y5 ) @ ( power_power @ real @ ( abs_abs @ real @ Y5 ) @ N ) )
@ ( top_top @ ( set @ real ) ) ) ) ).
% inj_sgn_power
thf(fact_7754_signed__drop__bit_Orsp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) )
@ ^ [K3: int,L4: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K3 )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L4 ) ) )
@ ^ [N6: nat] : ( comp @ int @ int @ int @ ( bit_se4197421643247451524op_bit @ int @ N6 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ^ [N6: nat] : ( comp @ int @ int @ int @ ( bit_se4197421643247451524op_bit @ int @ N6 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% signed_drop_bit.rsp
thf(fact_7755_ex__subset__image__inj,axiom,
! [A: $tType,B: $tType,F3: B > A,S3: set @ B,P2: ( set @ A ) > $o] :
( ( ? [T11: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T11 @ ( image @ B @ A @ F3 @ S3 ) )
& ( P2 @ T11 ) ) )
= ( ? [T11: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ T11 @ S3 )
& ( inj_on @ B @ A @ F3 @ T11 )
& ( P2 @ ( image @ B @ A @ F3 @ T11 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_7756_all__subset__image__inj,axiom,
! [A: $tType,B: $tType,F3: B > A,S3: set @ B,P2: ( set @ A ) > $o] :
( ( ! [T11: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T11 @ ( image @ B @ A @ F3 @ S3 ) )
=> ( P2 @ T11 ) ) )
= ( ! [T11: set @ B] :
( ( ( ord_less_eq @ ( set @ B ) @ T11 @ S3 )
& ( inj_on @ B @ A @ F3 @ T11 ) )
=> ( P2 @ ( image @ B @ A @ F3 @ T11 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_7757_unset__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ ( bit_se2638667681897837118et_bit @ nat )
@ ( bit_se2638667681897837118et_bit @ nat ) ) ).
% unset_bit_natural.rsp
thf(fact_7758_flip__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ ( bit_se8732182000553998342ip_bit @ nat )
@ ( bit_se8732182000553998342ip_bit @ nat ) ) ).
% flip_bit_natural.rsp
thf(fact_7759_set__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
@ ( bit_se5668285175392031749et_bit @ nat )
@ ( bit_se5668285175392031749et_bit @ nat ) ) ).
% set_bit_natural.rsp
thf(fact_7760_signed__drop__bit_Otransfer,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( ( word @ A ) > ( word @ A ) )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( bNF_rel_fun @ int @ ( word @ A ) @ int @ ( word @ A ) @ ( pcr_word @ A ) @ ( pcr_word @ A ) )
@ ^ [N6: nat] : ( comp @ int @ int @ int @ ( bit_se4197421643247451524op_bit @ int @ N6 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( signed_drop_bit @ A ) ) ) ).
% signed_drop_bit.transfer
thf(fact_7761_signed_Otransfer,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ( bNF_rel_fun @ int @ ( word @ B ) @ A @ A @ ( pcr_word @ B )
@ ^ [Y6: A,Z4: A] : Y6 = Z4
@ ( comp @ int @ A @ int @ ( ring_1_of_int @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( ring_1_signed @ B @ A ) ) ) ).
% signed.transfer
thf(fact_7762_less__eq__word_Otransfer,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ int @ ( word @ A ) @ ( int > $o ) @ ( ( word @ A ) > $o ) @ ( pcr_word @ A )
@ ( bNF_rel_fun @ int @ ( word @ A ) @ $o @ $o @ ( pcr_word @ A )
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ^ [A8: int,B8: int] : ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ A8 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ B8 ) )
@ ( ord_less_eq @ ( word @ A ) ) ) ) ).
% less_eq_word.transfer
thf(fact_7763_less__word_Otransfer,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ int @ ( word @ A ) @ ( int > $o ) @ ( ( word @ A ) > $o ) @ ( pcr_word @ A )
@ ( bNF_rel_fun @ int @ ( word @ A ) @ $o @ $o @ ( pcr_word @ A )
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ^ [A8: int,B8: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ A8 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ B8 ) )
@ ( ord_less @ ( word @ A ) ) ) ) ).
% less_word.transfer
thf(fact_7764_bit__word_Otransfer,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ int @ ( word @ A ) @ ( nat > $o ) @ ( nat > $o ) @ ( pcr_word @ A )
@ ^ [Y6: nat > $o,Z4: nat > $o] : Y6 = Z4
@ ^ [K3: int,N6: nat] :
( ( ord_less @ nat @ N6 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ K3 @ N6 ) )
@ ( bit_se5641148757651400278ts_bit @ ( word @ A ) ) ) ) ).
% bit_word.transfer
thf(fact_7765_word__test__bit__transfer,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ int @ ( word @ A ) @ ( nat > $o ) @ ( nat > $o ) @ ( pcr_word @ A )
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ^ [X4: int,N6: nat] :
( ( ord_less @ nat @ N6 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ X4 @ N6 ) )
@ ( bit_se5641148757651400278ts_bit @ ( word @ A ) ) ) ) ).
% word_test_bit_transfer
thf(fact_7766_word__sle_Otransfer,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ int @ ( word @ A ) @ ( int > $o ) @ ( ( word @ A ) > $o ) @ ( pcr_word @ A )
@ ( bNF_rel_fun @ int @ ( word @ A ) @ $o @ $o @ ( pcr_word @ A )
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ^ [K3: int,L4: int] : ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ K3 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L4 ) )
@ ( word_sle @ A ) ) ) ).
% word_sle.transfer
thf(fact_7767_word__sless_Otransfer,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ int @ ( word @ A ) @ ( int > $o ) @ ( ( word @ A ) > $o ) @ ( pcr_word @ A )
@ ( bNF_rel_fun @ int @ ( word @ A ) @ $o @ $o @ ( pcr_word @ A )
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ^ [K3: int,L4: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ K3 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L4 ) )
@ ( word_sless @ A ) ) ) ).
% word_sless.transfer
thf(fact_7768_signed__divide__word_Otransfer,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ int @ ( word @ A ) @ ( int > int ) @ ( ( word @ A ) > ( word @ A ) ) @ ( pcr_word @ A ) @ ( bNF_rel_fun @ int @ ( word @ A ) @ int @ ( word @ A ) @ ( pcr_word @ A ) @ ( pcr_word @ A ) )
@ ^ [K3: int,L4: int] : ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ K3 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L4 ) )
@ ( signed7115095781618012415divide @ ( word @ A ) ) ) ) ).
% signed_divide_word.transfer
thf(fact_7769_sshiftr__transfer,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ int @ ( word @ A ) @ ( nat > int ) @ ( nat > ( word @ A ) ) @ ( pcr_word @ A )
@ ( bNF_rel_fun @ nat @ nat @ int @ ( word @ A )
@ ^ [Y6: nat,Z4: nat] : Y6 = Z4
@ ( pcr_word @ A ) )
@ ^ [K3: int,N6: nat] : ( bit_se4197421643247451524op_bit @ int @ N6 @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ K3 ) )
@ ( bit_Sh8784991116023147202shiftr @ A ) ) ) ).
% sshiftr_transfer
thf(fact_7770_signed__cast_Otransfer,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ( bNF_rel_fun @ int @ ( word @ A ) @ int @ ( word @ B ) @ ( pcr_word @ A ) @ ( pcr_word @ B ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) @ ( signed_cast @ A @ B ) ) ) ).
% signed_cast.transfer
thf(fact_7771_the__signed__int_Otransfer,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ int @ ( word @ A ) @ int @ int @ ( pcr_word @ A )
@ ^ [Y6: int,Z4: int] : Y6 = Z4
@ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( the_signed_int @ A ) ) ) ).
% the_signed_int.transfer
thf(fact_7772_signed__modulo__word_Otransfer,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( bNF_rel_fun @ int @ ( word @ A ) @ ( int > int ) @ ( ( word @ A ) > ( word @ A ) ) @ ( pcr_word @ A ) @ ( bNF_rel_fun @ int @ ( word @ A ) @ int @ ( word @ A ) @ ( pcr_word @ A ) @ ( pcr_word @ A ) )
@ ^ [K3: int,L4: int] : ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ K3 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L4 ) )
@ ( signed6721504322012087516modulo @ ( word @ A ) ) ) ) ).
% signed_modulo_word.transfer
thf(fact_7773_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G3: A > B,C4: set @ A,B5: set @ A,X: A] :
( ( inj_on @ A @ B @ G3 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ C4 @ ( sup_sup @ ( set @ A ) @ B5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ ( B > A )
@ ^ [I3: B] : ( if @ A @ ( member @ B @ I3 @ ( image @ A @ B @ G3 @ C4 ) ) @ ( the_inv_into @ A @ B @ C4 @ G3 @ I3 ) @ X )
@ ( bNF_Wellorder_Func @ B @ A @ ( top_top @ ( set @ B ) ) @ ( sup_sup @ ( set @ A ) @ B5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_7774_refl__ge__eq,axiom,
! [A: $tType,R3: A > A > $o] :
( ! [X3: A] : ( R3 @ X3 @ X3 )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y6: A,Z4: A] : Y6 = Z4
@ R3 ) ) ).
% refl_ge_eq
thf(fact_7775_ge__eq__refl,axiom,
! [A: $tType,R3: A > A > $o,X: A] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y6: A,Z4: A] : Y6 = Z4
@ R3 )
=> ( R3 @ X @ X ) ) ).
% ge_eq_refl
thf(fact_7776_Ball__Collect,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A6: set @ A,P5: A > $o] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( collect @ A @ P5 ) ) ) ) ).
% Ball_Collect
thf(fact_7777_Func__map__surj,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F1: B > A,A19: set @ B,B14: set @ A,F22: C > D,B23: set @ C,A26: set @ D] :
( ( ( image @ B @ A @ F1 @ A19 )
= B14 )
=> ( ( inj_on @ C @ D @ F22 @ B23 )
=> ( ( ord_less_eq @ ( set @ D ) @ ( image @ C @ D @ F22 @ B23 ) @ A26 )
=> ( ( ( B23
= ( bot_bot @ ( set @ C ) ) )
=> ( A26
= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( bNF_Wellorder_Func @ C @ A @ B23 @ B14 )
= ( image @ ( D > B ) @ ( C > A ) @ ( bNF_We4925052301507509544nc_map @ C @ B @ A @ D @ B23 @ F1 @ F22 ) @ ( bNF_Wellorder_Func @ D @ B @ A26 @ A19 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_7778_lexord__take__index__conv,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R2 ) )
= ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) )
& ( ( take @ A @ ( size_size @ ( list @ A ) @ X ) @ Y )
= X ) )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) ) )
& ( ( take @ A @ I3 @ X )
= ( take @ A @ I3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X @ I3 ) @ ( nth @ A @ Y @ I3 ) ) @ R2 ) ) ) ) ).
% lexord_take_index_conv
thf(fact_7779_lexord__cons__cons,axiom,
! [A: $tType,A2: A,X: list @ A,B2: A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A2 @ X ) @ ( cons @ A @ B2 @ Y ) ) @ ( lexord @ A @ R2 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
| ( ( A2 = B2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R2 ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_7780_lexord__linear,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: list @ A,Y: list @ A] :
( ! [A4: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ R2 )
| ( A4 = B4 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A4 ) @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R2 ) )
| ( X = Y )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_linear
thf(fact_7781_lexord__irreflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_irreflexive
thf(fact_7782_lexord__partial__trans,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A ),Ys: list @ A,Zs: list @ A] :
( ! [X3: A,Y3: A,Z3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z3 ) @ R2 ) ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( lexord @ A @ R2 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_7783_lexord__append__leftD,axiom,
! [A: $tType,X: list @ A,U: list @ A,V3: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V3 ) ) @ ( lexord @ A @ R2 ) )
=> ( ! [A4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V3 ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_append_leftD
thf(fact_7784_lexord__append__left__rightI,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),U: list @ A,X: list @ A,Y: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A2 @ X ) ) @ ( append @ A @ U @ ( cons @ A @ B2 @ Y ) ) ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_append_left_rightI
thf(fact_7785_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R2 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R2 ) )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_7786_lexord__sufI,axiom,
! [A: $tType,U: list @ A,W2: list @ A,R2: set @ ( product_prod @ A @ A ),V3: list @ A,Z: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W2 ) @ ( lexord @ A @ R2 ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W2 ) @ ( size_size @ ( list @ A ) @ U ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V3 ) @ ( append @ A @ W2 @ Z ) ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_sufI
thf(fact_7787_Func__map,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,G3: A > B,A26: set @ A,A19: set @ B,F1: B > C,B14: set @ C,F22: D > A,B23: set @ D] :
( ( member @ ( A > B ) @ G3 @ ( bNF_Wellorder_Func @ A @ B @ A26 @ A19 ) )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ F1 @ A19 ) @ B14 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ D @ A @ F22 @ B23 ) @ A26 )
=> ( member @ ( D > C ) @ ( bNF_We4925052301507509544nc_map @ D @ B @ C @ A @ B23 @ F1 @ F22 @ G3 ) @ ( bNF_Wellorder_Func @ D @ C @ B23 @ B14 ) ) ) ) ) ).
% Func_map
thf(fact_7788_Func__non__emp,axiom,
! [A: $tType,B: $tType,B5: set @ A,A5: set @ B] :
( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bNF_Wellorder_Func @ B @ A @ A5 @ B5 )
!= ( bot_bot @ ( set @ ( B > A ) ) ) ) ) ).
% Func_non_emp
thf(fact_7789_Func__is__emp,axiom,
! [A: $tType,B: $tType,A5: set @ A,B5: set @ B] :
( ( ( bNF_Wellorder_Func @ A @ B @ A5 @ B5 )
= ( bot_bot @ ( set @ ( A > B ) ) ) )
= ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Func_is_emp
thf(fact_7790_lexord__def,axiom,
! [A: $tType] :
( ( lexord @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [X4: list @ A,Y5: list @ A] :
? [A8: A,V4: list @ A] :
( ( Y5
= ( append @ A @ X4 @ ( cons @ A @ A8 @ V4 ) ) )
| ? [U3: list @ A,B8: A,C7: A,W3: list @ A,Z2: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ C7 ) @ R5 )
& ( X4
= ( append @ A @ U3 @ ( cons @ A @ B8 @ W3 ) ) )
& ( Y5
= ( append @ A @ U3 @ ( cons @ A @ C7 @ Z2 ) ) ) ) ) ) ) ) ) ).
% lexord_def
thf(fact_7791_msb__big,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( most_s684356279273892711sb_msb @ ( word @ A ) )
= ( ord_less_eq @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% msb_big
thf(fact_7792_integer__of__num_I3_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit1 @ N ) )
= ( plus_plus @ code_integer @ ( plus_plus @ code_integer @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% integer_of_num(3)
thf(fact_7793_word__sless__msb__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sless @ A )
= ( ^ [X4: word @ A,Y5: word @ A] :
( ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ Y5 )
=> ( most_s684356279273892711sb_msb @ ( word @ A ) @ X4 ) )
& ( ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ X4 )
& ~ ( most_s684356279273892711sb_msb @ ( word @ A ) @ Y5 ) )
| ( ord_less @ ( word @ A ) @ X4 @ Y5 ) ) ) ) ) ) ).
% word_sless_msb_less
thf(fact_7794_integer__of__num__def,axiom,
( code_integer_of_num
= ( numeral_numeral @ code_integer ) ) ).
% integer_of_num_def
thf(fact_7795_word__msb__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( most_s684356279273892711sb_msb @ ( word @ A ) )
= ( ^ [W3: word @ A] : ( ord_less @ int @ ( ring_1_signed @ A @ int @ W3 ) @ ( zero_zero @ int ) ) ) ) ) ).
% word_msb_sint
thf(fact_7796_word__sle__msb__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sle @ A )
= ( ^ [X4: word @ A,Y5: word @ A] :
( ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ Y5 )
=> ( most_s684356279273892711sb_msb @ ( word @ A ) @ X4 ) )
& ( ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ X4 )
& ~ ( most_s684356279273892711sb_msb @ ( word @ A ) @ Y5 ) )
| ( ord_less_eq @ ( word @ A ) @ X4 @ Y5 ) ) ) ) ) ) ).
% word_sle_msb_le
thf(fact_7797_msb__word__iff__bit,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( most_s684356279273892711sb_msb @ ( word @ A ) )
= ( ^ [W3: word @ A] : ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W3 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% msb_word_iff_bit
thf(fact_7798_integer__of__num__triv_I1_J,axiom,
( ( code_integer_of_num @ one2 )
= ( one_one @ code_integer ) ) ).
% integer_of_num_triv(1)
thf(fact_7799_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_7800_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_7801_not__msb__from__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V3: word @ A] :
( ( ord_less @ ( word @ A ) @ V3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ~ ( most_s684356279273892711sb_msb @ ( word @ A ) @ V3 ) ) ) ).
% not_msb_from_less
thf(fact_7802_eventually__filtercomap__at__topological,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ B )
=> ! [P2: A > $o,F3: A > B,A5: B,B5: set @ B] :
( ( eventually @ A @ P2 @ ( filtercomap @ A @ B @ F3 @ ( topolo174197925503356063within @ B @ A5 @ B5 ) ) )
= ( ? [S6: set @ B] :
( ( topolo1002775350975398744n_open @ B @ S6 )
& ( member @ B @ A5 @ S6 )
& ! [X4: A] :
( ( member @ B @ ( F3 @ X4 ) @ ( minus_minus @ ( set @ B ) @ ( inf_inf @ ( set @ B ) @ S6 @ B5 ) @ ( insert @ B @ A5 @ ( bot_bot @ ( set @ B ) ) ) ) )
=> ( P2 @ X4 ) ) ) ) ) ) ).
% eventually_filtercomap_at_topological
thf(fact_7803_take__hd__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( append @ A @ ( take @ A @ N @ Xs ) @ ( cons @ A @ ( hd @ A @ ( drop @ A @ N @ Xs ) ) @ ( nil @ A ) ) )
= ( take @ A @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_7804_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( hd @ nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_7805_hd__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( hd @ A @ ( replicate @ A @ N @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_7806_hd__take,axiom,
! [A: $tType,J: nat,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ J )
=> ( ( hd @ A @ ( take @ A @ J @ Xs ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_take
thf(fact_7807_hd__conv__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ Xs )
= ( nth @ A @ Xs @ ( zero_zero @ nat ) ) ) ) ).
% hd_conv_nth
thf(fact_7808_filtercomap__sup,axiom,
! [A: $tType,B: $tType,F3: A > B,F14: filter @ B,F24: filter @ B] : ( ord_less_eq @ ( filter @ A ) @ ( sup_sup @ ( filter @ A ) @ ( filtercomap @ A @ B @ F3 @ F14 ) @ ( filtercomap @ A @ B @ F3 @ F24 ) ) @ ( filtercomap @ A @ B @ F3 @ ( sup_sup @ ( filter @ B ) @ F14 @ F24 ) ) ) ).
% filtercomap_sup
thf(fact_7809_filtercomap__mono,axiom,
! [B: $tType,A: $tType,F5: filter @ A,F7: filter @ A,F3: B > A] :
( ( ord_less_eq @ ( filter @ A ) @ F5 @ F7 )
=> ( ord_less_eq @ ( filter @ B ) @ ( filtercomap @ B @ A @ F3 @ F5 ) @ ( filtercomap @ B @ A @ F3 @ F7 ) ) ) ).
% filtercomap_mono
thf(fact_7810_filterlim__iff__le__filtercomap,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F4: A > B,F11: filter @ B,G9: filter @ A] : ( ord_less_eq @ ( filter @ A ) @ G9 @ ( filtercomap @ A @ B @ F4 @ F11 ) ) ) ) ).
% filterlim_iff_le_filtercomap
thf(fact_7811_hd__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ( hd @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( product_Pair @ A @ B @ ( hd @ A @ Xs ) @ ( hd @ B @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_7812_eventually__filtercomap__at__top__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [P2: B > $o,F3: B > A] :
( ( eventually @ B @ P2 @ ( filtercomap @ B @ A @ F3 @ ( at_top @ A ) ) )
= ( ? [N7: A] :
! [X4: B] :
( ( ord_less_eq @ A @ N7 @ ( F3 @ X4 ) )
=> ( P2 @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_top_linorder
thf(fact_7813_eventually__filtercomap__at__top__dense,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P2: B > $o,F3: B > A] :
( ( eventually @ B @ P2 @ ( filtercomap @ B @ A @ F3 @ ( at_top @ A ) ) )
= ( ? [N7: A] :
! [X4: B] :
( ( ord_less @ A @ N7 @ ( F3 @ X4 ) )
=> ( P2 @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_top_dense
thf(fact_7814_eventually__filtercomap__at__bot__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [P2: B > $o,F3: B > A] :
( ( eventually @ B @ P2 @ ( filtercomap @ B @ A @ F3 @ ( at_bot @ A ) ) )
= ( ? [N7: A] :
! [X4: B] :
( ( ord_less_eq @ A @ ( F3 @ X4 ) @ N7 )
=> ( P2 @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_bot_linorder
thf(fact_7815_eventually__filtercomap__at__bot__dense,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P2: B > $o,F3: B > A] :
( ( eventually @ B @ P2 @ ( filtercomap @ B @ A @ F3 @ ( at_bot @ A ) ) )
= ( ? [N7: A] :
! [X4: B] :
( ( ord_less @ A @ ( F3 @ X4 ) @ N7 )
=> ( P2 @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_bot_dense
thf(fact_7816_sorted__hd__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( hd @ A @ Xs ) @ X5 ) ) ) ) ) ).
% sorted_hd_min
thf(fact_7817_hd__drop__conv__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( hd @ A @ ( drop @ A @ N @ Xs ) )
= ( nth @ A @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_7818_hd__butlast,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( hd @ A @ ( butlast @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_butlast
thf(fact_7819_sorted__hd__last,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( L
!= ( nil @ A ) )
=> ( ord_less_eq @ A @ ( hd @ A @ L ) @ ( last @ A @ L ) ) ) ) ) ).
% sorted_hd_last
thf(fact_7820_hd__last__singletonI,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( hd @ A @ Xs )
= ( last @ A @ Xs ) )
=> ( ( distinct @ A @ Xs )
=> ( Xs
= ( cons @ A @ ( hd @ A @ Xs ) @ ( nil @ A ) ) ) ) ) ) ).
% hd_last_singletonI
thf(fact_7821_filtercomap__SUP,axiom,
! [A: $tType,C: $tType,B: $tType,F3: A > C,F5: B > ( filter @ C ),B5: set @ B] :
( ord_less_eq @ ( filter @ A )
@ ( complete_Sup_Sup @ ( filter @ A )
@ ( image @ B @ ( filter @ A )
@ ^ [B8: B] : ( filtercomap @ A @ C @ F3 @ ( F5 @ B8 ) )
@ B5 ) )
@ ( filtercomap @ A @ C @ F3 @ ( complete_Sup_Sup @ ( filter @ C ) @ ( image @ B @ ( filter @ C ) @ F5 @ B5 ) ) ) ) ).
% filtercomap_SUP
thf(fact_7822_slice__head,axiom,
! [A: $tType,From: nat,To: nat,Xs: list @ A] :
( ( ord_less @ nat @ From @ To )
=> ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( hd @ A @ ( slice @ A @ From @ To @ Xs ) )
= ( nth @ A @ Xs @ From ) ) ) ) ).
% slice_head
thf(fact_7823_insort__key__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [A2: B,Xs: list @ B,F3: B > A] :
( ( member @ B @ A2 @ ( set2 @ B @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( ( hd @ B
@ ( filter2 @ B
@ ^ [X4: B] :
( ( F3 @ A2 )
= ( F3 @ X4 ) )
@ Xs ) )
= A2 )
=> ( ( linorder_insort_key @ B @ A @ F3 @ A2 @ ( remove1 @ B @ A2 @ Xs ) )
= Xs ) ) ) ) ) ).
% insort_key_remove1
thf(fact_7824_test__bit__rcat,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [Sw: nat,Wl: list @ ( word @ A ),Rc: word @ B,N: nat] :
( ( Sw
= ( size_size @ ( word @ A ) @ ( hd @ ( word @ A ) @ Wl ) ) )
=> ( ( Rc
= ( word_rcat @ A @ B @ Wl ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ Rc @ N )
= ( ( ord_less @ nat @ N @ ( size_size @ ( word @ B ) @ Rc ) )
& ( ord_less @ nat @ ( divide_divide @ nat @ N @ Sw ) @ ( size_size @ ( list @ ( word @ A ) ) @ Wl ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( nth @ ( word @ A ) @ ( rev @ ( word @ A ) @ Wl ) @ ( divide_divide @ nat @ N @ Sw ) ) @ ( modulo_modulo @ nat @ N @ Sw ) ) ) ) ) ) ) ).
% test_bit_rcat
thf(fact_7825_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_7826_fst__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F3: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F3 ) )
= ( product_fst @ A @ B ) ) ).
% fst_comp_apsnd
thf(fact_7827_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,X: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F3 @ X ) )
= ( product_fst @ A @ C @ X ) ) ).
% fst_apsnd
thf(fact_7828_img__fst,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,S3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S3 )
=> ( member @ A @ A2 @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S3 ) ) ) ).
% img_fst
thf(fact_7829_fst__divmod__nat,axiom,
! [M: nat,N: nat] :
( ( product_fst @ nat @ nat @ ( divmod_nat @ M @ N ) )
= ( divide_divide @ nat @ M @ N ) ) ).
% fst_divmod_nat
thf(fact_7830_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_7831_sorted__wrt__map__linord,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [L: list @ ( product_prod @ A @ B )] :
( ( sorted_wrt @ ( product_prod @ A @ B )
@ ^ [X4: product_prod @ A @ B,Y5: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ X4 ) @ ( product_fst @ A @ B @ Y5 ) )
@ L )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) ) ) ) ).
% sorted_wrt_map_linord
thf(fact_7832_map__fst__mk__snd,axiom,
! [B: $tType,A: $tType,K: B,L: list @ A] :
( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ K )
@ L ) )
= L ) ).
% map_fst_mk_snd
thf(fact_7833_sorted__wrt__map__rev__linord,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [L: list @ ( product_prod @ A @ B )] :
( ( sorted_wrt @ ( product_prod @ A @ B )
@ ^ [X4: product_prod @ A @ B,Y5: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ Y5 ) @ ( product_fst @ A @ B @ X4 ) )
@ L )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) ) ) ) ) ).
% sorted_wrt_map_rev_linord
thf(fact_7834_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_7835_map__fst__mk__fst,axiom,
! [B: $tType,A: $tType,K: A,L: list @ B] :
( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K ) @ L ) )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) ) ).
% map_fst_mk_fst
thf(fact_7836_fst__image__mp,axiom,
! [B: $tType,A: $tType,A5: set @ ( product_prod @ A @ B ),B5: set @ A,X: A,Y: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A5 ) @ B5 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ A5 )
=> ( member @ A @ X @ B5 ) ) ) ).
% fst_image_mp
thf(fact_7837_in__fst__imageE,axiom,
! [B: $tType,A: $tType,X: A,S3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ X @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S3 ) )
=> ~ ! [Y3: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ S3 ) ) ).
% in_fst_imageE
thf(fact_7838_inj__on__fst__map__to__set,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map_to_set @ A @ B @ M ) ) ).
% inj_on_fst_map_to_set
thf(fact_7839_fst__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% fst_diag_fst
thf(fact_7840_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_7841_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X16: A,X24: B] : X16 ) ) ).
% fst_def
thf(fact_7842_fn__fst__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F3: A > C] :
( ( ^ [X4: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ X4 ) ) )
= ( product_case_prod @ A @ B @ C
@ ^ [A8: A,Uu3: B] : ( F3 @ A8 ) ) ) ).
% fn_fst_conv
thf(fact_7843_fstE,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A2: A,B2: B,P2: A > $o] :
( ( X
= ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( ( P2 @ ( product_fst @ A @ B @ X ) )
=> ( P2 @ A2 ) ) ) ).
% fstE
thf(fact_7844_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A2: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_7845_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X2: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_7846_distinct__map__fst__filterI,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),P2: ( product_prod @ A @ B ) > $o] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( filter2 @ ( product_prod @ A @ B ) @ P2 @ Xs ) ) ) ) ).
% distinct_map_fst_filterI
thf(fact_7847_fstI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_fst @ A @ B @ X )
= Y ) ) ).
% fstI
thf(fact_7848_sorted__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) ) ).
% sorted_enumerate
thf(fact_7849_fst__foldl,axiom,
! [B: $tType,A: $tType,C: $tType,F3: A > C > A,G3: A > B > C > B,A2: A,B2: B,Xs: list @ C] :
( ( product_fst @ A @ B
@ ( foldl @ ( product_prod @ A @ B ) @ C
@ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ B ) )
@ ^ [A8: A,B8: B,X4: C] : ( product_Pair @ A @ B @ ( F3 @ A8 @ X4 ) @ ( G3 @ A8 @ B8 @ X4 ) ) )
@ ( product_Pair @ A @ B @ A2 @ B2 )
@ Xs ) )
= ( foldl @ A @ C @ F3 @ A2 @ Xs ) ) ).
% fst_foldl
thf(fact_7850_set__to__map__inverse,axiom,
! [B: $tType,A: $tType,S3: set @ ( product_prod @ A @ B )] :
( ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S3 )
=> ( ( map_to_set @ A @ B @ ( set_to_map @ A @ B @ S3 ) )
= S3 ) ) ).
% set_to_map_inverse
thf(fact_7851_eq__key__imp__eq__value,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ A @ B ),K: A,V1: B,V22: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V1 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V22 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( V1 = V22 ) ) ) ) ).
% eq_key_imp_eq_value
thf(fact_7852_distinct__map__fstD,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ A @ B ),X: A,Y: B,Z: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( Y = Z ) ) ) ) ).
% distinct_map_fstD
thf(fact_7853_set__to__map__simp,axiom,
! [B: $tType,A: $tType,S3: set @ ( product_prod @ A @ B ),K: A,V3: B] :
( ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S3 )
=> ( ( ( set_to_map @ A @ B @ S3 @ K )
= ( some @ B @ V3 ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V3 ) @ S3 ) ) ) ).
% set_to_map_simp
thf(fact_7854_map__of__distinct__upd4,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) ) )
=> ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ Ys ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys ) ) ) @ X @ ( none @ B ) ) ) ) ) ).
% map_of_distinct_upd4
thf(fact_7855_map__of__distinct__upd3,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B ),Y: B,Y8: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) ) )
=> ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y8 ) @ Ys ) ) ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% map_of_distinct_upd3
thf(fact_7856_fst__divmod__integer,axiom,
! [K: code_integer,L: code_integer] :
( ( product_fst @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
= ( divide_divide @ code_integer @ K @ L ) ) ).
% fst_divmod_integer
thf(fact_7857_fst__divmod__abs,axiom,
! [K: code_integer,L: code_integer] :
( ( product_fst @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
= ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).
% fst_divmod_abs
thf(fact_7858_map__of__rev__distinct,axiom,
! [B: $tType,A: $tType,M: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ M ) )
=> ( ( map_of @ A @ B @ ( rev @ ( product_prod @ A @ B ) @ M ) )
= ( map_of @ A @ B @ M ) ) ) ).
% map_of_rev_distinct
thf(fact_7859_map__of__is__SomeI,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ A @ B ),X: A,Y: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) )
=> ( ( map_of @ A @ B @ Xys @ X )
= ( some @ B @ Y ) ) ) ) ).
% map_of_is_SomeI
thf(fact_7860_Some__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),Y: B,X: A] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( some @ B @ Y )
= ( map_of @ A @ B @ Xys @ X ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% Some_eq_map_of_iff
thf(fact_7861_map__of__eq__Some__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),X: A,Y: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( map_of @ A @ B @ Xys @ X )
= ( some @ B @ Y ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% map_of_eq_Some_iff
thf(fact_7862_map__of__None__filterD,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),X: B,P2: ( product_prod @ B @ A ) > $o] :
( ( ( map_of @ B @ A @ Xs @ X )
= ( none @ A ) )
=> ( ( map_of @ B @ A @ ( filter2 @ ( product_prod @ B @ A ) @ P2 @ Xs ) @ X )
= ( none @ A ) ) ) ).
% map_of_None_filterD
thf(fact_7863_map__of__SomeD,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,Y: A] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ Y ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ Y ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs ) ) ) ).
% map_of_SomeD
thf(fact_7864_weak__map__of__SomeI,axiom,
! [A: $tType,B: $tType,K: A,X: B,L: list @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ X ) @ ( set2 @ ( product_prod @ A @ B ) @ L ) )
=> ? [X3: B] :
( ( map_of @ A @ B @ L @ K )
= ( some @ B @ X3 ) ) ) ).
% weak_map_of_SomeI
thf(fact_7865_map__of__Cons__code_I2_J,axiom,
! [C: $tType,B: $tType,L: B,K: B,V3: C,Ps2: list @ ( product_prod @ B @ C )] :
( ( ( L = K )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V3 ) @ Ps2 ) @ K )
= ( some @ C @ V3 ) ) )
& ( ( L != K )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V3 ) @ Ps2 ) @ K )
= ( map_of @ B @ C @ Ps2 @ K ) ) ) ) ).
% map_of_Cons_code(2)
thf(fact_7866_map__of__Some__split,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,V3: A] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ V3 ) )
=> ? [Ys5: list @ ( product_prod @ B @ A ),Zs2: list @ ( product_prod @ B @ A )] :
( ( Xs
= ( append @ ( product_prod @ B @ A ) @ Ys5 @ ( cons @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V3 ) @ Zs2 ) ) )
& ( ( map_of @ B @ A @ Ys5 @ K )
= ( none @ A ) ) ) ) ).
% map_of_Some_split
thf(fact_7867_map__of__map__to__set,axiom,
! [B: $tType,A: $tType,L: list @ ( product_prod @ A @ B ),M: A > ( option @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) )
=> ( ( ( map_of @ A @ B @ L )
= M )
= ( ( set2 @ ( product_prod @ A @ B ) @ L )
= ( map_to_set @ A @ B @ M ) ) ) ) ).
% map_of_map_to_set
thf(fact_7868_map__to__set__map__of,axiom,
! [B: $tType,A: $tType,L: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) )
=> ( ( map_to_set @ A @ B @ ( map_of @ A @ B @ L ) )
= ( set2 @ ( product_prod @ A @ B ) @ L ) ) ) ).
% map_to_set_map_of
thf(fact_7869_map__of__map__restrict,axiom,
! [B: $tType,A: $tType,F3: A > B,Ks: list @ A] :
( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K3: A] : ( product_Pair @ A @ B @ K3 @ ( F3 @ K3 ) )
@ Ks ) )
= ( restrict_map @ A @ B @ ( comp @ B @ ( option @ B ) @ A @ ( some @ B ) @ F3 ) @ ( set2 @ A @ Ks ) ) ) ).
% map_of_map_restrict
thf(fact_7870_map__of__zip__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,I: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ ( nth @ A @ Xs @ I ) )
= ( some @ B @ ( nth @ B @ Ys @ I ) ) ) ) ) ) ).
% map_of_zip_nth
thf(fact_7871_map__of__Some__filter__not__in,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,V3: A,P2: ( product_prod @ B @ A ) > $o] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ V3 ) )
=> ( ~ ( P2 @ ( product_Pair @ B @ A @ K @ V3 ) )
=> ( ( distinct @ B @ ( map @ ( product_prod @ B @ A ) @ B @ ( product_fst @ B @ A ) @ Xs ) )
=> ( ( map_of @ B @ A @ ( filter2 @ ( product_prod @ B @ A ) @ P2 @ Xs ) @ K )
= ( none @ A ) ) ) ) ) ).
% map_of_Some_filter_not_in
thf(fact_7872_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,T2: list @ ( product_prod @ A @ C ),K: A,X: C] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map_of @ A @ C @ T2 @ K )
= ( some @ C @ X ) )
=> ( ( map_of @ B @ C
@ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C )
@ ( product_case_prod @ A @ C @ ( product_prod @ B @ C )
@ ^ [K3: A] : ( product_Pair @ B @ C @ ( F3 @ K3 ) ) )
@ T2 )
@ ( F3 @ K ) )
= ( some @ C @ X ) ) ) ) ).
% map_of_mapk_SomeI
thf(fact_7873_map__of__distinct__lookup,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) ) )
=> ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys ) ) @ X )
= ( some @ B @ Y ) ) ) ) ).
% map_of_distinct_lookup
thf(fact_7874_map__of__distinct__upd2,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) ) )
=> ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ Ys ) ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% map_of_distinct_upd2
thf(fact_7875_bezw_Oelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa2 )
= Y )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa2 ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_7876_bezw_Osimps,axiom,
( bezw
= ( ^ [X4: nat,Y5: nat] :
( if @ ( product_prod @ int @ int )
@ ( Y5
= ( zero_zero @ nat ) )
@ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y5 @ ( modulo_modulo @ nat @ X4 @ Y5 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y5 @ ( modulo_modulo @ nat @ X4 @ Y5 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y5 @ ( modulo_modulo @ nat @ X4 @ Y5 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X4 @ Y5 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_7877_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,X: product_prod @ B @ C] :
( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F3 @ X ) )
= ( F3 @ ( product_snd @ B @ C @ X ) ) ) ).
% snd_apsnd
thf(fact_7878_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,X: product_prod @ A @ C,G3: C > B] :
( ( ( product_apsnd @ C @ B @ A @ F3 @ X )
= ( product_apsnd @ C @ B @ A @ G3 @ X ) )
= ( ( F3 @ ( product_snd @ A @ C @ X ) )
= ( G3 @ ( product_snd @ A @ C @ X ) ) ) ) ).
% apsnd_eq_conv
thf(fact_7879_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_7880_img__snd,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,S3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S3 )
=> ( member @ B @ B2 @ ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S3 ) ) ) ).
% img_snd
thf(fact_7881_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_7882_map__snd__mk__fst,axiom,
! [B: $tType,A: $tType,K: B,L: list @ A] :
( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( map @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K ) @ L ) )
= L ) ).
% map_snd_mk_fst
thf(fact_7883_snd__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F3: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F3 ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ F3 @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_apsnd
thf(fact_7884_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_7885_map__snd__mk__snd,axiom,
! [B: $tType,A: $tType,K: A,L: list @ B] :
( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( map @ B @ ( product_prod @ B @ A )
@ ^ [X4: B] : ( product_Pair @ B @ A @ X4 @ K )
@ L ) )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) ) ).
% map_snd_mk_snd
thf(fact_7886_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,F3: A > B > C,G3: D > A] :
( ( ^ [U3: product_prod @ D @ B] : ( F3 @ ( G3 @ ( product_fst @ D @ B @ U3 ) ) @ ( product_snd @ D @ B @ U3 ) ) )
= ( product_case_prod @ D @ B @ C
@ ^ [X4: D] : ( F3 @ ( G3 @ X4 ) ) ) ) ).
% split_comp_eq
thf(fact_7887_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F4: A > B > C,X4: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ X4 ) @ ( product_snd @ A @ B @ X4 ) ) ) ) ).
% case_prod_beta'
thf(fact_7888_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C7: A > B > C,P7: product_prod @ A @ B] : ( C7 @ ( product_fst @ A @ B @ P7 ) @ ( product_snd @ A @ B @ P7 ) ) ) ) ).
% case_prod_unfold
thf(fact_7889_split__beta,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F4: A > B > C,Prod2: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ Prod2 ) @ ( product_snd @ A @ B @ Prod2 ) ) ) ) ).
% split_beta
thf(fact_7890_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F4: B > C > A,P7: product_prod @ B @ C] : ( F4 @ ( product_fst @ B @ C @ P7 ) @ ( product_snd @ B @ C @ P7 ) ) ) ) ).
% case_prod_beta
thf(fact_7891_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A5: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A5 ) ) )
=> ( A5 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_7892_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P2: C > $o,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( P2 @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P2 @ ( F3 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_7893_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P2: C > $o,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( P2 @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P2 @ ( F3 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_7894_surjective__pairing,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( T2
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% surjective_pairing
thf(fact_7895_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_7896_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,X: A,Y: B,A2: product_prod @ A @ B] :
( ( P2 @ X @ Y )
=> ( ( A2
= ( product_Pair @ A @ B @ X @ Y ) )
=> ( P2 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_7897_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod3: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod3 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod3 ) ) )
=> ( Prod = Prod3 ) ) ).
% prod.expand
thf(fact_7898_prod__eqI,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B,Q3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P3 )
= ( product_fst @ A @ B @ Q3 ) )
=> ( ( ( product_snd @ A @ B @ P3 )
= ( product_snd @ A @ B @ Q3 ) )
=> ( P3 = Q3 ) ) ) ).
% prod_eqI
thf(fact_7899_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y6: product_prod @ A @ B,Z4: product_prod @ A @ B] : Y6 = Z4 )
= ( ^ [S8: product_prod @ A @ B,T3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S8 )
= ( product_fst @ A @ B @ T3 ) )
& ( ( product_snd @ A @ B @ S8 )
= ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_7900_Ex__prod__contract,axiom,
! [B: $tType,A: $tType,P2: A > B > $o] :
( ( ? [A8: A,X8: B] : ( P2 @ A8 @ X8 ) )
= ( ? [Z2: product_prod @ A @ B] : ( P2 @ ( product_fst @ A @ B @ Z2 ) @ ( product_snd @ A @ B @ Z2 ) ) ) ) ).
% Ex_prod_contract
thf(fact_7901_All__prod__contract,axiom,
! [B: $tType,A: $tType,P2: A > B > $o] :
( ( ! [A8: A,X8: B] : ( P2 @ A8 @ X8 ) )
= ( ! [Z2: product_prod @ A @ B] : ( P2 @ ( product_fst @ A @ B @ Z2 ) @ ( product_snd @ A @ B @ Z2 ) ) ) ) ).
% All_prod_contract
thf(fact_7902_fn__snd__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F3: B > C] :
( ( ^ [X4: product_prod @ A @ B] : ( F3 @ ( product_snd @ A @ B @ X4 ) ) )
= ( product_case_prod @ A @ B @ C
@ ^ [Uu3: A] : F3 ) ) ).
% fn_snd_conv
thf(fact_7903_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X16: A,X24: B] : X24 ) ) ).
% snd_def
thf(fact_7904_snd__image__mp,axiom,
! [B: $tType,A: $tType,A5: set @ ( product_prod @ B @ A ),B5: set @ A,X: B,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A5 ) @ B5 )
=> ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A5 )
=> ( member @ A @ Y @ B5 ) ) ) ).
% snd_image_mp
thf(fact_7905_in__snd__imageE,axiom,
! [A: $tType,B: $tType,Y: A,S3: set @ ( product_prod @ B @ A )] :
( ( member @ A @ Y @ ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ S3 ) )
=> ~ ! [X3: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y ) @ S3 ) ) ).
% in_snd_imageE
thf(fact_7906_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_7907_sndE,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,A2: A,B2: B,P2: B > $o] :
( ( X
= ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( ( P2 @ ( product_snd @ A @ B @ X ) )
=> ( P2 @ B2 ) ) ) ).
% sndE
thf(fact_7908_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y: A,A2: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_7909_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_7910_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_7911_sndI,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_snd @ A @ B @ X )
= Z ) ) ).
% sndI
thf(fact_7912_fs__contract,axiom,
! [B: $tType,C: $tType,A: $tType,F3: A > B > C,S3: set @ C] :
( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [P7: product_prod @ A @ B] :
( ( Uu3 = P7 )
& ( member @ C @ ( F3 @ ( product_fst @ A @ B @ P7 ) @ ( product_snd @ A @ B @ P7 ) ) @ S3 ) ) ) )
= ( collect @ A
@ ^ [A8: A] :
? [B8: B] : ( member @ C @ ( F3 @ A8 @ B8 ) @ S3 ) ) ) ).
% fs_contract
thf(fact_7913_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F3: D > C > A,G3: B > D,X: product_prod @ B @ C] :
( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F3 @ G3 ) @ X )
= ( F3 @ ( G3 @ ( product_fst @ B @ C @ X ) ) @ ( product_snd @ B @ C @ X ) ) ) ).
% case_prod_comp
thf(fact_7914_map__to__set__ran,axiom,
! [A: $tType,B: $tType] :
( ( ran @ B @ A )
= ( ^ [M5: B > ( option @ A )] : ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( map_to_set @ B @ A @ M5 ) ) ) ) ).
% map_to_set_ran
thf(fact_7915_snd__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X4: B] : ( product_Pair @ B @ B @ X4 @ X4 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% snd_diag_snd
thf(fact_7916_The__case__prod,axiom,
! [B: $tType,A: $tType,P2: A > B > $o] :
( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) )
= ( the @ ( product_prod @ A @ B )
@ ^ [Xy2: product_prod @ A @ B] : ( P2 @ ( product_fst @ A @ B @ Xy2 ) @ ( product_snd @ A @ B @ Xy2 ) ) ) ) ).
% The_case_prod
thf(fact_7917_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X6: set @ A,A5: set @ ( product_prod @ A @ B ),Y7: set @ B,P2: A > B > $o,Q2: A > B > $o] :
( ( X6
= ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A5 ) )
=> ( ( Y7
= ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A5 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ! [Xa4: B] :
( ( member @ B @ Xa4 @ Y7 )
=> ( ( P2 @ X3 @ Xa4 )
=> ( Q2 @ X3 @ Xa4 ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A5 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A5 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q2 ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_7918_set__to__map__ran,axiom,
! [A: $tType,B: $tType,S3: set @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( ran @ B @ A @ ( set_to_map @ B @ A @ S3 ) ) @ ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ S3 ) ) ).
% set_to_map_ran
thf(fact_7919_snd__fst__flip,axiom,
! [A: $tType,B: $tType] :
( ( product_snd @ B @ A )
= ( comp @ ( product_prod @ A @ B ) @ A @ ( product_prod @ B @ A ) @ ( product_fst @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X4: B,Y5: A] : ( product_Pair @ A @ B @ Y5 @ X4 ) ) ) ) ).
% snd_fst_flip
thf(fact_7920_fst__snd__flip,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( comp @ ( product_prod @ B @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [X4: A,Y5: B] : ( product_Pair @ B @ A @ Y5 @ X4 ) ) ) ) ).
% fst_snd_flip
thf(fact_7921_snd__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% snd_diag_fst
thf(fact_7922_fst__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_fst @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X4: B] : ( product_Pair @ B @ B @ X4 @ X4 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% fst_diag_snd
thf(fact_7923_rel__fun__Collect__case__prodD,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,A5: A > B > $o,B5: C > D > $o,F3: A > C,G3: B > D,X6: set @ ( product_prod @ A @ B ),X: product_prod @ A @ B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A5 @ B5 @ F3 @ G3 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A5 ) ) )
=> ( ( member @ ( product_prod @ A @ B ) @ X @ X6 )
=> ( B5 @ ( comp @ A @ C @ ( product_prod @ A @ B ) @ F3 @ ( product_fst @ A @ B ) @ X ) @ ( comp @ B @ D @ ( product_prod @ A @ B ) @ G3 @ ( product_snd @ A @ B ) @ X ) ) ) ) ) ).
% rel_fun_Collect_case_prodD
thf(fact_7924_ran__map__of,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( ran @ B @ A @ ( map_of @ B @ A @ Xs ) ) @ ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs ) ) ) ).
% ran_map_of
thf(fact_7925_predicate2__transferD,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R12: A > B > $o,R23: C > D > $o,P2: A > C > $o,Q2: B > D > $o,A2: product_prod @ A @ B,A5: set @ ( product_prod @ A @ B ),B2: product_prod @ C @ D,B5: set @ ( product_prod @ C @ D )] :
( ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ R12
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ R23
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ P2
@ Q2 )
=> ( ( member @ ( product_prod @ A @ B ) @ A2 @ A5 )
=> ( ( member @ ( product_prod @ C @ D ) @ B2 @ B5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A5 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R12 ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ C @ D ) ) @ B5 @ ( collect @ ( product_prod @ C @ D ) @ ( product_case_prod @ C @ D @ $o @ R23 ) ) )
=> ( ( P2 @ ( product_fst @ A @ B @ A2 ) @ ( product_fst @ C @ D @ B2 ) )
= ( Q2 @ ( product_snd @ A @ B @ A2 ) @ ( product_snd @ C @ D @ B2 ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_7926_in__set__zip,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ P3 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
= ( ? [N6: nat] :
( ( ( nth @ A @ Xs @ N6 )
= ( product_fst @ A @ B @ P3 ) )
& ( ( nth @ B @ Ys @ N6 )
= ( product_snd @ A @ B @ P3 ) )
& ( ord_less @ nat @ N6 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ord_less @ nat @ N6 @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ).
% in_set_zip
thf(fact_7927_set__to__map__insert,axiom,
! [B: $tType,A: $tType,Kv: product_prod @ A @ B,S3: set @ ( product_prod @ A @ B )] :
( ~ ( member @ A @ ( product_fst @ A @ B @ Kv ) @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S3 ) )
=> ( ( set_to_map @ A @ B @ ( insert @ ( product_prod @ A @ B ) @ Kv @ S3 ) )
= ( fun_upd @ A @ ( option @ B ) @ ( set_to_map @ A @ B @ S3 ) @ ( product_fst @ A @ B @ Kv ) @ ( some @ B @ ( product_snd @ A @ B @ Kv ) ) ) ) ) ).
% set_to_map_insert
thf(fact_7928_finite__range__prod,axiom,
! [A: $tType,C: $tType,B: $tType,F3: B > ( product_prod @ A @ C )] :
( ( finite_finite2 @ A @ ( image @ B @ A @ ( comp @ ( product_prod @ A @ C ) @ A @ B @ ( product_fst @ A @ C ) @ F3 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( ( finite_finite2 @ C @ ( image @ B @ C @ ( comp @ ( product_prod @ A @ C ) @ C @ B @ ( product_snd @ A @ C ) @ F3 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ C ) @ ( image @ B @ ( product_prod @ A @ C ) @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% finite_range_prod
thf(fact_7929_Misc_Oran__distinct,axiom,
! [B: $tType,A: $tType,Al: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Al ) )
=> ( ( ran @ A @ B @ ( map_of @ A @ B @ Al ) )
= ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ ( set2 @ ( product_prod @ A @ B ) @ Al ) ) ) ) ).
% Misc.ran_distinct
thf(fact_7930_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y )
=> ( ( bezw @ X @ Y )
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_7931_bezw_Opelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) )
=> ~ ( ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa2 ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) ) ) ) ) ).
% bezw.pelims
thf(fact_7932_fun_Oin__rel,axiom,
! [B: $tType,A: $tType,D: $tType,R3: A > B > $o,A2: D > A,B2: D > B] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y6: D,Z4: D] : Y6 = Z4
@ R3
@ A2
@ B2 )
= ( ? [Z2: D > ( product_prod @ A @ B )] :
( ( member @ ( D > ( product_prod @ A @ B ) ) @ Z2
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X4: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ D @ ( product_prod @ A @ B ) @ X4 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) ) ) )
& ( ( comp @ ( product_prod @ A @ B ) @ A @ D @ ( product_fst @ A @ B ) @ Z2 )
= A2 )
& ( ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) @ Z2 )
= B2 ) ) ) ) ).
% fun.in_rel
thf(fact_7933_snd__divmod__nat,axiom,
! [M: nat,N: nat] :
( ( product_snd @ nat @ nat @ ( divmod_nat @ M @ N ) )
= ( modulo_modulo @ nat @ M @ N ) ) ).
% snd_divmod_nat
thf(fact_7934_snd__divmod__integer,axiom,
! [K: code_integer,L: code_integer] :
( ( product_snd @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
= ( modulo_modulo @ code_integer @ K @ L ) ) ).
% snd_divmod_integer
thf(fact_7935_mod__pure,axiom,
! [B2: $o,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( pure_assn @ B2 ) @ H2 )
= ( ( ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 )
= ( bot_bot @ ( set @ nat ) ) )
& B2 ) ) ).
% mod_pure
thf(fact_7936_snd__divmod__abs,axiom,
! [K: code_integer,L: code_integer] :
( ( product_snd @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
= ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).
% snd_divmod_abs
thf(fact_7937_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_7938_mod__emp,axiom,
! [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( one_one @ assn ) @ H2 )
= ( ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 )
= ( bot_bot @ ( set @ nat ) ) ) ) ).
% mod_emp
thf(fact_7939_nths__shift__lemma__Suc,axiom,
! [A: $tType,P2: nat > $o,Xs: list @ A,Is: list @ nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P7: product_prod @ A @ nat] : ( P2 @ ( suc @ ( product_snd @ A @ nat @ P7 ) ) )
@ ( zip @ A @ nat @ Xs @ Is ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P7: product_prod @ A @ nat] : ( P2 @ ( product_snd @ A @ nat @ P7 ) )
@ ( zip @ A @ nat @ Xs @ ( map @ nat @ nat @ suc @ Is ) ) ) ) ) ).
% nths_shift_lemma_Suc
thf(fact_7940_mod__star__trueE_H,axiom,
! [P2: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P2 @ ( top_top @ assn ) ) @ H2 )
=> ~ ! [H6: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( ( product_fst @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 )
= ( product_fst @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 ) )
=> ( ( ord_less_eq @ ( set @ nat ) @ ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 ) @ ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 ) )
=> ~ ( rep_assn @ P2 @ H6 ) ) ) ) ).
% mod_star_trueE'
thf(fact_7941_nths__shift__lemma,axiom,
! [A: $tType,A5: set @ nat,Xs: list @ A,I: nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P7: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P7 ) @ A5 )
@ ( zip @ A @ nat @ Xs @ ( upt @ I @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P7: product_prod @ A @ nat] : ( member @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P7 ) @ I ) @ A5 )
@ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nths_shift_lemma
thf(fact_7942_fun_Orel__mono,axiom,
! [D: $tType,B: $tType,A: $tType,R3: A > B > $o,Ra2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R3 @ Ra2 )
=> ( ord_less_eq @ ( ( D > A ) > ( D > B ) > $o )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y6: D,Z4: D] : Y6 = Z4
@ R3 )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y6: D,Z4: D] : Y6 = Z4
@ Ra2 ) ) ) ).
% fun.rel_mono
thf(fact_7943_in__set__enumerate__eq,axiom,
! [A: $tType,P3: product_prod @ nat @ A,N: nat,Xs: list @ A] :
( ( member @ ( product_prod @ nat @ A ) @ P3 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) )
= ( ( ord_less_eq @ nat @ N @ ( product_fst @ nat @ A @ P3 ) )
& ( ord_less @ nat @ ( product_fst @ nat @ A @ P3 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
& ( ( nth @ A @ Xs @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P3 ) @ N ) )
= ( product_snd @ nat @ A @ P3 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_7944_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_7945_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_7946_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_7947_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_7948_Divides_Oadjust__mod__def,axiom,
( adjust_mod
= ( ^ [L4: int,R5: int] :
( if @ int
@ ( R5
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ L4 @ R5 ) ) ) ) ).
% Divides.adjust_mod_def
thf(fact_7949_size__prod__simp,axiom,
! [B: $tType,A: $tType] :
( ( basic_BNF_size_prod @ A @ B )
= ( ^ [F4: A > nat,G4: B > nat,P7: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F4 @ ( product_fst @ A @ B @ P7 ) ) @ ( G4 @ ( product_snd @ A @ B @ P7 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% size_prod_simp
thf(fact_7950_fun__mono,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,C4: A > B > $o,A5: A > B > $o,B5: C > D > $o,D5: C > D > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ C4 @ A5 )
=> ( ( ord_less_eq @ ( C > D > $o ) @ B5 @ D5 )
=> ( ord_less_eq @ ( ( A > C ) > ( B > D ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A5 @ B5 ) @ ( bNF_rel_fun @ A @ B @ C @ D @ C4 @ D5 ) ) ) ) ).
% fun_mono
thf(fact_7951_nths__def,axiom,
! [A: $tType] :
( ( nths @ A )
= ( ^ [Xs2: list @ A,A6: set @ nat] :
( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P7: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P7 ) @ A6 )
@ ( zip @ A @ nat @ Xs2 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ) ).
% nths_def
thf(fact_7952_exI__realizer,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Y: A,X: B] :
( ( P2 @ Y @ X )
=> ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_7953_nths__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( nths @ A @ Xs @ ( bot_bot @ ( set @ nat ) ) )
= ( nil @ A ) ) ).
% nths_empty
thf(fact_7954_nths__singleton,axiom,
! [A: $tType,A5: set @ nat,X: A] :
( ( ( member @ nat @ ( zero_zero @ nat ) @ A5 )
=> ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A5 )
= ( cons @ A @ X @ ( nil @ A ) ) ) )
& ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A5 )
=> ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A5 )
= ( nil @ A ) ) ) ) ).
% nths_singleton
thf(fact_7955_nths__all,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ nat @ I2 @ I6 ) )
=> ( ( nths @ A @ Xs @ I6 )
= Xs ) ) ).
% nths_all
thf(fact_7956_drop__eq__nths,axiom,
! [A: $tType] :
( ( drop @ A )
= ( ^ [N6: nat,Xs2: list @ A] : ( nths @ A @ Xs2 @ ( collect @ nat @ ( ord_less_eq @ nat @ N6 ) ) ) ) ) ).
% drop_eq_nths
thf(fact_7957_set__nths__subset,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( nths @ A @ Xs @ I6 ) ) @ ( set2 @ A @ Xs ) ) ).
% set_nths_subset
thf(fact_7958_sorted__nths,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I6: set @ nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nths @ A @ Xs @ I6 ) ) ) ) ).
% sorted_nths
thf(fact_7959_length__nths,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs @ I6 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I3 @ I6 ) ) ) ) ) ).
% length_nths
thf(fact_7960_nths__nths,axiom,
! [A: $tType,Xs: list @ A,A5: set @ nat,B5: set @ nat] :
( ( nths @ A @ ( nths @ A @ Xs @ A5 ) @ B5 )
= ( nths @ A @ Xs
@ ( collect @ nat
@ ^ [I3: nat] :
( ( member @ nat @ I3 @ A5 )
& ( member @ nat
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [I9: nat] :
( ( member @ nat @ I9 @ A5 )
& ( ord_less @ nat @ I9 @ I3 ) ) ) )
@ B5 ) ) ) ) ) ).
% nths_nths
thf(fact_7961_filter__eq__nths,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P5: A > $o,Xs2: list @ A] :
( nths @ A @ Xs2
@ ( collect @ nat
@ ^ [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P5 @ ( nth @ A @ Xs2 @ I3 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_7962_nths__Cons,axiom,
! [A: $tType,X: A,L: list @ A,A5: set @ nat] :
( ( nths @ A @ ( cons @ A @ X @ L ) @ A5 )
= ( append @ A @ ( if @ ( list @ A ) @ ( member @ nat @ ( zero_zero @ nat ) @ A5 ) @ ( cons @ A @ X @ ( nil @ A ) ) @ ( nil @ A ) )
@ ( nths @ A @ L
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( suc @ J3 ) @ A5 ) ) ) ) ) ).
% nths_Cons
thf(fact_7963_set__nths,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ( set2 @ A @ ( nths @ A @ Xs @ I6 ) )
= ( collect @ A
@ ^ [Uu3: A] :
? [I3: nat] :
( ( Uu3
= ( nth @ A @ Xs @ I3 ) )
& ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I3 @ I6 ) ) ) ) ).
% set_nths
thf(fact_7964_conjI__realizer,axiom,
! [A: $tType,B: $tType,P2: A > $o,P3: A,Q2: B > $o,Q3: B] :
( ( P2 @ P3 )
=> ( ( Q2 @ Q3 )
=> ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P3 @ Q3 ) ) )
& ( Q2 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P3 @ Q3 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_7965_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S3: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,X6: set @ ( product_prod @ C @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 )
=> ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert @ ( product_prod @ C @ A ) @ X @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S3 ) @ X6 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z2: B,A11: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W3 )
@ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z2 ) @ A11 )
@ A11 ) )
@ X6
@ S3 ) ) ) ).
% insert_relcomp_union_fold
thf(fact_7966_mergesort__by__rel__split__length,axiom,
! [A: $tType,Xs12: list @ A,Xs23: list @ A,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) @ Xs ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs12 ) @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( modulo_modulo @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ A ) @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) @ Xs ) ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs23 ) @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% mergesort_by_rel_split_length
thf(fact_7967_trancl__Int__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 ) @ S ) @ R2 ) @ S )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 ) @ S ) ) ) ).
% trancl_Int_subset
thf(fact_7968_relcomp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R6: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C ),S: set @ ( product_prod @ B @ C )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R6 @ R2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S2 @ S )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R6 @ S2 ) @ ( relcomp @ A @ B @ C @ R2 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_7969_max__ext__compat,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R3 @ S3 ) @ 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 @ S3 ) @ ( 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_7970_min__ext__compat,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R3 @ S3 ) @ 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 @ S3 ) @ ( 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_7971_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A2: A,C3: B,R2: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ C3 ) @ ( relcomp @ A @ C @ B @ R2 @ S ) )
=> ~ ! [B4: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A2 @ B4 ) @ R2 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B4 @ C3 ) @ S ) ) ) ).
% relcompEpair
thf(fact_7972_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R2: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R2 @ S ) )
=> ~ ! [X3: A,Y3: C,Z3: B] :
( ( Xz
= ( product_Pair @ A @ B @ X3 @ Z3 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Y3 ) @ R2 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y3 @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_7973_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),C3: C,S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ C3 ) @ S )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A2 @ C3 ) @ ( relcomp @ A @ B @ C @ R2 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_7974_relcomp_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,A12: A,A23: C,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A12 @ A23 ) @ ( relcomp @ A @ B @ C @ R2 @ S ) )
= ( ? [A8: A,B8: B,C7: C] :
( ( A12 = A8 )
& ( A23 = C7 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A8 @ B8 ) @ R2 )
& ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B8 @ C7 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_7975_relcomp_Ocases,axiom,
! [A: $tType,C: $tType,B: $tType,A12: A,A23: C,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A12 @ A23 ) @ ( relcomp @ A @ B @ C @ R2 @ S ) )
=> ~ ! [B4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A12 @ B4 ) @ R2 )
=> ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B4 @ A23 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_7976_mergesort__by__rel__split_Osimps_I1_J,axiom,
! [A: $tType,Xs12: list @ A,Xs23: list @ A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) @ ( nil @ A ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) ) ).
% mergesort_by_rel_split.simps(1)
thf(fact_7977_mergesort__by__rel__split_Osimps_I3_J,axiom,
! [A: $tType,Xs12: list @ A,Xs23: list @ A,X1: A,X2: A,Xs: list @ A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) @ ( cons @ A @ X1 @ ( cons @ A @ X2 @ Xs ) ) )
= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ Xs12 ) @ ( cons @ A @ X2 @ Xs23 ) ) @ Xs ) ) ).
% mergesort_by_rel_split.simps(3)
thf(fact_7978_relcomp__unfold,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcomp @ A @ C @ B )
= ( ^ [R5: set @ ( product_prod @ A @ C ),S8: set @ ( product_prod @ C @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Z2: B] :
? [Y5: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Y5 ) @ R5 )
& ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y5 @ Z2 ) @ S8 ) ) ) ) ) ) ).
% relcomp_unfold
thf(fact_7979_mergesort__by__rel__split_Oelims,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A ),Xa2: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( merges295452479951948502_split @ A @ X @ Xa2 )
= Y )
=> ( ! [Xs1: list @ A,Xs22: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
=> ( ( Xa2
= ( nil @ A ) )
=> ( Y
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) ) ) )
=> ( ! [Xs1: list @ A,Xs22: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
=> ! [X3: A] :
( ( Xa2
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( Y
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs1 ) @ Xs22 ) ) ) )
=> ~ ! [Xs1: list @ A,Xs22: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
=> ! [X15: A,X23: A,Xs3: list @ A] :
( ( Xa2
= ( cons @ A @ X15 @ ( cons @ A @ X23 @ Xs3 ) ) )
=> ( Y
!= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X15 @ Xs1 ) @ ( cons @ A @ X23 @ Xs22 ) ) @ Xs3 ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.elims
thf(fact_7980_mergesort__by__rel__split_Osimps_I2_J,axiom,
! [A: $tType,Xs12: list @ A,Xs23: list @ A,X: A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs12 ) @ Xs23 ) ) ).
% mergesort_by_rel_split.simps(2)
thf(fact_7981_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R3 )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S3 )
=> ( ( relcomp @ A @ B @ C @ R3 @ S3 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [X4: A,Y5: B,A6: set @ ( product_prod @ A @ C )] :
( finite_fold @ ( product_prod @ B @ C ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ B @ C @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [W3: B,Z2: C,A11: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y5 = W3 ) @ ( insert @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Z2 ) @ A11 ) @ A11 ) )
@ A6
@ S3 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
@ R3 ) ) ) ) ).
% relcomp_fold
thf(fact_7982_set__relcomp,axiom,
! [B: $tType,C: $tType,A: $tType,Xys: list @ ( product_prod @ A @ C ),Yzs: list @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( set2 @ ( product_prod @ A @ C ) @ Xys ) @ ( set2 @ ( product_prod @ C @ B ) @ Yzs ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ A @ C ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Xy2: product_prod @ A @ C] :
( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ C @ B ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Yz: product_prod @ C @ B] :
( if @ ( list @ ( product_prod @ A @ B ) )
@ ( ( product_snd @ A @ C @ Xy2 )
= ( product_fst @ C @ B @ Yz ) )
@ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Xy2 ) @ ( product_snd @ C @ B @ Yz ) ) @ ( nil @ ( product_prod @ A @ B ) ) )
@ ( nil @ ( product_prod @ A @ B ) ) )
@ Yzs ) )
@ Xys ) ) ) ) ).
% set_relcomp
thf(fact_7983_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S3: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,R3: set @ ( product_prod @ C @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 )
=> ( ( relcomp @ C @ A @ B @ ( insert @ ( product_prod @ C @ A ) @ X @ R3 ) @ S3 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z2: B,A11: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W3 )
@ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z2 ) @ A11 )
@ A11 ) )
@ ( relcomp @ C @ A @ B @ R3 @ S3 )
@ S3 ) ) ) ).
% insert_relcomp_fold
thf(fact_7984_mergesort__by__rel_Opinduct,axiom,
! [A: $tType,A0: A > A > $o,A12: list @ A,P2: ( A > A > $o ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ A0 @ A12 ) )
=> ( ! [R8: A > A > $o,Xs3: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R8 @ Xs3 ) )
=> ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( P2 @ R8 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) )
=> ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( P2 @ R8 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) )
=> ( P2 @ R8 @ Xs3 ) ) ) )
=> ( P2 @ A0 @ A12 ) ) ) ).
% mergesort_by_rel.pinduct
thf(fact_7985_mergesort__by__rel__split_Opelims,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A ),Xa2: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( merges295452479951948502_split @ A @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ X @ Xa2 ) )
=> ( ! [Xs1: list @ A,Xs22: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
=> ( ( Xa2
= ( nil @ A ) )
=> ( ( Y
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( nil @ A ) ) ) ) ) )
=> ( ! [Xs1: list @ A,Xs22: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
=> ! [X3: A] :
( ( Xa2
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ( Y
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs1 ) @ Xs22 ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ) )
=> ~ ! [Xs1: list @ A,Xs22: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
=> ! [X15: A,X23: A,Xs3: list @ A] :
( ( Xa2
= ( cons @ A @ X15 @ ( cons @ A @ X23 @ Xs3 ) ) )
=> ( ( Y
= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X15 @ Xs1 ) @ ( cons @ A @ X23 @ Xs22 ) ) @ Xs3 ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X15 @ ( cons @ A @ X23 @ Xs3 ) ) ) ) ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.pelims
thf(fact_7986_quickcheck__narrowing__samples_Opartial__term__of__sample__def,axiom,
! [A: $tType] :
( ( ( code_term_of @ A )
& ( quickc6926020345158392990erm_of @ A ) )
=> ( ( code_T4081349890594273596sample @ A )
= ( ^ [A_of_integer: code_integer > ( product_prod @ A @ A ),Zero: A,I3: code_integer] :
( if @ A @ ( ord_less @ code_integer @ I3 @ ( zero_zero @ code_integer ) ) @ ( undefined @ A )
@ ( if @ A
@ ( I3
= ( zero_zero @ code_integer ) )
@ Zero
@ ( if @ A
@ ( ( modulo_modulo @ code_integer @ I3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
= ( zero_zero @ code_integer ) )
@ ( product_snd @ A @ A @ ( A_of_integer @ ( divide_divide @ code_integer @ I3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) )
@ ( product_fst @ A @ A @ ( A_of_integer @ ( plus_plus @ code_integer @ ( divide_divide @ code_integer @ I3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ) ) ) ).
% quickcheck_narrowing_samples.partial_term_of_sample_def
thf(fact_7987_normalize__def,axiom,
( normalize
= ( ^ [P7: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P7 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P7 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P7 ) @ ( product_snd @ int @ int @ P7 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P7 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P7 ) @ ( product_snd @ int @ int @ P7 ) ) ) )
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_snd @ int @ int @ P7 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P7 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P7 ) @ ( product_snd @ int @ int @ P7 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P7 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P7 ) @ ( product_snd @ int @ int @ P7 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_7988_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_7989_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_7990_Gcd__2,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_Gcd @ A @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ).
% Gcd_2
thf(fact_7991_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_7992_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_7993_gcd__cases__int,axiom,
! [X: int,Y: int,P2: int > $o] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( P2 @ ( gcd_gcd @ int @ X @ Y ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) )
=> ( P2 @ ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ Y ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( P2 @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ Y ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) )
=> ( P2 @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ ( uminus_uminus @ int @ Y ) ) ) ) )
=> ( P2 @ ( gcd_gcd @ int @ X @ Y ) ) ) ) ) ) ).
% gcd_cases_int
thf(fact_7994_Gcd__fin_Osubset,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B5: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( gcd_gcd @ A @ ( semiring_gcd_Gcd_fin @ A @ B5 ) @ ( semiring_gcd_Gcd_fin @ A @ A5 ) )
= ( semiring_gcd_Gcd_fin @ A @ A5 ) ) ) ) ).
% Gcd_fin.subset
thf(fact_7995_gcd__ge__0__int,axiom,
! [X: int,Y: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ X @ Y ) ) ).
% gcd_ge_0_int
thf(fact_7996_gcd__unique__int,axiom,
! [D2: int,A2: int,B2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
& ( dvd_dvd @ int @ D2 @ A2 )
& ( dvd_dvd @ int @ D2 @ B2 )
& ! [E4: int] :
( ( ( dvd_dvd @ int @ E4 @ A2 )
& ( dvd_dvd @ int @ E4 @ B2 ) )
=> ( dvd_dvd @ int @ E4 @ D2 ) ) )
= ( D2
= ( gcd_gcd @ int @ A2 @ B2 ) ) ) ).
% gcd_unique_int
thf(fact_7997_gcd__non__0__int,axiom,
! [Y: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Y )
=> ( ( gcd_gcd @ int @ X @ Y )
= ( gcd_gcd @ int @ Y @ ( modulo_modulo @ int @ X @ Y ) ) ) ) ).
% gcd_non_0_int
thf(fact_7998_Gcd__fin_Oremove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A5 )
= ( gcd_gcd @ A @ A2 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Gcd_fin.remove
thf(fact_7999_Gcd__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,A5: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( insert @ A @ A2 @ A5 ) )
= ( gcd_gcd @ A @ A2 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% Gcd_fin.insert_remove
thf(fact_8000_Func__empty,axiom,
! [B: $tType,A: $tType,B5: set @ B] :
( ( bNF_Wellorder_Func @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B5 )
= ( insert @ ( A > B )
@ ^ [X4: A] : ( undefined @ B )
@ ( bot_bot @ ( set @ ( A > B ) ) ) ) ) ).
% Func_empty
thf(fact_8001_Gcd__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( ^ [A6: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A6 ) @ ( finite_fold @ A @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ A6 ) @ ( one_one @ A ) ) ) ) ) ).
% Gcd_fin.eq_fold
thf(fact_8002_mod__h__bot__normalize,axiom,
! [A: $tType,H2: heap_ext @ product_unit,P2: assn] :
( ( syntax7388354845996824322omatch @ A @ ( heap_ext @ product_unit ) @ ( undefined @ A ) @ H2 )
=> ( ( rep_assn @ P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( rep_assn @ P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ ( undefined @ ( heap_ext @ product_unit ) ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_normalize
thf(fact_8003_integer__set__bit__code,axiom,
( bits_integer_set_bit
= ( ^ [X4: code_integer,N6: code_integer,B8: $o] : ( if @ code_integer @ ( ord_less @ code_integer @ N6 @ ( zero_zero @ code_integer ) ) @ ( undefined @ ( code_integer > code_integer > $o > code_integer ) @ X4 @ N6 @ B8 ) @ ( if @ code_integer @ B8 @ ( bit_se1065995026697491101ons_or @ code_integer @ X4 @ ( bit_se4730199178511100633sh_bit @ code_integer @ ( code_nat_of_integer @ N6 ) @ ( one_one @ code_integer ) ) ) @ ( bit_se5824344872417868541ns_and @ code_integer @ X4 @ ( bit_ri4277139882892585799ns_not @ code_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ ( code_nat_of_integer @ N6 ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ) ).
% integer_set_bit_code
thf(fact_8004_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_8005_gcd__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ A2 )
= A2 ) ).
% gcd_nat.left_neutral
thf(fact_8006_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_8007_gcd__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( gcd_gcd @ nat @ A2 @ ( zero_zero @ nat ) )
= A2 ) ).
% gcd_nat.right_neutral
thf(fact_8008_gcd__0__nat,axiom,
! [X: nat] :
( ( gcd_gcd @ nat @ X @ ( zero_zero @ nat ) )
= X ) ).
% gcd_0_nat
thf(fact_8009_gcd__0__left__nat,axiom,
! [X: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ X )
= X ) ).
% gcd_0_left_nat
thf(fact_8010_gcd__Suc__0,axiom,
! [M: nat] :
( ( gcd_gcd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% gcd_Suc_0
thf(fact_8011_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_8012_Gcd__in,axiom,
! [A5: set @ nat] :
( ! [A4: nat,B4: nat] :
( ( member @ nat @ A4 @ A5 )
=> ( ( member @ nat @ B4 @ A5 )
=> ( member @ nat @ ( gcd_gcd @ nat @ A4 @ B4 ) @ A5 ) ) )
=> ( ( A5
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( gcd_Gcd @ nat @ A5 ) @ A5 ) ) ) ).
% Gcd_in
thf(fact_8013_bezout__gcd__nat_H,axiom,
! [B2: nat,A2: nat] :
? [X3: nat,Y3: nat] :
( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ ( times_times @ nat @ A2 @ X3 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ A2 @ X3 ) @ ( times_times @ nat @ B2 @ Y3 ) )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) )
| ( ( ord_less_eq @ nat @ ( times_times @ nat @ A2 @ Y3 ) @ ( times_times @ nat @ B2 @ X3 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A2 @ Y3 ) )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_8014_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_8015_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_8016_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_8017_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_8018_bezout__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ? [X3: nat,Y3: nat] :
( ( times_times @ nat @ A2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ).
% bezout_nat
thf(fact_8019_gcd__nat_Oelims,axiom,
! [X: nat,Xa2: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa2 )
= Y )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y = X ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) ) ) ) ).
% gcd_nat.elims
thf(fact_8020_gcd__nat_Osimps,axiom,
( ( gcd_gcd @ nat )
= ( ^ [X4: nat,Y5: nat] :
( if @ nat
@ ( Y5
= ( zero_zero @ nat ) )
@ X4
@ ( gcd_gcd @ nat @ Y5 @ ( modulo_modulo @ nat @ X4 @ Y5 ) ) ) ) ) ).
% gcd_nat.simps
thf(fact_8021_gcd__non__0__nat,axiom,
! [Y: nat,X: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ( ( gcd_gcd @ nat @ X @ Y )
= ( gcd_gcd @ nat @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) ) ).
% gcd_non_0_nat
thf(fact_8022_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) @ ( dvd_dvd @ nat )
@ ^ [M5: nat,N6: nat] :
( ( dvd_dvd @ nat @ M5 @ N6 )
& ( M5 != N6 ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_8023_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
@ ^ [D6: nat] :
( ( dvd_dvd @ nat @ D6 @ M )
& ( dvd_dvd @ nat @ D6 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_8024_gcd__nat_Opelims,axiom,
! [X: nat,Xa2: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) )
=> ~ ( ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y = X ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_8025_uint32__shiftl__def,axiom,
( uint32_shiftl
= ( ^ [X4: uint32,N6: code_integer] :
( if @ uint32
@ ( ( ord_less @ code_integer @ N6 @ ( zero_zero @ code_integer ) )
| ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N6 ) )
@ ( undefined @ ( ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ) @ ( bit_se4730199178511100633sh_bit @ uint32 ) @ X4 @ N6 )
@ ( bit_se4730199178511100633sh_bit @ uint32 @ ( code_nat_of_integer @ N6 ) @ X4 ) ) ) ) ).
% uint32_shiftl_def
thf(fact_8026_uint32__shiftr__def,axiom,
( uint32_shiftr
= ( ^ [X4: uint32,N6: code_integer] :
( if @ uint32
@ ( ( ord_less @ code_integer @ N6 @ ( zero_zero @ code_integer ) )
| ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N6 ) )
@ ( undefined @ ( ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ) @ ( bit_se4197421643247451524op_bit @ uint32 ) @ X4 @ N6 )
@ ( bit_se4197421643247451524op_bit @ uint32 @ ( code_nat_of_integer @ N6 ) @ X4 ) ) ) ) ).
% uint32_shiftr_def
thf(fact_8027_uint32__test__bit__def,axiom,
( uint32_test_bit
= ( ^ [X4: uint32,N6: code_integer] :
( ( ( ( ord_less @ code_integer @ N6 @ ( zero_zero @ code_integer ) )
| ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N6 ) )
=> ( undefined @ ( ( uint32 > nat > $o ) > uint32 > code_integer > $o ) @ ( bit_se5641148757651400278ts_bit @ uint32 ) @ X4 @ N6 ) )
& ( ~ ( ( ord_less @ code_integer @ N6 @ ( zero_zero @ code_integer ) )
| ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N6 ) )
=> ( bit_se5641148757651400278ts_bit @ uint32 @ X4 @ ( code_nat_of_integer @ N6 ) ) ) ) ) ) ).
% uint32_test_bit_def
thf(fact_8028_integer__set__bit__def,axiom,
( bits_integer_set_bit
= ( ^ [X4: code_integer,N6: code_integer,B8: $o] : ( if @ code_integer @ ( ord_less @ code_integer @ N6 @ ( zero_zero @ code_integer ) ) @ ( undefined @ ( code_integer > code_integer > $o > code_integer ) @ X4 @ N6 @ B8 ) @ ( generi7602027413899671122et_bit @ code_integer @ X4 @ ( code_nat_of_integer @ N6 ) @ B8 ) ) ) ) ).
% integer_set_bit_def
thf(fact_8029_uint32__shiftr__code,axiom,
! [N: code_integer,W2: uint32] :
( ( ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
| ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N ) )
=> ( ( rep_uint32 @ ( uint32_shiftr @ W2 @ N ) )
= ( rep_uint32 @ ( undefined @ ( ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ) @ ( bit_se4197421643247451524op_bit @ uint32 ) @ W2 @ N ) ) ) )
& ( ~ ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
| ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N ) )
=> ( ( rep_uint32 @ ( uint32_shiftr @ W2 @ N ) )
= ( bit_se4197421643247451524op_bit @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( code_nat_of_integer @ N ) @ ( rep_uint32 @ W2 ) ) ) ) ) ).
% uint32_shiftr_code
thf(fact_8030_uint32_Oless__iff__word__of,axiom,
( ( ord_less @ uint32 )
= ( ^ [P7: uint32,Q8: uint32] : ( ord_less @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint32 @ P7 ) @ ( rep_uint32 @ Q8 ) ) ) ) ).
% uint32.less_iff_word_of
thf(fact_8031_less__uint32_Orep__eq,axiom,
( ( ord_less @ uint32 )
= ( ^ [X4: uint32,Xa5: uint32] : ( ord_less @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint32 @ X4 ) @ ( rep_uint32 @ Xa5 ) ) ) ) ).
% less_uint32.rep_eq
thf(fact_8032_uint32_Oless__eq__iff__word__of,axiom,
( ( ord_less_eq @ uint32 )
= ( ^ [P7: uint32,Q8: uint32] : ( ord_less_eq @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint32 @ P7 ) @ ( rep_uint32 @ Q8 ) ) ) ) ).
% uint32.less_eq_iff_word_of
thf(fact_8033_less__eq__uint32_Orep__eq,axiom,
( ( ord_less_eq @ uint32 )
= ( ^ [X4: uint32,Xa5: uint32] : ( ord_less_eq @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint32 @ X4 ) @ ( rep_uint32 @ Xa5 ) ) ) ) ).
% less_eq_uint32.rep_eq
thf(fact_8034_uint32__test__bit__code,axiom,
( uint32_test_bit
= ( ^ [W3: uint32,N6: code_integer] :
( ( ( ( ord_less @ code_integer @ N6 @ ( zero_zero @ code_integer ) )
| ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N6 ) )
=> ( undefined @ ( ( uint32 > nat > $o ) > uint32 > code_integer > $o ) @ ( bit_se5641148757651400278ts_bit @ uint32 ) @ W3 @ N6 ) )
& ( ~ ( ( ord_less @ code_integer @ N6 @ ( zero_zero @ code_integer ) )
| ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N6 ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint32 @ W3 ) @ ( code_nat_of_integer @ N6 ) ) ) ) ) ) ).
% uint32_test_bit_code
thf(fact_8035_uint32__shiftl__code,axiom,
! [N: code_integer,W2: uint32] :
( ( ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
| ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N ) )
=> ( ( rep_uint32 @ ( uint32_shiftl @ W2 @ N ) )
= ( rep_uint32 @ ( undefined @ ( ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ) @ ( bit_se4730199178511100633sh_bit @ uint32 ) @ W2 @ N ) ) ) )
& ( ~ ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
| ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N ) )
=> ( ( rep_uint32 @ ( uint32_shiftl @ W2 @ N ) )
= ( bit_se4730199178511100633sh_bit @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( code_nat_of_integer @ N ) @ ( rep_uint32 @ W2 ) ) ) ) ) ).
% uint32_shiftl_code
thf(fact_8036_bit__set__bit__iff__2n,axiom,
! [A: $tType] :
( ( generic_set_set_bit @ A )
=> ! [A2: A,M: nat,B2: $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( generi7602027413899671122et_bit @ A @ A2 @ M @ B2 ) @ N )
= ( ( ( M = N )
=> B2 )
& ( ( M != N )
=> ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) )
& ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_set_bit_iff_2n
thf(fact_8037_integer__shiftl__def,axiom,
( bits_integer_shiftl
= ( ^ [X4: code_integer,N6: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ N6 @ ( zero_zero @ code_integer ) ) @ ( undefined @ ( code_integer > code_integer > code_integer ) @ X4 @ N6 ) @ ( bit_se4730199178511100633sh_bit @ code_integer @ ( code_nat_of_integer @ N6 ) @ X4 ) ) ) ) ).
% integer_shiftl_def
thf(fact_8038_test__bit__set__gen,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat,X: $o,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W2 @ N @ X ) @ M )
= ( ( ( M = N )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W2 ) )
& X ) )
& ( ( M != N )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ M ) ) ) ) ) ).
% test_bit_set_gen
thf(fact_8039_test__bit__set,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat,X: $o] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W2 @ N @ X ) @ N )
= ( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W2 ) )
& X ) ) ) ).
% test_bit_set
thf(fact_8040_bit__set__bit__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,M: nat,B2: $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W2 @ M @ B2 ) @ N )
= ( ( ( M = N )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& B2 ) )
& ( ( M != N )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ N ) ) ) ) ) ).
% bit_set_bit_word_iff
thf(fact_8041_word__set__nth__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat,B2: $o] :
( ( ( generi7602027413899671122et_bit @ ( word @ A ) @ W2 @ N @ B2 )
= W2 )
= ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ N )
= B2 )
| ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ W2 ) @ N ) ) ) ) ).
% word_set_nth_iff
thf(fact_8042_set__bit__beyond,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,B2: $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ X ) @ N )
=> ( ( generi7602027413899671122et_bit @ ( word @ A ) @ X @ N @ B2 )
= X ) ) ) ).
% set_bit_beyond
thf(fact_8043_word__set__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat] : ( ord_less_eq @ ( word @ A ) @ W2 @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W2 @ N @ $true ) ) ) ).
% word_set_ge
thf(fact_8044_word__clr__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W2: word @ A,N: nat] : ( ord_less_eq @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W2 @ N @ $false ) @ W2 ) ) ).
% word_clr_le
thf(fact_8045_less__eq__uint32_Orsp,axiom,
( bNF_rel_fun @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) > $o ) @ ( ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) > $o )
@ ^ [Y6: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ),Z4: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) )] : Y6 = Z4
@ ( bNF_rel_fun @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ $o @ $o
@ ^ [Y6: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ),Z4: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) )] : Y6 = Z4
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ( ord_less_eq @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) )
@ ( ord_less_eq @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) ) ) ).
% less_eq_uint32.rsp
thf(fact_8046_less__uint32_Orsp,axiom,
( bNF_rel_fun @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) > $o ) @ ( ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) > $o )
@ ^ [Y6: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ),Z4: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) )] : Y6 = Z4
@ ( bNF_rel_fun @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ $o @ $o
@ ^ [Y6: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ),Z4: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) )] : Y6 = Z4
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
@ ( ord_less @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) )
@ ( ord_less @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) ) ) ).
% less_uint32.rsp
thf(fact_8047_uint32__set__bit__code,axiom,
! [N: code_integer,W2: uint32,B2: $o] :
( ( ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
| ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N ) )
=> ( ( rep_uint32 @ ( uint32_set_bit @ W2 @ N @ B2 ) )
= ( rep_uint32 @ ( undefined @ ( ( uint32 > nat > $o > uint32 ) > uint32 > code_integer > $o > uint32 ) @ ( generi7602027413899671122et_bit @ uint32 ) @ W2 @ N @ B2 ) ) ) )
& ( ~ ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
| ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N ) )
=> ( ( rep_uint32 @ ( uint32_set_bit @ W2 @ N @ B2 ) )
= ( generi7602027413899671122et_bit @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint32 @ W2 ) @ ( code_nat_of_integer @ N ) @ B2 ) ) ) ) ).
% uint32_set_bit_code
thf(fact_8048_uint32__set__bit__def,axiom,
( uint32_set_bit
= ( ^ [X4: uint32,N6: code_integer,B8: $o] :
( if @ uint32
@ ( ( ord_less @ code_integer @ N6 @ ( zero_zero @ code_integer ) )
| ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N6 ) )
@ ( undefined @ ( ( uint32 > nat > $o > uint32 ) > uint32 > code_integer > $o > uint32 ) @ ( generi7602027413899671122et_bit @ uint32 ) @ X4 @ N6 @ B8 )
@ ( generi7602027413899671122et_bit @ uint32 @ X4 @ ( code_nat_of_integer @ N6 ) @ B8 ) ) ) ) ).
% uint32_set_bit_def
thf(fact_8049_integer__shiftr__def,axiom,
( bits_integer_shiftr
= ( ^ [X4: code_integer,N6: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ N6 @ ( zero_zero @ code_integer ) ) @ ( undefined @ ( code_integer > code_integer > code_integer ) @ X4 @ N6 ) @ ( bit_se4197421643247451524op_bit @ code_integer @ ( code_nat_of_integer @ N6 ) @ X4 ) ) ) ) ).
% integer_shiftr_def
thf(fact_8050_uint32__divmod__code,axiom,
( uint32_divmod
= ( ^ [X4: uint32,Y5: uint32] :
( if @ ( product_prod @ uint32 @ uint32 ) @ ( ord_less_eq @ uint32 @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ Y5 ) @ ( if @ ( product_prod @ uint32 @ uint32 ) @ ( ord_less @ uint32 @ X4 @ Y5 ) @ ( product_Pair @ uint32 @ uint32 @ ( zero_zero @ uint32 ) @ X4 ) @ ( product_Pair @ uint32 @ uint32 @ ( one_one @ uint32 ) @ ( minus_minus @ uint32 @ X4 @ Y5 ) ) )
@ ( if @ ( product_prod @ uint32 @ uint32 )
@ ( Y5
= ( zero_zero @ uint32 ) )
@ ( product_Pair @ uint32 @ uint32 @ ( div0_uint32 @ X4 ) @ ( mod0_uint32 @ X4 ) )
@ ( if @ ( product_prod @ uint32 @ uint32 ) @ ( ord_less_eq @ uint32 @ Y5 @ ( minus_minus @ uint32 @ X4 @ ( times_times @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X4 ) @ Y5 ) ) @ Y5 ) ) ) @ ( product_Pair @ uint32 @ uint32 @ ( plus_plus @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X4 ) @ Y5 ) ) @ ( one_one @ uint32 ) ) @ ( minus_minus @ uint32 @ ( minus_minus @ uint32 @ X4 @ ( times_times @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X4 ) @ Y5 ) ) @ Y5 ) ) @ Y5 ) ) @ ( product_Pair @ uint32 @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X4 ) @ Y5 ) ) @ ( minus_minus @ uint32 @ X4 @ ( times_times @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X4 ) @ Y5 ) ) @ Y5 ) ) ) ) ) ) ) ) ).
% uint32_divmod_code
thf(fact_8051_uint32_Oset__bits__aux__code,axiom,
( set_bits_aux_uint32
= ( ^ [F4: nat > $o,N6: nat,W3: uint32] :
( if @ uint32
@ ( N6
= ( zero_zero @ nat ) )
@ W3
@ ( set_bits_aux_uint32 @ F4 @ ( minus_minus @ nat @ N6 @ ( one_one @ nat ) ) @ ( bit_se1065995026697491101ons_or @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ W3 ) @ ( if @ uint32 @ ( F4 @ ( minus_minus @ nat @ N6 @ ( one_one @ nat ) ) ) @ ( one_one @ uint32 ) @ ( zero_zero @ uint32 ) ) ) ) ) ) ) ).
% uint32.set_bits_aux_code
thf(fact_8052_shiftr__uint32__code,axiom,
( ( bit_se4197421643247451524op_bit @ uint32 )
= ( ^ [N6: nat,X4: uint32] : ( if @ uint32 @ ( ord_less @ nat @ N6 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( uint32_shiftr @ X4 @ ( code_integer_of_nat @ N6 ) ) @ ( zero_zero @ uint32 ) ) ) ) ).
% shiftr_uint32_code
thf(fact_8053_nat__of__integer__integer__of__nat,axiom,
! [N: nat] :
( ( code_nat_of_integer @ ( code_integer_of_nat @ N ) )
= N ) ).
% nat_of_integer_integer_of_nat
thf(fact_8054_integer__of__nat_Orep__eq,axiom,
! [X: nat] :
( ( code_int_of_integer @ ( code_integer_of_nat @ X ) )
= ( semiring_1_of_nat @ int @ X ) ) ).
% integer_of_nat.rep_eq
thf(fact_8055_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_8056_integer__of__nat__0,axiom,
( ( code_integer_of_nat @ ( zero_zero @ nat ) )
= ( zero_zero @ code_integer ) ) ).
% integer_of_nat_0
thf(fact_8057_integer__of__nat__1,axiom,
( ( code_integer_of_nat @ ( one_one @ nat ) )
= ( one_one @ code_integer ) ) ).
% integer_of_nat_1
thf(fact_8058_integer__of__nat__less__0__conv,axiom,
! [N: nat] :
~ ( ord_less @ code_integer @ ( code_integer_of_nat @ N ) @ ( zero_zero @ code_integer ) ) ).
% integer_of_nat_less_0_conv
thf(fact_8059_integer__of__nat__eq__of__nat,axiom,
( code_integer_of_nat
= ( semiring_1_of_nat @ code_integer ) ) ).
% integer_of_nat_eq_of_nat
thf(fact_8060_integer__of__nat_Oabs__eq,axiom,
( code_integer_of_nat
= ( ^ [X4: nat] : ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X4 ) ) ) ) ).
% integer_of_nat.abs_eq
thf(fact_8061_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_8062_test__bit__uint32__code,axiom,
( ( bit_se5641148757651400278ts_bit @ uint32 )
= ( ^ [X4: uint32,N6: nat] :
( ( ord_less @ nat @ N6 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( uint32_test_bit @ X4 @ ( code_integer_of_nat @ N6 ) ) ) ) ) ).
% test_bit_uint32_code
thf(fact_8063_set__bit__uint32__code,axiom,
( ( generi7602027413899671122et_bit @ uint32 )
= ( ^ [X4: uint32,N6: nat,B8: $o] : ( if @ uint32 @ ( ord_less @ nat @ N6 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( uint32_set_bit @ X4 @ ( code_integer_of_nat @ N6 ) @ B8 ) @ X4 ) ) ) ).
% set_bit_uint32_code
thf(fact_8064_shiftl__uint32__code,axiom,
( ( bit_se4730199178511100633sh_bit @ uint32 )
= ( ^ [N6: nat,X4: uint32] : ( if @ uint32 @ ( ord_less @ nat @ N6 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( uint32_shiftl @ X4 @ ( code_integer_of_nat @ N6 ) ) @ ( zero_zero @ uint32 ) ) ) ) ).
% shiftl_uint32_code
thf(fact_8065_sshiftr__uint32__code,axiom,
( signed489701013188660438uint32
= ( ^ [N6: nat,X4: uint32] : ( if @ uint32 @ ( ord_less @ nat @ N6 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( uint32_sshiftr @ X4 @ ( code_integer_of_nat @ N6 ) ) @ ( if @ uint32 @ ( bit_se5641148757651400278ts_bit @ uint32 @ X4 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( uminus_uminus @ uint32 @ ( one_one @ uint32 ) ) @ ( zero_zero @ uint32 ) ) ) ) ) ).
% sshiftr_uint32_code
thf(fact_8066_mergesort__by__rel_Opelims,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A,Y: list @ A] :
( ( ( mergesort_by_rel @ A @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa2 ) )
=> ~ ( ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( Y = Xa2 ) )
& ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( Y
= ( merges9089515139780605204_merge @ A @ X @ ( mergesort_by_rel @ A @ X @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa2 ) ) ) @ ( mergesort_by_rel @ A @ X @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa2 ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa2 ) ) ) ) ) ).
% mergesort_by_rel.pelims
thf(fact_8067_mergesort__by__rel__simps_I1_J,axiom,
! [A: $tType,R3: A > A > $o] :
( ( mergesort_by_rel @ A @ R3 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% mergesort_by_rel_simps(1)
thf(fact_8068_set__mergesort__by__rel,axiom,
! [A: $tType,R3: A > A > $o,Xs: list @ A] :
( ( set2 @ A @ ( mergesort_by_rel @ A @ R3 @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_mergesort_by_rel
thf(fact_8069_mergesort__by__rel__merge__simps_I3_J,axiom,
! [A: $tType,R3: A > A > $o,Ys: list @ A] :
( ( merges9089515139780605204_merge @ A @ R3 @ ( nil @ A ) @ Ys )
= Ys ) ).
% mergesort_by_rel_merge_simps(3)
thf(fact_8070_sorted__wrt__mergesort__by__rel__merge,axiom,
! [A: $tType,R3: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ! [X3: A,Y3: A] :
( ( R3 @ X3 @ Y3 )
| ( R3 @ Y3 @ X3 ) )
=> ( ! [X3: A,Y3: A,Z3: A] :
( ( R3 @ X3 @ Y3 )
=> ( ( R3 @ Y3 @ Z3 )
=> ( R3 @ X3 @ Z3 ) ) )
=> ( ( sorted_wrt @ A @ R3 @ ( merges9089515139780605204_merge @ A @ R3 @ Xs @ Ys ) )
= ( ( sorted_wrt @ A @ R3 @ Xs )
& ( sorted_wrt @ A @ R3 @ Ys ) ) ) ) ) ).
% sorted_wrt_mergesort_by_rel_merge
thf(fact_8071_mergesort__by__rel__simps_I2_J,axiom,
! [A: $tType,R3: A > A > $o,X: A] :
( ( mergesort_by_rel @ A @ R3 @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% mergesort_by_rel_simps(2)
thf(fact_8072_set__mergesort__by__rel__merge,axiom,
! [A: $tType,R3: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( set2 @ A @ ( merges9089515139780605204_merge @ A @ R3 @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% set_mergesort_by_rel_merge
thf(fact_8073_mergesort__by__rel__simps_I3_J,axiom,
! [A: $tType,R3: A > A > $o,X1: A,X2: A,Xs: list @ A] :
( ( mergesort_by_rel @ A @ R3 @ ( cons @ A @ X1 @ ( cons @ A @ X2 @ Xs ) ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs13: list @ A,Xs24: list @ A] : ( merges9089515139780605204_merge @ A @ R3 @ ( mergesort_by_rel @ A @ R3 @ Xs13 ) @ ( mergesort_by_rel @ A @ R3 @ Xs24 ) )
@ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ ( nil @ A ) ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ Xs ) ) ) ).
% mergesort_by_rel_simps(3)
thf(fact_8074_sorted__wrt__mergesort__by__rel,axiom,
! [X19: $tType,R3: X19 > X19 > $o,Xs: list @ X19] :
( ! [X3: X19,Y3: X19] :
( ( R3 @ X3 @ Y3 )
| ( R3 @ Y3 @ X3 ) )
=> ( ! [X3: X19,Y3: X19,Z3: X19] :
( ( R3 @ X3 @ Y3 )
=> ( ( R3 @ Y3 @ Z3 )
=> ( R3 @ X3 @ Z3 ) ) )
=> ( sorted_wrt @ X19 @ R3 @ ( mergesort_by_rel @ X19 @ R3 @ Xs ) ) ) ) ).
% sorted_wrt_mergesort_by_rel
thf(fact_8075_mergesort__by__rel__merge_Osimps_I3_J,axiom,
! [A: $tType,R3: A > A > $o,V3: A,Va2: list @ A] :
( ( merges9089515139780605204_merge @ A @ R3 @ ( nil @ A ) @ ( cons @ A @ V3 @ Va2 ) )
= ( cons @ A @ V3 @ Va2 ) ) ).
% mergesort_by_rel_merge.simps(3)
thf(fact_8076_mergesort__by__rel__merge_Oelims,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( merges9089515139780605204_merge @ A @ X @ Xa2 @ Xb )
= Y )
=> ( ! [X3: A,Xs3: list @ A] :
( ( Xa2
= ( cons @ A @ X3 @ Xs3 ) )
=> ! [Y3: A,Ys5: list @ A] :
( ( Xb
= ( cons @ A @ Y3 @ Ys5 ) )
=> ~ ( ( ( X @ X3 @ Y3 )
=> ( Y
= ( cons @ A @ X3 @ ( merges9089515139780605204_merge @ A @ X @ Xs3 @ ( cons @ A @ Y3 @ Ys5 ) ) ) ) )
& ( ~ ( X @ X3 @ Y3 )
=> ( Y
= ( cons @ A @ Y3 @ ( merges9089515139780605204_merge @ A @ X @ ( cons @ A @ X3 @ Xs3 ) @ Ys5 ) ) ) ) ) ) )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( Y != Xa2 ) )
=> ~ ( ( Xa2
= ( nil @ A ) )
=> ! [V: A,Va: list @ A] :
( ( Xb
= ( cons @ A @ V @ Va ) )
=> ( Y
!= ( cons @ A @ V @ Va ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.elims
thf(fact_8077_mergesort__by__rel__merge__simps_I1_J,axiom,
! [A: $tType,R3: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( ( R3 @ X @ Y )
=> ( ( merges9089515139780605204_merge @ A @ R3 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( cons @ A @ X @ ( merges9089515139780605204_merge @ A @ R3 @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ) )
& ( ~ ( R3 @ X @ Y )
=> ( ( merges9089515139780605204_merge @ A @ R3 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( cons @ A @ Y @ ( merges9089515139780605204_merge @ A @ R3 @ ( cons @ A @ X @ Xs ) @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_simps(1)
thf(fact_8078_mergesort__by__rel__merge__simps_I2_J,axiom,
! [A: $tType,R3: A > A > $o,Xs: list @ A] :
( ( merges9089515139780605204_merge @ A @ R3 @ Xs @ ( nil @ A ) )
= Xs ) ).
% mergesort_by_rel_merge_simps(2)
thf(fact_8079_sort__mergesort__by__rel,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4 )
= ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).
% sort_mergesort_by_rel
thf(fact_8080_sorted__mergesort__by__rel,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted_mergesort_by_rel
thf(fact_8081_mergesort__by__rel_Oelims,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A,Y: list @ A] :
( ( ( mergesort_by_rel @ A @ X @ Xa2 )
= Y )
=> ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( Y = Xa2 ) )
& ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( Y
= ( merges9089515139780605204_merge @ A @ X @ ( mergesort_by_rel @ A @ X @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa2 ) ) ) @ ( mergesort_by_rel @ A @ X @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa2 ) ) ) ) ) ) ) ) ).
% mergesort_by_rel.elims
thf(fact_8082_mergesort__by__rel_Osimps,axiom,
! [A: $tType] :
( ( mergesort_by_rel @ A )
= ( ^ [R4: A > A > $o,Xs2: list @ A] : ( if @ ( list @ A ) @ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Xs2 @ ( merges9089515139780605204_merge @ A @ R4 @ ( mergesort_by_rel @ A @ R4 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs2 ) ) ) @ ( mergesort_by_rel @ A @ R4 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs2 ) ) ) ) ) ) ) ).
% mergesort_by_rel.simps
thf(fact_8083_uint32__sshiftr__def,axiom,
( uint32_sshiftr
= ( ^ [X4: uint32,N6: code_integer] :
( if @ uint32
@ ( ( ord_less @ code_integer @ N6 @ ( zero_zero @ code_integer ) )
| ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N6 ) )
@ ( undefined @ ( ( nat > uint32 > uint32 ) > code_integer > uint32 > uint32 ) @ signed489701013188660438uint32 @ N6 @ X4 )
@ ( signed489701013188660438uint32 @ ( code_nat_of_integer @ N6 ) @ X4 ) ) ) ) ).
% uint32_sshiftr_def
thf(fact_8084_uint32__sshiftr__code,axiom,
! [N: code_integer,W2: uint32] :
( ( ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
| ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N ) )
=> ( ( rep_uint32 @ ( uint32_sshiftr @ W2 @ N ) )
= ( rep_uint32 @ ( undefined @ ( ( nat > uint32 > uint32 ) > code_integer > uint32 > uint32 ) @ signed489701013188660438uint32 @ N @ W2 ) ) ) )
& ( ~ ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
| ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N ) )
=> ( ( rep_uint32 @ ( uint32_sshiftr @ W2 @ N ) )
= ( signed_drop_bit @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) @ ( code_nat_of_integer @ N ) @ ( rep_uint32 @ W2 ) ) ) ) ) ).
% uint32_sshiftr_code
thf(fact_8085_mergesort__by__rel_Opsimps,axiom,
! [A: $tType,R3: A > A > $o,Xs: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R3 @ Xs ) )
=> ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( mergesort_by_rel @ A @ R3 @ Xs )
= Xs ) )
& ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( mergesort_by_rel @ A @ R3 @ Xs )
= ( merges9089515139780605204_merge @ A @ R3 @ ( mergesort_by_rel @ A @ R3 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs ) ) ) @ ( mergesort_by_rel @ A @ R3 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs ) ) ) ) ) ) ) ) ).
% mergesort_by_rel.psimps
thf(fact_8086_mergesort__by__rel__merge_Opelims,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( merges9089515139780605204_merge @ A @ X @ Xa2 @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa2 @ Xb ) ) )
=> ( ! [X3: A,Xs3: list @ A] :
( ( Xa2
= ( cons @ A @ X3 @ Xs3 ) )
=> ! [Y3: A,Ys5: list @ A] :
( ( Xb
= ( cons @ A @ Y3 @ Ys5 ) )
=> ( ( ( ( X @ X3 @ Y3 )
=> ( Y
= ( cons @ A @ X3 @ ( merges9089515139780605204_merge @ A @ X @ Xs3 @ ( cons @ A @ Y3 @ Ys5 ) ) ) ) )
& ( ~ ( X @ X3 @ Y3 )
=> ( Y
= ( cons @ A @ Y3 @ ( merges9089515139780605204_merge @ A @ X @ ( cons @ A @ X3 @ Xs3 ) @ Ys5 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ A @ Y3 @ Ys5 ) ) ) ) ) ) )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( ( Y = Xa2 )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa2 @ ( nil @ A ) ) ) ) ) )
=> ~ ( ( Xa2
= ( nil @ A ) )
=> ! [V: A,Va: list @ A] :
( ( Xb
= ( cons @ A @ V @ Va ) )
=> ( ( Y
= ( cons @ A @ V @ Va ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V @ Va ) ) ) ) ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.pelims
thf(fact_8087_mergesort__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( mergesort @ A )
= ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).
% mergesort_def
thf(fact_8088_sort__mergesort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4 )
= ( mergesort @ A ) ) ) ).
% sort_mergesort
thf(fact_8089_arg__min__list_Oelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa2: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa2 )
= Y )
=> ( ! [X3: A] :
( ( Xa2
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( Y != X3 ) )
=> ( ! [X3: A,Y3: A,Zs2: list @ A] :
( ( Xa2
= ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs2 ) ) )
=> ( Y
!= ( if @ A @ ( ord_less_eq @ B @ ( X @ X3 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) )
=> ~ ( ( Xa2
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_8090_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( ^ [A6: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y5: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( sup_sup @ A @ X4 ) @ Y5 ) )
@ ( none @ A )
@ A6 ) ) ) ) ) ).
% Sup_fin.eq_fold'
thf(fact_8091_Sup__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Sup_fin.singleton
thf(fact_8092_inf__Sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ( inf_inf @ A @ A2 @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= A2 ) ) ) ) ).
% inf_Sup_absorb
thf(fact_8093_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_8094_Sup__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( lattic5882676163264333800up_fin @ A @ A5 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Sup_fin.infinite
thf(fact_8095_Sup__fin__Max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% Sup_fin_Max
thf(fact_8096_Sup__fin__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A5 )
= ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% Sup_fin_Sup
thf(fact_8097_cSup__eq__Sup__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= ( lattic5882676163264333800up_fin @ A @ X6 ) ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_8098_Sup__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_8099_Sup__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ord_less_eq @ A @ A2 @ ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_8100_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ X )
=> ! [A9: A] :
( ( member @ A @ A9 @ A5 )
=> ( ord_less_eq @ A @ A9 @ X ) ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_8101_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( ord_less_eq @ A @ A4 @ X ) )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ X ) ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_8102_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ X )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X4 @ X ) ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_8103_arg__min__list_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,X: A,Y: A,Zs: list @ A] :
( ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) )
= ( if @ A @ ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ Y @ Zs ) ) ) ) @ X @ ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_8104_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ B5 ) ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_8105_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [H2: A > A,N8: set @ A] :
( ! [X3: A,Y3: A] :
( ( H2 @ ( sup_sup @ A @ X3 @ Y3 ) )
= ( sup_sup @ A @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N8 )
=> ( ( N8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic5882676163264333800up_fin @ A @ N8 ) )
= ( lattic5882676163264333800up_fin @ A @ ( image @ A @ A @ H2 @ N8 ) ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_8106_Sup__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B5 ) @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_8107_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] : ( member @ A @ ( sup_sup @ A @ X3 @ Y3 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Sup_fin.closed
thf(fact_8108_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_8109_Sup__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ B5 ) ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_8110_Sup__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ X @ A5 ) ) ) ) ).
% Sup_fin.eq_fold
thf(fact_8111_inf__Sup2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ B5 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A8: A,B8: A] :
( ( Uu3
= ( inf_inf @ A @ A8 @ B8 ) )
& ( member @ A @ A8 @ A5 )
& ( member @ A @ B8 @ B5 ) ) ) ) ) ) ) ) ) ) ).
% inf_Sup2_distrib
thf(fact_8112_inf__Sup1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A8: A] :
( ( Uu3
= ( inf_inf @ A @ X @ A8 ) )
& ( member @ A @ A8 @ A5 ) ) ) ) ) ) ) ) ).
% inf_Sup1_distrib
thf(fact_8113_Sup__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A5 )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_8114_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_8115_arg__min__list_Opelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa2: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ Xa2 ) )
=> ( ! [X3: A] :
( ( Xa2
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ( Y = X3 )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) )
=> ( ! [X3: A,Y3: A,Zs2: list @ A] :
( ( Xa2
= ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs2 ) ) )
=> ( ( Y
= ( if @ A @ ( ord_less_eq @ B @ ( X @ X3 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) ) )
=> ~ ( ( Xa2
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) ) ) ) ) ) ) ).
% arg_min_list.pelims
thf(fact_8116_mset__size2elem,axiom,
! [A: $tType,P2: multiset @ A,Q3: A,Q7: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ P2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Q3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ Q7 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ P2 )
=> ( P2
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Q3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ Q7 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% mset_size2elem
thf(fact_8117_size__add__mset,axiom,
! [A: $tType,A2: A,A5: multiset @ A] :
( ( size_size @ ( multiset @ A ) @ ( add_mset @ A @ A2 @ A5 ) )
= ( suc @ ( size_size @ ( multiset @ A ) @ A5 ) ) ) ).
% size_add_mset
thf(fact_8118_mset__le__subtract__add__mset__right,axiom,
! [A: $tType,X: A,B5: multiset @ A,X6: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B5 ) @ X6 )
=> ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ X6 @ B5 ) )
& ( subseteq_mset @ A @ B5 @ X6 ) ) ) ).
% mset_le_subtract_add_mset_right
thf(fact_8119_mset__le__subtract__add__mset__left,axiom,
! [A: $tType,X: A,B5: multiset @ A,X6: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B5 ) @ X6 )
=> ( ( subseteq_mset @ A @ B5 @ ( minus_minus @ ( multiset @ A ) @ X6 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
& ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ X6 ) ) ) ).
% mset_le_subtract_add_mset_left
thf(fact_8120_mset__le__add__mset__decr__left2,axiom,
! [A: $tType,C3: A,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ C3 @ A2 ) @ B2 )
=> ( subseteq_mset @ A @ ( add_mset @ A @ C3 @ ( zero_zero @ ( multiset @ A ) ) ) @ B2 ) ) ).
% mset_le_add_mset_decr_left2
thf(fact_8121_mset__le__single__cases,axiom,
! [A: $tType,M7: multiset @ A,A2: A] :
( ( subseteq_mset @ A @ M7 @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) )
=> ( ( M7
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( M7
= ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_le_single_cases
thf(fact_8122_mset__le__add__mset,axiom,
! [A: $tType,X: A,B5: multiset @ A,C4: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B5 ) @ C4 )
=> ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ C4 )
& ( subseteq_mset @ A @ B5 @ C4 ) ) ) ).
% mset_le_add_mset
thf(fact_8123_mset__union__2__elem,axiom,
! [A: $tType,A2: A,B2: A,C3: A,M7: multiset @ A] :
( ( ( add_mset @ A @ A2 @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( add_mset @ A @ C3 @ M7 ) )
=> ( ( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= M7 )
& ( B2 = C3 ) )
| ( ( A2 = C3 )
& ( ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
= M7 ) ) ) ) ).
% mset_union_2_elem
thf(fact_8124_mset__unplusm__dist__cases2,axiom,
! [A: $tType,B5: multiset @ A,C4: multiset @ A,S: A,A5: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ B5 @ C4 )
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ A5 ) )
=> ( ( ( B5
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ B5 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B5 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C4 ) ) )
=> ~ ( ( C4
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C4 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ B5 @ ( minus_minus @ ( multiset @ A ) @ C4 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_unplusm_dist_cases2
thf(fact_8125_mset__unplusm__dist__cases,axiom,
! [A: $tType,S: A,A5: multiset @ A,B5: multiset @ A,C4: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ A5 )
= ( plus_plus @ ( multiset @ A ) @ B5 @ C4 ) )
=> ( ( ( B5
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ B5 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B5 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C4 ) ) )
=> ~ ( ( C4
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C4 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ B5 @ ( minus_minus @ ( multiset @ A ) @ C4 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_unplusm_dist_cases
thf(fact_8126_mset__single__cases2_H,axiom,
! [A: $tType,S: A,C3: multiset @ A,R6: A,C8: multiset @ A] :
( ( ( add_mset @ A @ S @ C3 )
= ( add_mset @ A @ R6 @ C8 ) )
=> ( ( ( S = R6 )
=> ( C3 != C8 ) )
=> ~ ! [Cc: multiset @ A] :
( ( C8
= ( plus_plus @ ( multiset @ A ) @ Cc @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( C3
= ( plus_plus @ ( multiset @ A ) @ Cc @ ( add_mset @ A @ R6 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( ( minus_minus @ ( multiset @ A ) @ C8 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) )
= Cc )
=> ( ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ R6 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= Cc ) ) ) ) ) ) ).
% mset_single_cases2'
thf(fact_8127_mset__single__cases2,axiom,
! [A: $tType,S: A,C3: multiset @ A,R6: A,C8: multiset @ A] :
( ( ( add_mset @ A @ S @ C3 )
= ( add_mset @ A @ R6 @ C8 ) )
=> ( ( ( S = R6 )
=> ( C3 != C8 ) )
=> ~ ( ( C8
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ C8 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( C3
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ R6 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ R6 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ R6 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= ( minus_minus @ ( multiset @ A ) @ C8 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_single_cases2
thf(fact_8128_mset__single__cases_H,axiom,
! [A: $tType,S: A,C3: multiset @ A,R6: A,C8: multiset @ A] :
( ( ( add_mset @ A @ S @ C3 )
= ( add_mset @ A @ R6 @ C8 ) )
=> ( ( ( S = R6 )
=> ( C3 != C8 ) )
=> ~ ! [Cc: multiset @ A] :
( ( C8
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ Cc ) )
=> ( ( C3
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ R6 @ ( zero_zero @ ( multiset @ A ) ) ) @ Cc ) )
=> ( ( ( minus_minus @ ( multiset @ A ) @ C8 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) )
= Cc )
=> ( ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ R6 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= Cc ) ) ) ) ) ) ).
% mset_single_cases'
thf(fact_8129_mset__single__cases,axiom,
! [A: $tType,S: A,C3: multiset @ A,R6: A,C8: multiset @ A] :
( ( ( add_mset @ A @ S @ C3 )
= ( add_mset @ A @ R6 @ C8 ) )
=> ( ( ( S = R6 )
=> ( C3 != C8 ) )
=> ~ ( ( C8
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C8 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( C3
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ R6 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ R6 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ R6 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= ( minus_minus @ ( multiset @ A ) @ C8 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_single_cases
thf(fact_8130_size__eq__Suc__imp__eq__union,axiom,
! [A: $tType,M7: multiset @ A,N: nat] :
( ( ( size_size @ ( multiset @ A ) @ M7 )
= ( suc @ N ) )
=> ? [A4: A,N11: multiset @ A] :
( M7
= ( add_mset @ A @ A4 @ N11 ) ) ) ).
% size_eq_Suc_imp_eq_union
thf(fact_8131_mset__le__add__mset__decr__left1,axiom,
! [A: $tType,C3: A,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ C3 @ A2 ) @ B2 )
=> ( subseteq_mset @ A @ A2 @ B2 ) ) ).
% mset_le_add_mset_decr_left1
thf(fact_8132_size__mset__SucE,axiom,
! [A: $tType,A5: multiset @ A,N: nat] :
( ( ( size_size @ ( multiset @ A ) @ A5 )
= ( suc @ N ) )
=> ~ ! [A4: A,B9: multiset @ A] :
( ( A5
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) @ B9 ) )
=> ( ( size_size @ ( multiset @ A ) @ B9 )
!= N ) ) ) ).
% size_mset_SucE
thf(fact_8133_size__Diff1__le,axiom,
! [A: $tType,M7: multiset @ A,X: A] : ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M7 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) ) @ ( size_size @ ( multiset @ A ) @ M7 ) ) ).
% size_Diff1_le
thf(fact_8134_mset__size__le1__cases,axiom,
! [A: $tType,M7: multiset @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ M7 ) @ ( suc @ ( zero_zero @ nat ) ) )
=> ( ( M7
!= ( zero_zero @ ( multiset @ A ) ) )
=> ~ ! [M3: A] :
( M7
!= ( add_mset @ A @ M3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_size_le1_cases
thf(fact_8135_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( ^ [A6: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y5: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( inf_inf @ A @ X4 ) @ Y5 ) )
@ ( none @ A )
@ A6 ) ) ) ) ) ).
% Inf_fin.eq_fold'
thf(fact_8136_is__up_Otransfer,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ( bNF_rel_fun @ ( int > int ) @ ( ( word @ A ) > ( word @ B ) ) @ $o @ $o @ ( bNF_rel_fun @ int @ ( word @ A ) @ int @ ( word @ B ) @ ( pcr_word @ A ) @ ( pcr_word @ B ) )
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4
@ ^ [Uu3: int > int] : ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
@ ( is_up @ A @ B ) ) ) ).
% is_up.transfer
thf(fact_8137_Inf__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A] :
( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Inf_fin.singleton
thf(fact_8138_sup__Inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ A2 )
= A2 ) ) ) ) ).
% sup_Inf_absorb
thf(fact_8139_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_8140_Inf__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( lattic7752659483105999362nf_fin @ A @ A5 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Inf_fin.infinite
thf(fact_8141_Inf__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_8142_Inf__fin__Min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% Inf_fin_Min
thf(fact_8143_Inf__fin__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A5 )
= ( complete_Inf_Inf @ A @ A5 ) ) ) ) ) ).
% Inf_fin_Inf
thf(fact_8144_cInf__eq__Inf__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X6 )
= ( lattic7752659483105999362nf_fin @ A @ X6 ) ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_8145_Inf__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ A2 ) ) ) ) ).
% Inf_fin.coboundedI
thf(fact_8146_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
=> ! [A9: A] :
( ( member @ A @ A9 @ A5 )
=> ( ord_less_eq @ A @ X @ A9 ) ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_8147_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( ord_less_eq @ A @ X @ A4 ) )
=> ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_8148_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X @ X4 ) ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_8149_is__up_Orep__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ( ( is_up @ A @ B )
= ( ^ [X4: ( word @ A ) > ( word @ B )] : ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ) ).
% is_up.rep_eq
thf(fact_8150_is__up,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ( ( is_up @ A @ B )
= ( ^ [C7: ( word @ A ) > ( word @ B )] : ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ) ).
% is_up
thf(fact_8151_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B5 ) @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_8152_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [H2: A > A,N8: set @ A] :
( ! [X3: A,Y3: A] :
( ( H2 @ ( inf_inf @ A @ X3 @ Y3 ) )
= ( inf_inf @ A @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N8 )
=> ( ( N8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic7752659483105999362nf_fin @ A @ N8 ) )
= ( lattic7752659483105999362nf_fin @ A @ ( image @ A @ A @ H2 @ N8 ) ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_8153_Inf__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B5 ) @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_8154_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] : ( member @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Inf_fin.closed
thf(fact_8155_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_8156_Inf__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ ( lattic7752659483105999362nf_fin @ A @ B5 ) ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_8157_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ X @ A5 ) ) ) ) ).
% Inf_fin.eq_fold
thf(fact_8158_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_8159_sup__Inf1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A8: A] :
( ( Uu3
= ( sup_sup @ A @ X @ A8 ) )
& ( member @ A @ A8 @ A5 ) ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_8160_sup__Inf2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ ( lattic7752659483105999362nf_fin @ A @ B5 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A8: A,B8: A] :
( ( Uu3
= ( sup_sup @ A @ A8 @ B8 ) )
& ( member @ A @ A8 @ A5 )
& ( member @ A @ B8 @ B5 ) ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_8161_Inf__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A5 )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_8162_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_8163_is__down_Otransfer,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ( bNF_rel_fun @ ( int > int ) @ ( ( word @ A ) > ( word @ B ) ) @ $o @ $o @ ( bNF_rel_fun @ int @ ( word @ A ) @ int @ ( word @ B ) @ ( pcr_word @ A ) @ ( pcr_word @ B ) )
@ ^ [Y6: $o,Z4: $o] : Y6 = Z4
@ ^ [Uu3: int > int] : ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
@ ( is_down @ A @ B ) ) ) ).
% is_down.transfer
thf(fact_8164_map__of__distinct__upd,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ( map_add @ A @ B
@ ( fun_upd @ A @ ( option @ B )
@ ^ [X4: A] : ( none @ B )
@ X
@ ( some @ B @ Y ) )
@ ( map_of @ A @ B @ Xs ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Xs ) @ X @ ( some @ B @ Y ) ) ) ) ).
% map_of_distinct_upd
thf(fact_8165_map__add__first__le,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [M: A > ( option @ B ),M6: A > ( option @ B ),N: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M @ M6 )
=> ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M @ N ) @ ( map_add @ A @ B @ M6 @ N ) ) ) ) ).
% map_add_first_le
thf(fact_8166_map__add__find__left,axiom,
! [A: $tType,B: $tType,G3: B > ( option @ A ),K: B,F3: B > ( option @ A )] :
( ( ( G3 @ K )
= ( none @ A ) )
=> ( ( map_add @ B @ A @ F3 @ G3 @ K )
= ( F3 @ K ) ) ) ).
% map_add_find_left
thf(fact_8167_map__add__left__None,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),K: B,G3: B > ( option @ A )] :
( ( ( F3 @ K )
= ( none @ A ) )
=> ( ( map_add @ B @ A @ F3 @ G3 @ K )
= ( G3 @ K ) ) ) ).
% map_add_left_None
thf(fact_8168_is__down,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ( ( is_down @ A @ B )
= ( ^ [C7: ( word @ A ) > ( word @ B )] : ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% is_down
thf(fact_8169_is__down_Orep__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ( ( is_down @ A @ B )
= ( ^ [X4: ( word @ A ) > ( word @ B )] : ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% is_down.rep_eq
thf(fact_8170_map__of__concat,axiom,
! [B: $tType,A: $tType,Xss: list @ ( list @ ( product_prod @ A @ B ) )] :
( ( map_of @ A @ B @ ( concat @ ( product_prod @ A @ B ) @ Xss ) )
= ( foldr @ ( list @ ( product_prod @ A @ B ) ) @ ( A > ( option @ B ) )
@ ^ [Xs2: list @ ( product_prod @ A @ B ),F4: A > ( option @ B )] : ( map_add @ A @ B @ F4 @ ( map_of @ A @ B @ Xs2 ) )
@ Xss
@ ^ [X4: A] : ( none @ B ) ) ) ).
% map_of_concat
thf(fact_8171_signed__divide__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa2: int,X: int] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( word2 @ A @ Xa2 ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa2 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ) ).
% signed_divide_word.abs_eq
thf(fact_8172_signed__modulo__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa2: int,X: int] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( word2 @ A @ Xa2 ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa2 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ) ).
% signed_modulo_word.abs_eq
thf(fact_8173_less__eq__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa2: int,X: int] :
( ( ord_less_eq @ ( word @ A ) @ ( word2 @ A @ Xa2 ) @ ( word2 @ A @ X ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ Xa2 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X ) ) ) ) ).
% less_eq_word.abs_eq
thf(fact_8174_less__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa2: int,X: int] :
( ( ord_less @ ( word @ A ) @ ( word2 @ A @ Xa2 ) @ ( word2 @ A @ X ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ Xa2 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X ) ) ) ) ).
% less_word.abs_eq
thf(fact_8175_bit__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word2 @ A @ X ) )
= ( ^ [N6: nat] :
( ( ord_less @ nat @ N6 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ X @ N6 ) ) ) ) ) ).
% bit_word.abs_eq
thf(fact_8176_the__signed__int_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( the_signed_int @ A @ ( word2 @ A @ X ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ).
% the_signed_int.abs_eq
thf(fact_8177_signed__cast_Oabs__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: int] :
( ( signed_cast @ A @ B @ ( word2 @ A @ X ) )
= ( word2 @ B @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ).
% signed_cast.abs_eq
thf(fact_8178_word__sle_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa2: int,X: int] :
( ( word_sle @ A @ ( word2 @ A @ Xa2 ) @ ( word2 @ A @ X ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa2 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ).
% word_sle.abs_eq
thf(fact_8179_word__sless_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa2: int,X: int] :
( ( word_sless @ A @ ( word2 @ A @ Xa2 ) @ ( word2 @ A @ X ) )
= ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa2 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ).
% word_sless.abs_eq
thf(fact_8180_signed_Oabs__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [X: int] :
( ( ring_1_signed @ B @ A @ ( word2 @ B @ X ) )
= ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ).
% signed.abs_eq
thf(fact_8181_signed__drop__bit_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa2: nat,X: int] :
( ( signed_drop_bit @ A @ Xa2 @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( bit_se4197421643247451524op_bit @ int @ Xa2 @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ) ).
% signed_drop_bit.abs_eq
thf(fact_8182_at__within__eq,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [X4: A,S8: set @ A] :
( complete_Inf_Inf @ ( filter @ A )
@ ( image @ ( set @ A ) @ ( filter @ A )
@ ^ [S6: set @ A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S6 @ S8 ) @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [S6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S6 )
& ( member @ A @ X4 @ S6 ) ) ) ) ) ) ) ) ).
% at_within_eq
thf(fact_8183_list__collect__set__alt,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F4: B > ( set @ A ),L4: list @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [I3: nat] :
( ( Uu3
= ( F4 @ ( nth @ B @ L4 @ I3 ) ) )
& ( ord_less @ nat @ I3 @ ( size_size @ ( list @ B ) @ L4 ) ) ) ) ) ) ) ).
% list_collect_set_alt
thf(fact_8184_list__collect__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,F3: B > ( set @ A ),A2: B] :
( ( list_collect_set @ B @ A @ F3 @ ( cons @ B @ A2 @ ( nil @ B ) ) )
= ( F3 @ A2 ) ) ).
% list_collect_set_simps(2)
thf(fact_8185_list__collect__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,F3: B > ( set @ A )] :
( ( list_collect_set @ B @ A @ F3 @ ( nil @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% list_collect_set_simps(1)
thf(fact_8186_principal__le__iff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ ( principal @ A @ A5 ) @ ( principal @ A @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% principal_le_iff
% Type constructors (1188)
thf(tcon_VEBT__BuildupMemImp_OVEBTi___Typerep_Otyperep,axiom,
typerep @ vEBT_VEBTi ).
thf(tcon_VEBT__Definitions_OVEBT___Typerep_Otyperep_1,axiom,
typerep @ vEBT_VEBT ).
thf(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep_2,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( typerep @ ( heap_Time_Heap @ A17 ) ) ) ).
thf(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_3,axiom,
typerep @ code_integer ).
thf(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_4,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( typerep @ ( heap_ext @ A17 ) ) ) ).
thf(tcon_Product__Type_Ounit___Typerep_Otyperep_5,axiom,
typerep @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top,axiom,
bounded_lattice_top @ product_unit ).
thf(tcon_Product__Type_Oprod___Typerep_Otyperep_6,axiom,
! [A17: $tType,A20: $tType] :
( ( ( typerep @ A17 )
& ( typerep @ A20 ) )
=> ( typerep @ ( product_prod @ A17 @ A20 ) ) ) ).
thf(tcon_Numeral__Type_Onum1___Typerep_Otyperep_7,axiom,
typerep @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Type__Length_Olen0,axiom,
type_len0 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum0___Typerep_Otyperep_8,axiom,
typerep @ numeral_num0 ).
thf(tcon_Numeral__Type_Onum0___Type__Length_Olen0_9,axiom,
type_len0 @ numeral_num0 ).
thf(tcon_Numeral__Type_Obit1___Typerep_Otyperep_10,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( typerep @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Type__Length_Olen0_11,axiom,
! [A17: $tType] :
( ( type_len0 @ A17 )
=> ( type_len0 @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Typerep_Otyperep_12,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( typerep @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Type__Length_Olen0_13,axiom,
! [A17: $tType] :
( ( type_len0 @ A17 )
=> ( type_len0 @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Multiset_Omultiset___Typerep_Otyperep_14,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( typerep @ ( multiset @ A17 ) ) ) ).
thf(tcon_Extended__Nat_Oenat___Typerep_Otyperep_15,axiom,
typerep @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__top_16,axiom,
bounded_lattice_top @ extended_enat ).
thf(tcon_Complex_Ocomplex___Typerep_Otyperep_17,axiom,
typerep @ complex ).
thf(tcon_Assertions_Oassn___Typerep_Otyperep_18,axiom,
typerep @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_19,axiom,
bounded_lattice_top @ assn ).
thf(tcon_String_Oliteral___Typerep_Otyperep_20,axiom,
typerep @ literal ).
thf(tcon_Uint32_Ouint32___Typerep_Otyperep_21,axiom,
typerep @ uint32 ).
thf(tcon_Option_Ooption___Typerep_Otyperep_22,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( typerep @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice__top_23,axiom,
! [A17: $tType] :
( ( bounded_lattice_top @ A17 )
=> ( bounded_lattice_top @ ( option @ A17 ) ) ) ).
thf(tcon_Filter_Ofilter___Typerep_Otyperep_24,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( typerep @ ( filter @ A17 ) ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_25,axiom,
! [A17: $tType] : ( bounded_lattice_top @ ( filter @ A17 ) ) ).
thf(tcon_Sum__Type_Osum___Typerep_Otyperep_26,axiom,
! [A17: $tType,A20: $tType] :
( ( ( typerep @ A17 )
& ( typerep @ A20 ) )
=> ( typerep @ ( sum_sum @ A17 @ A20 ) ) ) ).
thf(tcon_String_Ochar___Typerep_Otyperep_27,axiom,
typerep @ char ).
thf(tcon_Heap_Oarray___Typerep_Otyperep_28,axiom,
! [A17: $tType] : ( typerep @ ( array @ A17 ) ) ).
thf(tcon_Word_Oword___Typerep_Otyperep_29,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( typerep @ ( word @ A17 ) ) ) ).
thf(tcon_Real_Oreal___Typerep_Otyperep_30,axiom,
typerep @ real ).
thf(tcon_List_Olist___Typerep_Otyperep_31,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( typerep @ ( list @ A17 ) ) ) ).
thf(tcon_Heap_Oref___Typerep_Otyperep_32,axiom,
! [A17: $tType] : ( typerep @ ( ref @ A17 ) ) ).
thf(tcon_HOL_Obool___Typerep_Otyperep_33,axiom,
typerep @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_34,axiom,
bounded_lattice_top @ $o ).
thf(tcon_Set_Oset___Typerep_Otyperep_35,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( typerep @ ( set @ A17 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_36,axiom,
! [A17: $tType] : ( bounded_lattice_top @ ( set @ A17 ) ) ).
thf(tcon_Rat_Orat___Typerep_Otyperep_37,axiom,
typerep @ rat ).
thf(tcon_Num_Onum___Typerep_Otyperep_38,axiom,
typerep @ num ).
thf(tcon_Nat_Onat___Typerep_Otyperep_39,axiom,
typerep @ nat ).
thf(tcon_Int_Oint___Typerep_Otyperep_40,axiom,
typerep @ int ).
thf(tcon_fun___Typerep_Otyperep_41,axiom,
! [A17: $tType,A20: $tType] :
( ( ( typerep @ A17 )
& ( typerep @ A20 ) )
=> ( typerep @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top_42,axiom,
! [A17: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounded_lattice_top @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A17: $tType,A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( condit1219197933456340205attice @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
! [A17: $tType,A20: $tType] :
( ( counta3822494911875563373attice @ A20 )
=> ( counta3822494911875563373attice @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A17: $tType,A20: $tType] :
( ( comple592849572758109894attice @ A20 )
=> ( comple592849572758109894attice @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A17: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounde4967611905675639751up_bot @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A17: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounde4346867609351753570nf_top @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A17: $tType,A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( comple6319245703460814977attice @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A17: $tType,A20: $tType] :
( ( boolea8198339166811842893lgebra @ A20 )
=> ( boolea8198339166811842893lgebra @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A17: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounded_lattice_bot @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A17: $tType,A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( comple9053668089753744459l_ccpo @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A17: $tType,A20: $tType] :
( ( semilattice_sup @ A20 )
=> ( semilattice_sup @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A17: $tType,A20: $tType] :
( ( semilattice_inf @ A20 )
=> ( semilattice_inf @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A17: $tType,A20: $tType] :
( ( distrib_lattice @ A20 )
=> ( distrib_lattice @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A17: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounded_lattice @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Code__Evaluation_Oterm__of,axiom,
! [A17: $tType,A20: $tType] :
( ( ( typerep @ A17 )
& ( typerep @ A20 ) )
=> ( code_term_of @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A17: $tType,A20: $tType] :
( ( order_top @ A20 )
=> ( order_top @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A17: $tType,A20: $tType] :
( ( order_bot @ A20 )
=> ( order_bot @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A17: $tType,A20: $tType] :
( ( preorder @ A20 )
=> ( preorder @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A17: $tType,A20: $tType] :
( ( ( finite_finite @ A17 )
& ( finite_finite @ A20 ) )
=> ( finite_finite @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A17: $tType,A20: $tType] :
( ( lattice @ A20 )
=> ( lattice @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A17: $tType,A20: $tType] :
( ( order @ A20 )
=> ( order @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A17: $tType,A20: $tType] :
( ( ord @ A20 )
=> ( ord @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A17: $tType,A20: $tType] :
( ( bot @ A20 )
=> ( bot @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A17: $tType,A20: $tType] :
( ( uminus @ A20 )
=> ( uminus @ ( A17 > A20 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A17: $tType,A20: $tType] :
( ( minus @ A20 )
=> ( minus @ ( A17 > A20 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_43,axiom,
condit1219197933456340205attice @ int ).
thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
bit_un5681908812861735899ations @ int ).
thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri1453513574482234551roduct @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
euclid5411537665997757685th_nat @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere1937475149494474687imp_le @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
euclid4440199948858584721cancel @ int ).
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri6575147826004484403cancel @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict9044650504122735259up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere580206878836729694up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere2412721322843649153imp_le @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
bit_se359711467146920520ations @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord2810124833399127020strict @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
topolo4958980785337419405_space @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
topolo1944317154257567458pology @ int ).
thf(tcon_Int_Oint___Quickcheck__Narrowing_Opartial__term__of,axiom,
quickc6926020345158392990erm_of @ int ).
thf(tcon_Int_Oint___Bit__Comprehension_Obit__comprehension,axiom,
bit_bi6583157726757044596ension @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord715952674999750819strict @ int ).
thf(tcon_Int_Oint___Limits_Otopological__comm__monoid__add,axiom,
topolo5987344860129210374id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
linord4140545234300271783up_add @ int ).
thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
topolo2564578578187576103pology @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
semiri2026040879449505780visors @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring @ int ).
thf(tcon_Int_Oint___Limits_Otopological__semigroup__mult,axiom,
topolo4211221413907600880p_mult @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord8928482502909563296strict @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
semiri3467727345109120633visors @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere6658533253407199908up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere166539214618696060dd_abs @ int ).
thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere6911136660526730532id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord5086331880401160121up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel2418104881723323429up_add @ int ).
thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_15535105094025558882visors @ int ).
thf(tcon_Int_Oint___Limits_Otopological__monoid__add,axiom,
topolo6943815403480290642id_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1802427076303600483id_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord4710134922213307826strict @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
bit_semiring_bits @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
topological_t2_space @ int ).
thf(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere2520102378445227354miring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord6961819062388156250ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__sup_44,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_45,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Lattices_Odistrib__lattice_46,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___Generic__set__bit_Oset__bit,axiom,
generic_set_set_bit @ int ).
thf(tcon_Int_Oint___Code__Evaluation_Oterm__of_47,axiom,
code_term_of @ 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___Orderings_Opreorder_48,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0 @ int ).
thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top @ int ).
thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot @ int ).
thf(tcon_Int_Oint___Lattices_Olattice_49,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_50,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_Osemidom,axiom,
semidom @ int ).
thf(tcon_Int_Oint___Orderings_Oord_51,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_52,axiom,
uminus @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Rings_Oabs__if,axiom,
abs_if @ int ).
thf(tcon_Int_Oint___Groups_Otimes,axiom,
times @ int ).
thf(tcon_Int_Oint___Groups_Ominus_53,axiom,
minus @ int ).
thf(tcon_Int_Oint___Power_Opower,axiom,
power @ int ).
thf(tcon_Int_Oint___Num_Onumeral,axiom,
numeral @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Groups_Oplus,axiom,
plus @ int ).
thf(tcon_Int_Oint___Rings_Oidom,axiom,
idom @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Int_Oint___Rings_Odvd,axiom,
dvd @ int ).
thf(tcon_Int_Oint___Heap_Oheap,axiom,
heap @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_54,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_55,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_56,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_57,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_58,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_59,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_60,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_61,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_62,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_63,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_64,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_65,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_66,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_67,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_68,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_69,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_70,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_71,axiom,
topolo4958980785337419405_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_72,axiom,
topolo1944317154257567458pology @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Narrowing_Opartial__term__of_73,axiom,
quickc6926020345158392990erm_of @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_74,axiom,
topolo5987344860129210374id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_75,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_76,axiom,
topolo2564578578187576103pology @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_77,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_78,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_79,axiom,
topolo4211221413907600880p_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_80,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_81,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_82,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_83,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_84,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_85,axiom,
topolo6943815403480290642id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_86,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_87,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_88,axiom,
topological_t2_space @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_89,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_90,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_91,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_92,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_93,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_94,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_95,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_96,axiom,
distrib_lattice @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_97,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_98,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_99,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_100,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_101,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Code__Evaluation_Oterm__of_102,axiom,
code_term_of @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_103,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_104,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_105,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_106,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_107,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_108,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_109,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_110,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_111,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_112,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_113,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_114,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_115,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_116,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_117,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_118,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_119,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_120,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_121,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_122,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_123,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_124,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_125,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_126,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_127,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_128,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_129,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom_130,axiom,
semidom @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_131,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_132,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Groups_Otimes_133,axiom,
times @ nat ).
thf(tcon_Nat_Onat___Groups_Ominus_134,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Power_Opower_135,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_136,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_137,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_138,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_139,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_140,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Heap_Oheap_141,axiom,
heap @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Quickcheck__Narrowing_Opartial__term__of_142,axiom,
quickc6926020345158392990erm_of @ num ).
thf(tcon_Num_Onum___Code__Evaluation_Oterm__of_143,axiom,
code_term_of @ num ).
thf(tcon_Num_Onum___Orderings_Opreorder_144,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_145,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_146,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_147,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Otimes_148,axiom,
times @ num ).
thf(tcon_Num_Onum___Groups_Oplus_149,axiom,
plus @ num ).
thf(tcon_Num_Onum___Nat_Osize_150,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_151,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_152,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_153,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_154,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_155,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_156,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_157,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_158,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_159,axiom,
ordere8940638589300402666id_add @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Narrowing_Opartial__term__of_160,axiom,
quickc6926020345158392990erm_of @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
archim462609752435547400_field @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_161,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_162,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_163,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_164,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_165,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_166,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_167,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_168,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_169,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_170,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_171,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_172,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_173,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_174,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_175,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_176,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_177,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_178,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_179,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_180,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_181,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_182,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_183,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_184,axiom,
distrib_lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_185,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_186,axiom,
semiring_1_cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_187,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_188,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Code__Evaluation_Oterm__of_189,axiom,
code_term_of @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_190,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_191,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_192,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_193,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_194,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_195,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_196,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_197,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_198,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_199,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_200,axiom,
semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__less__one_201,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_202,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_203,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_204,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_205,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_206,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_207,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_208,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_209,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_210,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_211,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_212,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_213,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_214,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_215,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_216,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_217,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_218,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_219,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_220,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_221,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_222,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom_223,axiom,
semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_224,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_225,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_226,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_227,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Groups_Otimes_228,axiom,
times @ rat ).
thf(tcon_Rat_Orat___Groups_Ominus_229,axiom,
minus @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_230,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_231,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_232,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_233,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_234,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_235,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_236,axiom,
dvd @ rat ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_237,axiom,
! [A17: $tType] : ( condit1219197933456340205attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_238,axiom,
! [A17: $tType] : ( counta3822494911875563373attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_239,axiom,
! [A17: $tType] : ( comple592849572758109894attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Narrowing_Opartial__term__of_240,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( quickc6926020345158392990erm_of @ ( set @ A17 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_241,axiom,
! [A17: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_242,axiom,
! [A17: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_243,axiom,
! [A17: $tType] : ( comple6319245703460814977attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_244,axiom,
! [A17: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_245,axiom,
! [A17: $tType] : ( bounded_lattice_bot @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_246,axiom,
! [A17: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_247,axiom,
! [A17: $tType] : ( semilattice_sup @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_248,axiom,
! [A17: $tType] : ( semilattice_inf @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Odistrib__lattice_249,axiom,
! [A17: $tType] : ( distrib_lattice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_250,axiom,
! [A17: $tType] : ( bounded_lattice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Code__Evaluation_Oterm__of_251,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( code_term_of @ ( set @ A17 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_252,axiom,
! [A17: $tType] : ( order_top @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_253,axiom,
! [A17: $tType] : ( order_bot @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_254,axiom,
! [A17: $tType] : ( preorder @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_255,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( finite_finite @ ( set @ A17 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_256,axiom,
! [A17: $tType] : ( lattice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_257,axiom,
! [A17: $tType] : ( order @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_258,axiom,
! [A17: $tType] : ( ord @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_259,axiom,
! [A17: $tType] : ( bot @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_260,axiom,
! [A17: $tType] : ( uminus @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_261,axiom,
! [A17: $tType] : ( minus @ ( set @ A17 ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_262,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_263,axiom,
counta3822494911875563373attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_264,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_265,axiom,
topolo4958980785337419405_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_266,axiom,
topolo1944317154257567458pology @ $o ).
thf(tcon_HOL_Obool___Quickcheck__Narrowing_Opartial__term__of_267,axiom,
quickc6926020345158392990erm_of @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_268,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_269,axiom,
bounde4346867609351753570nf_top @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_270,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_271,axiom,
topolo2564578578187576103pology @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_272,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_273,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_274,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_275,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_276,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_277,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_278,axiom,
distrib_lattice @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_279,axiom,
bounded_lattice @ $o ).
thf(tcon_HOL_Obool___Code__Evaluation_Oterm__of_280,axiom,
code_term_of @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_281,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_282,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_283,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_284,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_285,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_286,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_287,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_288,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_289,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_290,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_291,axiom,
minus @ $o ).
thf(tcon_HOL_Obool___Heap_Oheap_292,axiom,
heap @ $o ).
thf(tcon_Heap_Oref___Quickcheck__Narrowing_Opartial__term__of_293,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( quickc6926020345158392990erm_of @ ( ref @ A17 ) ) ) ).
thf(tcon_Heap_Oref___Code__Evaluation_Oterm__of_294,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( code_term_of @ ( ref @ A17 ) ) ) ).
thf(tcon_Heap_Oref___Heap_Oheap_295,axiom,
! [A17: $tType] : ( heap @ ( ref @ A17 ) ) ).
thf(tcon_Heap_Oref___Nat_Osize_296,axiom,
! [A17: $tType] : ( size @ ( ref @ A17 ) ) ).
thf(tcon_List_Olist___Quickcheck__Narrowing_Opartial__term__of_297,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( quickc6926020345158392990erm_of @ ( list @ A17 ) ) ) ).
thf(tcon_List_Olist___Code__Evaluation_Oterm__of_298,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( code_term_of @ ( list @ A17 ) ) ) ).
thf(tcon_List_Olist___Heap_Oheap_299,axiom,
! [A17: $tType] :
( ( heap @ A17 )
=> ( heap @ ( list @ A17 ) ) ) ).
thf(tcon_List_Olist___Nat_Osize_300,axiom,
! [A17: $tType] : ( size @ ( list @ A17 ) ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_301,axiom,
condit6923001295902523014norder @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_302,axiom,
condit1219197933456340205attice @ real ).
thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_303,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_304,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_305,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_306,axiom,
strict9044650504122735259up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_307,axiom,
ordere580206878836729694up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_308,axiom,
ordere2412721322843649153imp_le @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_309,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_310,axiom,
strict7427464778891057005id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_311,axiom,
ordere8940638589300402666id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_312,axiom,
topolo4958980785337419405_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_313,axiom,
topolo1944317154257567458pology @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field,axiom,
real_V3459762299906320749_field @ real ).
thf(tcon_Real_Oreal___Quickcheck__Narrowing_Opartial__term__of_314,axiom,
quickc6926020345158392990erm_of @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom,
real_V5047593784448816457lgebra @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field_315,axiom,
archim462609752435547400_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_316,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_317,axiom,
unboun7993243217541854897norder @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_318,axiom,
topolo5987344860129210374id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_319,axiom,
linord4140545234300271783up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_320,axiom,
topolo2564578578187576103pology @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_321,axiom,
semiri2026040879449505780visors @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_322,axiom,
linord181362715937106298miring @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space,axiom,
real_V8037385150606011577_space @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__semigroup__mult_323,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_324,axiom,
linord8928482502909563296strict @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_325,axiom,
semiri3467727345109120633visors @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
real_V7819770556892013058_space @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__ab__group__add,axiom,
topolo1287966508704411220up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_326,axiom,
ordere6658533253407199908up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_327,axiom,
ordere166539214618696060dd_abs @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_328,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_329,axiom,
ordere6911136660526730532id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_330,axiom,
linord5086331880401160121up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_331,axiom,
cancel2418104881723323429up_add @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_332,axiom,
ring_15535105094025558882visors @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__monoid__add_333,axiom,
topolo6943815403480290642id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_334,axiom,
cancel1802427076303600483id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_335,axiom,
linord4710134922213307826strict @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_336,axiom,
topological_t2_space @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_337,axiom,
ordere2520102378445227354miring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_338,axiom,
linord6961819062388156250ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_339,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_340,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring_341,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_342,axiom,
ordered_semiring_0 @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_343,axiom,
linordered_semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder_344,axiom,
dense_linorder @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_345,axiom,
semilattice_sup @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__inf_346,axiom,
semilattice_inf @ real ).
thf(tcon_Real_Oreal___Lattices_Odistrib__lattice_347,axiom,
distrib_lattice @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_348,axiom,
ab_semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__cancel_349,axiom,
semiring_1_cancel @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_350,axiom,
comm_monoid_mult @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_351,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field_352,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Code__Evaluation_Oterm__of_353,axiom,
code_term_of @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring_354,axiom,
ordered_semiring @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring__abs_355,axiom,
ordered_ring_abs @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_356,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring_357,axiom,
linordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_358,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_359,axiom,
comm_semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_360,axiom,
comm_semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__order_361,axiom,
dense_order @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_362,axiom,
semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide_363,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Num_Osemiring__numeral_364,axiom,
semiring_numeral @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_365,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring_366,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one_367,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_368,axiom,
semiring_char_0 @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_369,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__char__0_370,axiom,
field_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_371,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring_372,axiom,
ordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_373,axiom,
idom_abs_sgn @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_374,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_375,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_376,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_377,axiom,
comm_ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_378,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_379,axiom,
semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__0_380,axiom,
semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_381,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot_382,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Lattices_Olattice_383,axiom,
lattice @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_384,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Rings_Omult__zero_385,axiom,
mult_zero @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring_386,axiom,
comm_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_387,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_388,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_389,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Fields_Oinverse_390,axiom,
inverse @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom_391,axiom,
semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_392,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_393,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_394,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Oabs__if_395,axiom,
abs_if @ real ).
thf(tcon_Real_Oreal___Groups_Otimes_396,axiom,
times @ real ).
thf(tcon_Real_Oreal___Groups_Ominus_397,axiom,
minus @ real ).
thf(tcon_Real_Oreal___Fields_Ofield_398,axiom,
field @ real ).
thf(tcon_Real_Oreal___Power_Opower_399,axiom,
power @ real ).
thf(tcon_Real_Oreal___Num_Onumeral_400,axiom,
numeral @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_401,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oplus_402,axiom,
plus @ real ).
thf(tcon_Real_Oreal___Rings_Oidom_403,axiom,
idom @ real ).
thf(tcon_Real_Oreal___Groups_Oone_404,axiom,
one @ real ).
thf(tcon_Real_Oreal___Rings_Odvd_405,axiom,
dvd @ real ).
thf(tcon_Word_Oword___Bit__Operations_Osemiring__bit__operations_406,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( bit_se359711467146920520ations @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Quickcheck__Narrowing_Opartial__term__of_407,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( quickc6926020345158392990erm_of @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Bit__Comprehension_Obit__comprehension_408,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( bit_bi6583157726757044596ension @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Bit__Operations_Oring__bit__operations_409,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( bit_ri3973907225187159222ations @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocancel__ab__semigroup__add_410,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( cancel2418104881723323429up_add @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocancel__comm__monoid__add_411,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( cancel1802427076303600483id_add @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Bit__Operations_Osemiring__bits_412,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( bit_semiring_bits @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocancel__semigroup__add_413,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( cancel_semigroup_add @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oab__semigroup__mult_414,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( ab_semigroup_mult @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__1__cancel_415,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( semiring_1_cancel @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocomm__monoid__mult_416,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( comm_monoid_mult @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oab__semigroup__add_417,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( ab_semigroup_add @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Generic__set__bit_Oset__bit_418,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( generic_set_set_bit @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Code__Evaluation_Oterm__of_419,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( code_term_of @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Parity_Osemiring__parity_420,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( semiring_parity @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocomm__monoid__add_421,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( comm_monoid_add @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__modulo_422,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( semiring_modulo @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring__1_423,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( comm_semiring_1 @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring__0_424,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( comm_semiring_0 @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Osemigroup__mult_425,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( semigroup_mult @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Num_Osemiring__numeral_426,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( semiring_numeral @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Osemigroup__add_427,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( semigroup_add @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Owellorder_428,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( wellorder @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oab__group__add_429,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( ab_group_add @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ozero__neq__one_430,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( zero_neq_one @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Opreorder_431,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( preorder @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Olinorder_432,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( linorder @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Omonoid__mult_433,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( monoid_mult @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__ring__1_434,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( comm_ring_1 @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Omonoid__add_435,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( monoid_add @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Finite__Set_Ofinite_436,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( finite_finite @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__1_437,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( semiring_1 @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__0_438,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( semiring_0 @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ogroup__add_439,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( group_add @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Rings_Omult__zero_440,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( mult_zero @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__ring_441,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( comm_ring @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Oorder_442,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( order @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Num_Oneg__numeral_443,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( neg_numeral @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Oord_444,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( ord @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ouminus_445,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( uminus @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Rings_Oring__1_446,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( ring_1 @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Otimes_447,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( times @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ominus_448,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( minus @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Power_Opower_449,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( power @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Num_Onumeral_450,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( numeral @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ozero_451,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( zero @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oplus_452,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( plus @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oone_453,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( one @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Rings_Odvd_454,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( dvd @ ( word @ A17 ) ) ) ).
thf(tcon_Word_Oword___Nat_Osize_455,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( size @ ( word @ A17 ) ) ) ).
thf(tcon_Heap_Oarray___Quickcheck__Narrowing_Opartial__term__of_456,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( quickc6926020345158392990erm_of @ ( array @ A17 ) ) ) ).
thf(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_457,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( code_term_of @ ( array @ A17 ) ) ) ).
thf(tcon_Heap_Oarray___Heap_Oheap_458,axiom,
! [A17: $tType] : ( heap @ ( array @ A17 ) ) ).
thf(tcon_Heap_Oarray___Nat_Osize_459,axiom,
! [A17: $tType] : ( size @ ( array @ A17 ) ) ).
thf(tcon_String_Ochar___Quickcheck__Narrowing_Opartial__term__of_460,axiom,
quickc6926020345158392990erm_of @ char ).
thf(tcon_String_Ochar___Lattices_Osemilattice__sup_461,axiom,
semilattice_sup @ char ).
thf(tcon_String_Ochar___Lattices_Osemilattice__inf_462,axiom,
semilattice_inf @ char ).
thf(tcon_String_Ochar___Lattices_Odistrib__lattice_463,axiom,
distrib_lattice @ char ).
thf(tcon_String_Ochar___Code__Evaluation_Oterm__of_464,axiom,
code_term_of @ char ).
thf(tcon_String_Ochar___Orderings_Opreorder_465,axiom,
preorder @ char ).
thf(tcon_String_Ochar___Orderings_Olinorder_466,axiom,
linorder @ char ).
thf(tcon_String_Ochar___Finite__Set_Ofinite_467,axiom,
finite_finite @ char ).
thf(tcon_String_Ochar___Lattices_Olattice_468,axiom,
lattice @ char ).
thf(tcon_String_Ochar___Orderings_Oorder_469,axiom,
order @ char ).
thf(tcon_String_Ochar___Orderings_Oord_470,axiom,
ord @ char ).
thf(tcon_String_Ochar___Heap_Oheap_471,axiom,
heap @ char ).
thf(tcon_String_Ochar___Nat_Osize_472,axiom,
size @ char ).
thf(tcon_Sum__Type_Osum___Quickcheck__Narrowing_Opartial__term__of_473,axiom,
! [A17: $tType,A20: $tType] :
( ( ( typerep @ A17 )
& ( typerep @ A20 ) )
=> ( quickc6926020345158392990erm_of @ ( sum_sum @ A17 @ A20 ) ) ) ).
thf(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_474,axiom,
! [A17: $tType,A20: $tType] :
( ( ( typerep @ A17 )
& ( typerep @ A20 ) )
=> ( code_term_of @ ( sum_sum @ A17 @ A20 ) ) ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_475,axiom,
! [A17: $tType,A20: $tType] :
( ( ( finite_finite @ A17 )
& ( finite_finite @ A20 ) )
=> ( finite_finite @ ( sum_sum @ A17 @ A20 ) ) ) ).
thf(tcon_Sum__Type_Osum___Heap_Oheap_476,axiom,
! [A17: $tType,A20: $tType] :
( ( ( heap @ A17 )
& ( heap @ A20 ) )
=> ( heap @ ( sum_sum @ A17 @ A20 ) ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_477,axiom,
! [A17: $tType,A20: $tType] : ( size @ ( sum_sum @ A17 @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_478,axiom,
! [A17: $tType] : ( condit1219197933456340205attice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_479,axiom,
! [A17: $tType] : ( counta3822494911875563373attice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Quickcheck__Narrowing_Opartial__term__of_480,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( quickc6926020345158392990erm_of @ ( filter @ A17 ) ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_481,axiom,
! [A17: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_482,axiom,
! [A17: $tType] : ( bounde4346867609351753570nf_top @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_483,axiom,
! [A17: $tType] : ( comple6319245703460814977attice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_484,axiom,
! [A17: $tType] : ( bounded_lattice_bot @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_485,axiom,
! [A17: $tType] : ( comple9053668089753744459l_ccpo @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_486,axiom,
! [A17: $tType] : ( semilattice_sup @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_487,axiom,
! [A17: $tType] : ( semilattice_inf @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_488,axiom,
! [A17: $tType] : ( distrib_lattice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_489,axiom,
! [A17: $tType] : ( bounded_lattice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_490,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( code_term_of @ ( filter @ A17 ) ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_491,axiom,
! [A17: $tType] : ( order_top @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_492,axiom,
! [A17: $tType] : ( order_bot @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_493,axiom,
! [A17: $tType] : ( preorder @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_494,axiom,
! [A17: $tType] : ( lattice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_495,axiom,
! [A17: $tType] : ( order @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_496,axiom,
! [A17: $tType] : ( ord @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Obot_497,axiom,
! [A17: $tType] : ( bot @ ( filter @ A17 ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_498,axiom,
! [A17: $tType] :
( ( comple5582772986160207858norder @ A17 )
=> ( condit6923001295902523014norder @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_499,axiom,
! [A17: $tType] :
( ( comple6319245703460814977attice @ A17 )
=> ( condit1219197933456340205attice @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Countable__Complete__Lattices_Ocountable__complete__lattice_500,axiom,
! [A17: $tType] :
( ( comple6319245703460814977attice @ A17 )
=> ( counta3822494911875563373attice @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_501,axiom,
! [A17: $tType] :
( ( comple592849572758109894attice @ A17 )
=> ( comple592849572758109894attice @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Quickcheck__Narrowing_Opartial__term__of_502,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( quickc6926020345158392990erm_of @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_503,axiom,
! [A17: $tType] :
( ( lattice @ A17 )
=> ( bounde4967611905675639751up_bot @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__semilattice__inf__top_504,axiom,
! [A17: $tType] :
( ( bounded_lattice_top @ A17 )
=> ( bounde4346867609351753570nf_top @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder,axiom,
! [A17: $tType] :
( ( comple5582772986160207858norder @ A17 )
=> ( comple5582772986160207858norder @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_505,axiom,
! [A17: $tType] :
( ( comple6319245703460814977attice @ A17 )
=> ( comple6319245703460814977attice @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice__bot_506,axiom,
! [A17: $tType] :
( ( lattice @ A17 )
=> ( bounded_lattice_bot @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Partial__Order_Occpo_507,axiom,
! [A17: $tType] :
( ( comple6319245703460814977attice @ A17 )
=> ( comple9053668089753744459l_ccpo @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__sup_508,axiom,
! [A17: $tType] :
( ( semilattice_sup @ A17 )
=> ( semilattice_sup @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__inf_509,axiom,
! [A17: $tType] :
( ( semilattice_inf @ A17 )
=> ( semilattice_inf @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Odistrib__lattice_510,axiom,
! [A17: $tType] :
( ( distrib_lattice @ A17 )
=> ( distrib_lattice @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice_511,axiom,
! [A17: $tType] :
( ( bounded_lattice_top @ A17 )
=> ( bounded_lattice @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Code__Evaluation_Oterm__of_512,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( code_term_of @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Owellorder_513,axiom,
! [A17: $tType] :
( ( wellorder @ A17 )
=> ( wellorder @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__top_514,axiom,
! [A17: $tType] :
( ( order_top @ A17 )
=> ( order_top @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__bot_515,axiom,
! [A17: $tType] :
( ( order @ A17 )
=> ( order_bot @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Opreorder_516,axiom,
! [A17: $tType] :
( ( preorder @ A17 )
=> ( preorder @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Olinorder_517,axiom,
! [A17: $tType] :
( ( linorder @ A17 )
=> ( linorder @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_518,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( finite_finite @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Olattice_519,axiom,
! [A17: $tType] :
( ( lattice @ A17 )
=> ( lattice @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder_520,axiom,
! [A17: $tType] :
( ( order @ A17 )
=> ( order @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oord_521,axiom,
! [A17: $tType] :
( ( preorder @ A17 )
=> ( ord @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Obot_522,axiom,
! [A17: $tType] :
( ( order @ A17 )
=> ( bot @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Heap_Oheap_523,axiom,
! [A17: $tType] :
( ( heap @ A17 )
=> ( heap @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_524,axiom,
! [A17: $tType] : ( size @ ( option @ A17 ) ) ).
thf(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bit__operations_525,axiom,
bit_se359711467146920520ations @ uint32 ).
thf(tcon_Uint32_Ouint32___Quickcheck__Narrowing_Opartial__term__of_526,axiom,
quickc6926020345158392990erm_of @ uint32 ).
thf(tcon_Uint32_Ouint32___Bit__Comprehension_Obit__comprehension_527,axiom,
bit_bi6583157726757044596ension @ uint32 ).
thf(tcon_Uint32_Ouint32___Bit__Operations_Oring__bit__operations_528,axiom,
bit_ri3973907225187159222ations @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocancel__ab__semigroup__add_529,axiom,
cancel2418104881723323429up_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocancel__comm__monoid__add_530,axiom,
cancel1802427076303600483id_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bits_531,axiom,
bit_semiring_bits @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocancel__semigroup__add_532,axiom,
cancel_semigroup_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oab__semigroup__mult_533,axiom,
ab_semigroup_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__1__cancel_534,axiom,
semiring_1_cancel @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__mult_535,axiom,
comm_monoid_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oab__semigroup__add_536,axiom,
ab_semigroup_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Generic__set__bit_Oset__bit_537,axiom,
generic_set_set_bit @ uint32 ).
thf(tcon_Uint32_Ouint32___Code__Evaluation_Oterm__of_538,axiom,
code_term_of @ uint32 ).
thf(tcon_Uint32_Ouint32___Parity_Osemiring__parity_539,axiom,
semiring_parity @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__add_540,axiom,
comm_monoid_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__modulo_541,axiom,
semiring_modulo @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1_542,axiom,
comm_semiring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__0_543,axiom,
comm_semiring_0 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Osemigroup__mult_544,axiom,
semigroup_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Num_Osemiring__numeral_545,axiom,
semiring_numeral @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Osemigroup__add_546,axiom,
semigroup_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oab__group__add_547,axiom,
ab_group_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ozero__neq__one_548,axiom,
zero_neq_one @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Opreorder_549,axiom,
preorder @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Olinorder_550,axiom,
linorder @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Omonoid__mult_551,axiom,
monoid_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__ring__1_552,axiom,
comm_ring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Omonoid__add_553,axiom,
monoid_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__1_554,axiom,
semiring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__0_555,axiom,
semiring_0 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ogroup__add_556,axiom,
group_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Omult__zero_557,axiom,
mult_zero @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__ring_558,axiom,
comm_ring @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Oorder_559,axiom,
order @ uint32 ).
thf(tcon_Uint32_Ouint32___Num_Oneg__numeral_560,axiom,
neg_numeral @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Oord_561,axiom,
ord @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ouminus_562,axiom,
uminus @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Oring__1_563,axiom,
ring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Otimes_564,axiom,
times @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ominus_565,axiom,
minus @ uint32 ).
thf(tcon_Uint32_Ouint32___Power_Opower_566,axiom,
power @ uint32 ).
thf(tcon_Uint32_Ouint32___Num_Onumeral_567,axiom,
numeral @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ozero_568,axiom,
zero @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oplus_569,axiom,
plus @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oone_570,axiom,
one @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Odvd_571,axiom,
dvd @ uint32 ).
thf(tcon_Uint32_Ouint32___Nat_Osize_572,axiom,
size @ uint32 ).
thf(tcon_String_Oliteral___Quickcheck__Narrowing_Opartial__term__of_573,axiom,
quickc6926020345158392990erm_of @ literal ).
thf(tcon_String_Oliteral___Code__Evaluation_Oterm__of_574,axiom,
code_term_of @ literal ).
thf(tcon_String_Oliteral___Groups_Osemigroup__add_575,axiom,
semigroup_add @ literal ).
thf(tcon_String_Oliteral___Orderings_Opreorder_576,axiom,
preorder @ literal ).
thf(tcon_String_Oliteral___Orderings_Olinorder_577,axiom,
linorder @ literal ).
thf(tcon_String_Oliteral___Groups_Omonoid__add_578,axiom,
monoid_add @ literal ).
thf(tcon_String_Oliteral___Orderings_Oorder_579,axiom,
order @ literal ).
thf(tcon_String_Oliteral___Orderings_Oord_580,axiom,
ord @ literal ).
thf(tcon_String_Oliteral___Groups_Ozero_581,axiom,
zero @ literal ).
thf(tcon_String_Oliteral___Groups_Oplus_582,axiom,
plus @ literal ).
thf(tcon_String_Oliteral___Heap_Oheap_583,axiom,
heap @ literal ).
thf(tcon_String_Oliteral___Nat_Osize_584,axiom,
size @ literal ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__sup__bot_585,axiom,
bounde4967611905675639751up_bot @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__inf__top_586,axiom,
bounde4346867609351753570nf_top @ assn ).
thf(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_587,axiom,
boolea8198339166811842893lgebra @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice__bot_588,axiom,
bounded_lattice_bot @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_589,axiom,
semilattice_sup @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_590,axiom,
semilattice_inf @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Odistrib__lattice_591,axiom,
distrib_lattice @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice_592,axiom,
bounded_lattice @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_593,axiom,
ab_semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_594,axiom,
comm_monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Osemigroup__mult_595,axiom,
semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__top_596,axiom,
order_top @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__bot_597,axiom,
order_bot @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Opreorder_598,axiom,
preorder @ assn ).
thf(tcon_Assertions_Oassn___Groups_Omonoid__mult_599,axiom,
monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Olattice_600,axiom,
lattice @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder_601,axiom,
order @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oord_602,axiom,
ord @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Obot_603,axiom,
bot @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ouminus_604,axiom,
uminus @ assn ).
thf(tcon_Assertions_Oassn___Groups_Otimes_605,axiom,
times @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ominus_606,axiom,
minus @ assn ).
thf(tcon_Assertions_Oassn___Power_Opower_607,axiom,
power @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oone_608,axiom,
one @ assn ).
thf(tcon_Assertions_Oassn___Rings_Odvd_609,axiom,
dvd @ assn ).
thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_610,axiom,
semiri1453513574482234551roduct @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_611,axiom,
topolo3112930676232923870pology @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_612,axiom,
real_V8999393235501362500lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_613,axiom,
real_V2822296259951069270ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_614,axiom,
semiri6575147826004484403cancel @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_615,axiom,
real_V4412858255891104859lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_616,axiom,
real_V822414075346904944vector @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_617,axiom,
topolo4958980785337419405_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_618,axiom,
real_V3459762299906320749_field @ complex ).
thf(tcon_Complex_Ocomplex___Quickcheck__Narrowing_Opartial__term__of_619,axiom,
quickc6926020345158392990erm_of @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_620,axiom,
real_V5047593784448816457lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_621,axiom,
real_V768167426530841204y_dist @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_622,axiom,
topolo5987344860129210374id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_623,axiom,
semiri2026040879449505780visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_624,axiom,
real_V8037385150606011577_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_625,axiom,
topolo4211221413907600880p_mult @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_626,axiom,
topolo7287701948861334536_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_627,axiom,
topolo8386298272705272623_space @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_628,axiom,
semiri3467727345109120633visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_629,axiom,
real_V7819770556892013058_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_630,axiom,
topolo1287966508704411220up_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_631,axiom,
real_V4867850818363320053vector @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_632,axiom,
cancel2418104881723323429up_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_633,axiom,
ring_15535105094025558882visors @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_634,axiom,
topolo6943815403480290642id_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_635,axiom,
cancel1802427076303600483id_add @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_636,axiom,
topological_t2_space @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_637,axiom,
cancel_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_638,axiom,
real_Vector_banach @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_639,axiom,
ab_semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_640,axiom,
semiring_1_cancel @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_641,axiom,
comm_monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_642,axiom,
ab_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Code__Evaluation_Oterm__of_643,axiom,
code_term_of @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_644,axiom,
comm_monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_645,axiom,
comm_semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_646,axiom,
comm_semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_647,axiom,
semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom__divide_648,axiom,
semidom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Num_Osemiring__numeral_649,axiom,
semiring_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__add_650,axiom,
semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Odivision__ring_651,axiom,
division_ring @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_652,axiom,
semiring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_653,axiom,
ab_group_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_654,axiom,
field_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_655,axiom,
zero_neq_one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_656,axiom,
idom_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_657,axiom,
monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_658,axiom,
comm_ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_659,axiom,
monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_660,axiom,
semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_661,axiom,
semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_662,axiom,
group_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Omult__zero_663,axiom,
mult_zero @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_664,axiom,
comm_ring @ complex ).
thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_665,axiom,
neg_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_666,axiom,
ring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Oinverse_667,axiom,
inverse @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom_668,axiom,
semidom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ouminus_669,axiom,
uminus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1_670,axiom,
ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Otimes_671,axiom,
times @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ominus_672,axiom,
minus @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield_673,axiom,
field @ complex ).
thf(tcon_Complex_Ocomplex___Power_Opower_674,axiom,
power @ complex ).
thf(tcon_Complex_Ocomplex___Num_Onumeral_675,axiom,
numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ozero_676,axiom,
zero @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oplus_677,axiom,
plus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom_678,axiom,
idom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oone_679,axiom,
one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Odvd_680,axiom,
dvd @ complex ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_681,axiom,
condit6923001295902523014norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_682,axiom,
condit1219197933456340205attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_683,axiom,
counta3822494911875563373attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_684,axiom,
comple592849572758109894attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_685,axiom,
strict9044650504122735259up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_686,axiom,
strict7427464778891057005id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_687,axiom,
canoni5634975068530333245id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Quickcheck__Narrowing_Opartial__term__of_688,axiom,
quickc6926020345158392990erm_of @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_689,axiom,
bounde4967611905675639751up_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_690,axiom,
bounde4346867609351753570nf_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder_691,axiom,
comple5582772986160207858norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_692,axiom,
linord4140545234300271783up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_693,axiom,
comple6319245703460814977attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_694,axiom,
linord181362715937106298miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_695,axiom,
semiri3467727345109120633visors @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_696,axiom,
ordere6658533253407199908up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_697,axiom,
ordere6911136660526730532id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__bot_698,axiom,
bounded_lattice_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_699,axiom,
ordere2520102378445227354miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_700,axiom,
comple9053668089753744459l_ccpo @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_701,axiom,
semilattice_sup @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_702,axiom,
semilattice_inf @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_703,axiom,
distrib_lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_704,axiom,
bounded_lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_705,axiom,
ab_semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_706,axiom,
comm_monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_707,axiom,
ab_semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Code__Evaluation_Oterm__of_708,axiom,
code_term_of @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_709,axiom,
ordered_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_710,axiom,
comm_monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_711,axiom,
comm_semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_712,axiom,
comm_semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_713,axiom,
semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_714,axiom,
semiring_numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_715,axiom,
semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_716,axiom,
zero_less_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_717,axiom,
wellorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_718,axiom,
order_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_719,axiom,
order_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_720,axiom,
semiring_char_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_721,axiom,
zero_neq_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_722,axiom,
preorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_723,axiom,
linorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_724,axiom,
monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_725,axiom,
monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_726,axiom,
semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_727,axiom,
semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Olattice_728,axiom,
lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_729,axiom,
mult_zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_730,axiom,
order @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_731,axiom,
ord @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Obot_732,axiom,
bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Otimes_733,axiom,
times @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ominus_734,axiom,
minus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Power_Opower_735,axiom,
power @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Onumeral_736,axiom,
numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_737,axiom,
zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oplus_738,axiom,
plus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oone_739,axiom,
one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Odvd_740,axiom,
dvd @ extended_enat ).
thf(tcon_Multiset_Omultiset___Quickcheck__Narrowing_Opartial__term__of_741,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( quickc6926020345158392990erm_of @ ( multiset @ A17 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_742,axiom,
! [A17: $tType] :
( ( preorder @ A17 )
=> ( ordere6658533253407199908up_add @ ( multiset @ A17 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_743,axiom,
! [A17: $tType] : ( cancel2418104881723323429up_add @ ( multiset @ A17 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_744,axiom,
! [A17: $tType] : ( cancel1802427076303600483id_add @ ( multiset @ A17 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_745,axiom,
! [A17: $tType] : ( cancel_semigroup_add @ ( multiset @ A17 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_746,axiom,
! [A17: $tType] : ( comm_monoid_diff @ ( multiset @ A17 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_747,axiom,
! [A17: $tType] : ( ab_semigroup_add @ ( multiset @ A17 ) ) ).
thf(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_748,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( code_term_of @ ( multiset @ A17 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_749,axiom,
! [A17: $tType] : ( comm_monoid_add @ ( multiset @ A17 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_750,axiom,
! [A17: $tType] : ( semigroup_add @ ( multiset @ A17 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_751,axiom,
! [A17: $tType] :
( ( preorder @ A17 )
=> ( preorder @ ( multiset @ A17 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Omonoid__add_752,axiom,
! [A17: $tType] : ( monoid_add @ ( multiset @ A17 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oorder_753,axiom,
! [A17: $tType] :
( ( preorder @ A17 )
=> ( order @ ( multiset @ A17 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_754,axiom,
! [A17: $tType] :
( ( preorder @ A17 )
=> ( ord @ ( multiset @ A17 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ominus_755,axiom,
! [A17: $tType] : ( minus @ ( multiset @ A17 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ozero_756,axiom,
! [A17: $tType] : ( zero @ ( multiset @ A17 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oplus_757,axiom,
! [A17: $tType] : ( plus @ ( multiset @ A17 ) ) ).
thf(tcon_Multiset_Omultiset___Nat_Osize_758,axiom,
! [A17: $tType] : ( size @ ( multiset @ A17 ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__ab__semigroup__add_759,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( cancel2418104881723323429up_add @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__comm__monoid__add_760,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( cancel1802427076303600483id_add @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__semigroup__add_761,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( cancel_semigroup_add @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__mult_762,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( ab_semigroup_mult @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__1__cancel_763,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( semiring_1_cancel @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__mult_764,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( comm_monoid_mult @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__add_765,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( ab_semigroup_add @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Code__Evaluation_Oterm__of_766,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( code_term_of @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__add_767,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( comm_monoid_add @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1_768,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( comm_semiring_1 @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__0_769,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( comm_semiring_0 @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Osemigroup__mult_770,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( semigroup_mult @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Num_Osemiring__numeral_771,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( semiring_numeral @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Osemigroup__add_772,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( semigroup_add @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Owellorder_773,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( wellorder @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oab__group__add_774,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( ab_group_add @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ozero__neq__one_775,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( zero_neq_one @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Opreorder_776,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( preorder @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Olinorder_777,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( linorder @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Omonoid__mult_778,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( monoid_mult @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring__1_779,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( comm_ring_1 @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Omonoid__add_780,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( monoid_add @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Finite__Set_Ofinite_781,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( finite_finite @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Cardinality_Ocard2,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( card2 @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__1_782,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( semiring_1 @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__0_783,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( semiring_0 @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ogroup__add_784,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( group_add @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Type__Length_Olen,axiom,
! [A17: $tType] :
( ( type_len @ A17 )
=> ( type_len @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Omult__zero_785,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( mult_zero @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring_786,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( comm_ring @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Oorder_787,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( order @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Num_Oneg__numeral_788,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( neg_numeral @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Oord_789,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( ord @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ouminus_790,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( uminus @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Oring__1_791,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( ring_1 @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Otimes_792,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( times @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ominus_793,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( minus @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Power_Opower_794,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( power @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Num_Onumeral_795,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( numeral @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ozero_796,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( zero @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oplus_797,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( plus @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oone_798,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( one @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Odvd_799,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( dvd @ ( numeral_bit0 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__ab__semigroup__add_800,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( cancel2418104881723323429up_add @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__comm__monoid__add_801,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( cancel1802427076303600483id_add @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__semigroup__add_802,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( cancel_semigroup_add @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__mult_803,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( ab_semigroup_mult @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__1__cancel_804,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( semiring_1_cancel @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__mult_805,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( comm_monoid_mult @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__add_806,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( ab_semigroup_add @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Code__Evaluation_Oterm__of_807,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( code_term_of @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__add_808,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( comm_monoid_add @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1_809,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( comm_semiring_1 @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__0_810,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( comm_semiring_0 @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Osemigroup__mult_811,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( semigroup_mult @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Num_Osemiring__numeral_812,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( semiring_numeral @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Osemigroup__add_813,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( semigroup_add @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Owellorder_814,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( wellorder @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oab__group__add_815,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( ab_group_add @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ozero__neq__one_816,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( zero_neq_one @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Opreorder_817,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( preorder @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Olinorder_818,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( linorder @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Omonoid__mult_819,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( monoid_mult @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring__1_820,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( comm_ring_1 @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Omonoid__add_821,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( monoid_add @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Finite__Set_Ofinite_822,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( finite_finite @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Cardinality_Ocard2_823,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( card2 @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__1_824,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( semiring_1 @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__0_825,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( semiring_0 @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ogroup__add_826,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( group_add @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Type__Length_Olen_827,axiom,
! [A17: $tType] :
( ( type_len0 @ A17 )
=> ( type_len @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Omult__zero_828,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( mult_zero @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring_829,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( comm_ring @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Oorder_830,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( order @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Num_Oneg__numeral_831,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( neg_numeral @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Oord_832,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( ord @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ouminus_833,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( uminus @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Oring__1_834,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( ring_1 @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Otimes_835,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( times @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ominus_836,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( minus @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Power_Opower_837,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( power @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Num_Onumeral_838,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( numeral @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ozero_839,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( zero @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oplus_840,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( plus @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oone_841,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( one @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Odvd_842,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( dvd @ ( numeral_bit1 @ A17 ) ) ) ).
thf(tcon_Numeral__Type_Onum0___Quickcheck__Narrowing_Opartial__term__of_843,axiom,
quickc6926020345158392990erm_of @ numeral_num0 ).
thf(tcon_Numeral__Type_Onum0___Code__Evaluation_Oterm__of_844,axiom,
code_term_of @ numeral_num0 ).
thf(tcon_Numeral__Type_Onum1___Quickcheck__Narrowing_Opartial__term__of_845,axiom,
quickc6926020345158392990erm_of @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__ab__semigroup__add_846,axiom,
cancel2418104881723323429up_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__comm__monoid__add_847,axiom,
cancel1802427076303600483id_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__semigroup__add_848,axiom,
cancel_semigroup_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__mult_849,axiom,
ab_semigroup_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__mult_850,axiom,
comm_monoid_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__add_851,axiom,
ab_semigroup_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Code__Evaluation_Oterm__of_852,axiom,
code_term_of @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__add_853,axiom,
comm_monoid_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring__0_854,axiom,
comm_semiring_0 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Osemigroup__mult_855,axiom,
semigroup_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Osemigroup__add_856,axiom,
semigroup_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Owellorder_857,axiom,
wellorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oab__group__add_858,axiom,
ab_group_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Opreorder_859,axiom,
preorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Olinorder_860,axiom,
linorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Omonoid__mult_861,axiom,
monoid_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Omonoid__add_862,axiom,
monoid_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Finite__Set_Ofinite_863,axiom,
finite_finite @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Osemiring__0_864,axiom,
semiring_0 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ogroup__add_865,axiom,
group_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Type__Length_Olen_866,axiom,
type_len @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Omult__zero_867,axiom,
mult_zero @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__ring_868,axiom,
comm_ring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Oorder_869,axiom,
order @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Oord_870,axiom,
ord @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ouminus_871,axiom,
uminus @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Otimes_872,axiom,
times @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ominus_873,axiom,
minus @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Power_Opower_874,axiom,
power @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Num_Onumeral_875,axiom,
numeral @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ozero_876,axiom,
zero @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oplus_877,axiom,
plus @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oone_878,axiom,
one @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Odvd_879,axiom,
dvd @ numeral_num1 ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_880,axiom,
! [A17: $tType,A20: $tType] :
( ( ( topolo4958980785337419405_space @ A17 )
& ( topolo4958980785337419405_space @ A20 ) )
=> ( topolo4958980785337419405_space @ ( product_prod @ A17 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Narrowing_Opartial__term__of_881,axiom,
! [A17: $tType,A20: $tType] :
( ( ( typerep @ A17 )
& ( typerep @ A20 ) )
=> ( quickc6926020345158392990erm_of @ ( product_prod @ A17 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_882,axiom,
! [A17: $tType,A20: $tType] :
( ( ( topological_t2_space @ A17 )
& ( topological_t2_space @ A20 ) )
=> ( topological_t2_space @ ( product_prod @ A17 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_883,axiom,
! [A17: $tType,A20: $tType] :
( ( ( typerep @ A17 )
& ( typerep @ A20 ) )
=> ( code_term_of @ ( product_prod @ A17 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_884,axiom,
! [A17: $tType,A20: $tType] :
( ( ( finite_finite @ A17 )
& ( finite_finite @ A20 ) )
=> ( finite_finite @ ( product_prod @ A17 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Heap_Oheap_885,axiom,
! [A17: $tType,A20: $tType] :
( ( ( heap @ A17 )
& ( heap @ A20 ) )
=> ( heap @ ( product_prod @ A17 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_886,axiom,
! [A17: $tType,A20: $tType] : ( size @ ( product_prod @ A17 @ A20 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_887,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_888,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_889,axiom,
counta3822494911875563373attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_890,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Quickcheck__Narrowing_Opartial__term__of_891,axiom,
quickc6926020345158392990erm_of @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_892,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_893,axiom,
bounde4346867609351753570nf_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_894,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_895,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_896,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_897,axiom,
bounded_lattice_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_898,axiom,
comple9053668089753744459l_ccpo @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_899,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_900,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_901,axiom,
distrib_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_902,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_903,axiom,
code_term_of @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_904,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_905,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_906,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_907,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_908,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_909,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_910,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_911,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_912,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_913,axiom,
bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_914,axiom,
uminus @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ominus_915,axiom,
minus @ product_unit ).
thf(tcon_Product__Type_Ounit___Heap_Oheap_916,axiom,
heap @ product_unit ).
thf(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Narrowing_Opartial__term__of_917,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( quickc6926020345158392990erm_of @ ( heap_ext @ A17 ) ) ) ).
thf(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_918,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( code_term_of @ ( heap_ext @ A17 ) ) ) ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_919,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_920,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_921,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_922,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_923,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_924,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_925,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_926,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_927,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_928,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_929,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_930,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_931,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_932,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_933,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_934,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Narrowing_Opartial__term__of_935,axiom,
quickc6926020345158392990erm_of @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_936,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_937,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_938,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_939,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_940,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_941,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_942,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_943,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_944,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_945,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_946,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_947,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_948,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_949,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_950,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_951,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_952,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_953,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_954,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_955,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_956,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_957,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_958,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_959,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_960,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_961,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_962,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_963,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Generic__set__bit_Oset__bit_964,axiom,
generic_set_set_bit @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_965,axiom,
code_term_of @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_966,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_967,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_968,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_969,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_970,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_971,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_972,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_973,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_974,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_975,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_976,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_977,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_978,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_979,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_980,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_981,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_982,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_983,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_984,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_985,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_986,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_987,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_988,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_989,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_990,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_991,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_992,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_993,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_994,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_995,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_996,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_997,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_998,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom_999,axiom,
semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_1000,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_1001,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_1002,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_1003,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Otimes_1004,axiom,
times @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ominus_1005,axiom,
minus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_1006,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_1007,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_1008,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_1009,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_1010,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_1011,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_1012,axiom,
dvd @ code_integer ).
thf(tcon_Heap__Time__Monad_OHeap___Quickcheck__Narrowing_Opartial__term__of_1013,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( quickc6926020345158392990erm_of @ ( heap_Time_Heap @ A17 ) ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of_1014,axiom,
! [A17: $tType] :
( ( typerep @ A17 )
=> ( code_term_of @ ( heap_Time_Heap @ A17 ) ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___Nat_Osize_1015,axiom,
! [A17: $tType] : ( size @ ( heap_Time_Heap @ A17 ) ) ).
thf(tcon_VEBT__Definitions_OVEBT___Quickcheck__Narrowing_Opartial__term__of_1016,axiom,
quickc6926020345158392990erm_of @ vEBT_VEBT ).
thf(tcon_VEBT__Definitions_OVEBT___Code__Evaluation_Oterm__of_1017,axiom,
code_term_of @ vEBT_VEBT ).
thf(tcon_VEBT__Definitions_OVEBT___Nat_Osize_1018,axiom,
size @ vEBT_VEBT ).
thf(tcon_VEBT__BuildupMemImp_OVEBTi___Quickcheck__Narrowing_Opartial__term__of_1019,axiom,
quickc6926020345158392990erm_of @ vEBT_VEBTi ).
thf(tcon_VEBT__BuildupMemImp_OVEBTi___Code__Evaluation_Oterm__of_1020,axiom,
code_term_of @ vEBT_VEBTi ).
thf(tcon_VEBT__BuildupMemImp_OVEBTi___Heap_Oheap_1021,axiom,
heap @ vEBT_VEBTi ).
thf(tcon_VEBT__BuildupMemImp_OVEBTi___Nat_Osize_1022,axiom,
size @ vEBT_VEBTi ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
% Conjectures (6)
thf(conj_0,hypothesis,
( s
= ( vEBT_set_vebt @ t ) ) ).
thf(conj_1,hypothesis,
vEBT_invar_vebt @ t @ n ).
thf(conj_2,hypothesis,
rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ t @ x ) @ r ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ a @ b ) ).
thf(conj_3,hypothesis,
member @ nat @ xa @ ( vEBT_set_vebt @ t ) ).
thf(conj_4,hypothesis,
~ ( member @ nat @ xa @ ( vEBT_set_vebt @ ( vEBT_vebt_delete @ t @ x ) ) ) ).
thf(conj_5,conjecture,
xa = x ).
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